Page 1 :
Downloaded from - Term 1, Books, Join us on Telegram – Click, Here

Page 5 :
Contents, þ One Day Revision, þ The Qualifiers, , 1-8, 9-20, , þ CBSE Question Bank, , 21-32, , þ Latest CBSE Sample Paper, , 33-45, , Sample Paper 1, , 49-63, , Sample Paper 2, , 64-79, , Sample Paper 3, , 80-94, , Sample Paper 4, , 95-107, , Sample Paper 5, , 108-120, , Sample Paper 6, , 121-132, , Sample Paper 7, , 133-145, , Sample Paper 8, , 146-157, , Sample Paper 9, , 158-172, , Sample Paper 10, , 173-185, , Watch Free Learning Videos, Video Solutions of CBSE Sample Papers, Chapterwise Important MCQs, CBSE Case Based MCQs, CBSE Updates, Much more valuable content will be available regularly

Page 6 :
Syllabus, Mathematics CBSE Class 12 (Term I ), , One Paper, Minutes, No., , Max. Marks, Units, , I., , Relations and Functions, , II., , Algebra, , III., , Calculus, , V, , Linear Programming, , Marks, , Total, Internal Assessment, Total, , UNIT I Relations and Functions, Chapter-, , Relations and Functions, Types of relations: reflexive, symmetric, transitive and equivalence relations., One to one and onto functions., , Chapter-, , Inverse Trigonometric Functions, Definition, range, domain, principal value branch., , UNIT II Algebra, Chapter-, , Matrices, Concept, notation, order, equality, types of matrices, zero and identity, matrix, transpose of a matrix, symmetric and skew symmetric matrices., Operation on matrices: Addition and multiplication and multiplication with a, scalar. Simple properties of addition, multiplication and scalar multiplication., Non- commutativity of multiplication of matrices, Invertible matrices;, Here all matrices will have real entries .

Page 7 :
Chapter-, , Determinants, Determinant of a square matrix up to x matrices , minors, co-factors, and applications of determinants in finding the area of a triangle. Adjoint, and inverse of a square matrix. Solving system of linear equations in two, or three variables having unique solution using inverse of a matrix., , UNIT III Calculus, Chapter-, , Continuity and Differentiability, Continuity and differentiability, derivative of composite functions, chain, rule, derivative of inverse trigonometric functions, derivative of implicit, functions. Concept of exponential and logarithmic functions., Derivatives of logarithmic and exponential functions. Logarithmic, differentiation, derivative of functions expressed in parametric forms., Second order derivatives., , Chapter-, , Applications of Derivatives, Applications of derivatives: increasing decreasing functions, tangents, and normals, maxima and minima first derivative test motivated, geometrically and second derivative test given as a provable tool . Simple, problems that illustrate basic principles and understanding of the, subject as well as real-life situations ., , UNIT IV Linear Programming, Chapter-, , Linear Programming, Introduction, related terminology such as constraints, objective function,, optimization, different types of linear programming L.P. problems., Graphical method of solution for problems in two variables, feasible and, infeasible regions bounded , feasible and infeasible solutions, optimal, feasible solutions up to three non-trivial constraints ., , Internal Assessment, Periodic Test, Mathematics Activities: Activity file record +Term, end assessment of one activity Viva, , Marks

Page 8 :
MCQs Preparation Tips, Focus on Theory, , Learn to Identify Wrong Answers, , MCQs can be formed from any part or, line of the chapter. So, strong command, on theory will increase your chances to, solve objective questions correctly and, quickly., , The simplest trick is, observe the options, first and take out the least possible one, and repeat the process until you reach, the correct option., , Practice of Solving MCQs, Cracking an MCQ-based examination, requires you to be familiar with the, question format, so continuous practice, will make you more efficient in solving, MCQs., , Speed & Accuracy, In MCQ-based examination, you need, both speed and accuracy, if your, accuracy is good but speed is slow then, you might attempt less questions,, resulting in low score., , Analyse your Performance, During the practice of MCQs you can, identify your weak & strong, topics/chapter by analysing of incorrect, answers, in this way you will get an, awareness about your weaker topics., , Practice through Sample Papers, Solving more & more papers will make, you more efficient and smarter for, exams. Solve lots of Sample Papers, given in a good Sample Papers book., , Attempting MCQs in Exams, 1. Read the paper from beginning to, end & attempt those questions first in, which you are confident. Now move, on to those questions which requires, thinking and in last attempt those, questions for which you need more, attention., , 2. Read instructions of objective, questions carefully and find out what, is being asked, a bit carelessness can, lead you to incorrect answer., , 3. Tick/Write down the correct option, only while filling the OMR sheet., Step by step solution is not required, , in MCQ type questions, it is a waste, of time, you will not get extra marks, for this., , 4. Most of the time, you need not to, solve the MCQ completely to get the, correct option. You can start thinking, in reverse order and choose the best, fit option., , 5. As there is no negative marking for, incorrect answers, so don't leave any, question unanswered. Use your, guess if you have not exact idea, about the correct answer.

Page 9 :
01, , CBSE Sample Paper Mathematics Class XII (Term I), , ONE DAY, , REVISION, Revise All the Concepts in a Day, Just Before the Examination..., ●, , Relations and Functions, , Relation, Let A be a non-empty set and R Í A ´ A. Then, R is, called a relation on A. If (a, b) Î R, then we say that a is, related to b and we write aRb and if (a, b) Ï R, then we, write a R/ b ., Domain, Range and Codomain of a Relation, Let R be a relation from set A to set B, such that, R = {(a, b): a Î A and b Î R}. The set of all first and, second elements of the ordered pairs in R is called the, domain and range respectively, i.e. Domain, (R ) = { a :(a, b) Î R } and Range (R ) = { b :(a, b) Î R }., The set B is called the codomain of relation R., Types of Relations, (i) Empty (void) Relation A relation R on a set A is, called empty relation, if no element of A is related to, any element of A,, i.e., , R = f Ì A ´ A., , (ii) Universal Relation A relation R on a set A is called, universal relation, if each element of A is related to, every element of A,, i.e., , R = A ´ A., , i.e., , I A = R = {(a, a): a Î A}, , Equivalence Relation, A relation R on a set A is called an equivalence relation,, if it is reflexive, symmetric and transitive., , Function, Let A and B be two non-empty sets. Then, a rule f from, A to B which associates each element x Î A, to a, unique element of f ( x ) Î B is called a function or, mapping from A to B and we write f : A ® B., Here, element of A is called the domain of f i.e. dom, (f ) = A and element of B is called the codomain of f., Also, { f ( x ) : x Î A} Í B is called the range of f., Every function is a relation but every relation is not a, function., Types of Functions, (i) One-one (Injective) Function A function f : A ® B, is said to be one-one, if distinct element of A have, distinct images in B,, i.e., f ( x 1) = f ( x 2 ) Þ x 1 = x 2, or, x 1 ¹ x 2 Þ f ( x 1) ¹ f ( x 2 ), where, x1, x2 Î A., , (iv) Reflexive Relation A relation R defined on set A is, said to be reflexive, if ( x, x ) Î R, " x Î A i.e., x R x," x Î A ., , (ii) Many-one Function A function f : A ® B is said to, be many-one, if two or more than two elements in A, have the same image in B., , (v) Symmetric Relation A relation R defined on set A, is said to be symmetric, if ( x, y ) Î R, , (iii) Onto (Surjective) Function A function f : A ® B is, said to be onto or surjective, if every element in B, have atleast one pre-image in A, i.e. if for each, y Î B, there exists an element x Î A, such that, f ( x ) = y., , Þ ( y, x ) Î R, " x, y Î A,, i.e. x R y Þ yR x, " x, y Î A ., , ONE DAY REVISION, , (iii) Identity Relation A relation R on a set A is called, an identity relation, if each element of A is related to, itself only. It is denoted by I A,, , (vi) Transitive Relation A relation R defined on set A is, said to be transitive, if ( x, y ) Î R and ( y, z ) Î R, Þ ( x, z ) Î R, " x, y, z Î A, i.e. x R y and yR z Þ x R z," x, y, z Î A.

Page 10 :
02, , CBSE Sample Paper Mathematics Class XII (Term I), (iv) Into Function A function f : A ® B is said to be into, if atleast one element of B do not have a pre-image in A ., (v) One-one and Onto (Bijective) Function A function f : X ® Y is said to be bijective, if f is both one-one and, onto., , ●, , Inverse Trigonometric Functions, , Trigonometric functions are not one-one on their natural domains, so their inverse does not exist for all values but, their inverse may exists in some interval of their restricted domains and codomains. Thus, we can say that, inverse, of trigonometric functions are defined within restricted domains of corresponding trigonometric functions. Inverse, of f is denoted by f -1., Domain and Principle Value Branch (Range), of Inverse Trigonometric Functions, , Function, , Domain, , Principle value branch (Range), , sin -1 x, , [–1, 1], , é p pù, êë - 2 , 2 úû, , cos -1 x, , [–1, 1], , [0, p ], , tan -1 x, , R, , cosec -1x, , (-¥, - 1 ] È [1, ¥ ), , sec -1 x, , (-¥, - 1 ] È [1, ¥ ), , cot -1 x, , R, , æ p pö, ç- , ÷, è 2 2ø, é p pù, êë - 2 , 2 úû - { 0}, ì pü, [0, p ] - í ý, î2þ, (0, p ), , 1, æ 1ö, Note sin -1 x ¹ (sin x )-1, sin -1 x ¹ sin -1ç ÷ , sin -1 x ¹, è xø, sin x, , Useful Results, é p pù, (i) sin -1 (sin q) = q, q Î ê - , ú, ë 2 2û, (ii) cos -1 (cos q) = q, q Î[0, p ], , (x) cot (cot -1 x ) = x, x Î R, (xi) cosec (cosec -1 x ) = x, x Î (- ¥, - 1] È [1, ¥ ), (xii) sec (sec -1 x ) = x, x Î (- ¥, - 1] È [1, ¥ ), , æ p pö, (iii) tan -1 (tan q) = q, q Î ç - , ÷, è 2 2ø, , (xiii) sin -1 (- x ) = - sin -1 x, x Î [-1 , 1], , (iv) cot -1(cot q) = q, q Î (0, p ), , (xiv) cos -1 (- x ) = p - cos -1 x, x Î [-1, 1], , é p pù, (v) cosec -1(cosec q) = q, q Î ê - , ú - {0}, ë 2 2û, , (xv) tan -1 (- x ) = - tan -1 x, x Î R, , ì pü, (vi) sec -1(sec q) = q, q Î [0, p ] - í ý, î2þ, (vii) sin (sin -1 x ) = x, x Î [- 1, 1], , ONE DAY REVISION, , (ix) tan (tan -1 x ) = x, x Î R, , (viii) cos (cos -1 x ) = x, x Î [-1, 1], , (xvi) cot -1 (- x ) = p - cot -1 x, x Î R, (xvii) cosec -1 (- x ) = - cosec -1 x,, | x | ³ 1or x Î (-¥,-1] È [1, ¥ ), (xviii) sec -1 (- x ) = p - sec -1 x,, | x | ³ 1or x Î (-¥,-1] È [1, ¥ )

Page 11 :
03, , CBSE Sample Paper Mathematics Class XII (Term I), , ●, , Matrices, , A matrix is an ordered rectangular array of numbers or, functions. The number or functions are called the, elements or the entries of the matrix., , Order of Matrix, If m represents number of rows and n represents, number of columns, then order of matrix is m ´ n., , Types of Matrices, (i) Row matrix A matrix having only one row, is called, a row matrix., (ii) Column matrix A matrix having only one column, is, called a column matrix., (iii) Zero or Null matrix If all the elements of a matrix, are zero, then it is called a zero matrix or null matrix., It is denoted by symbol O., (iv) Square matrix A matrix in which number of rows, and number of columns are equal, is called a square, matrix., (v) Diagonal matrix A square matrix is said to be a, diagonal matrix, if all the elements lying outside the, diagonal elements are zero and atleast one of the, diagonal element is not zero., (vi) Scalar matrix A diagonal matrix in which all, diagonal elements are equal, is called a scalar, matrix., (vii) Unit or Identity matrix A diagonal matrix in which all, the diagonal elements are equal to unity (one), is, called an identity matrix. It is denoted by I., , Equality of Matrices, Two matrices are said to be equal, if their order are, same and their corresponding elements are also equal,, i.e. aij = bij , " i , j ., , Addition of Matrices, , (ii) Matrix addition is associative,, i.e., ( A + B ) + C = A + (B + C ), (iii) Existence of additive identity Zero matrix (O ) of, order m ´ n (same as of A) is called additive, identity,, if, A + O = A = O + A., (iv) Existence of additive inverse For the square, matrix, the matrix (- A) is called additive inverse,, if, A + ( - A) = O = ( - A) + A ., , Let A = [aij ]m ´ n be a matrix and k be any scalar. Then,, kA is another matrix which is obtained by multiplying, each element of A by k,, i.e. kA = k[aij ] m ´ n = [k(aij )] m ´ n., Negative of a Matrix, If we multiply a matrix A by a scalar quantity (-1), then, the negative of a matrix (i.e. - A) is obtained., In negative of A, each element is multiplied by (-1)., Properties of Scalar Multiplication, Let A and B be the two matrices of same order, then, (i) k( A + B) = kA + kB, where k is scalar., (ii) (k1 + k2 )A = k1A + k2 A, where k1 and k2 are scalars., (iii) (kl ) A = k(lA) = l (kA), where l and k are scalars., , Difference (or Subtraction) of Matrices, If A = [aij ] and B = [bij ] are two matrices of the, same order m ´ n, then difference of these matrices, A - B is defined as a matrix D = [ d ij ],, where d ij = aij - bij , " i , j ., , Multiplication of Matrices, Let A = [aij ] m ´ n and B = [b jk ] n ´ p be two matrices, such that the number of columns of A is equal to the, number of rows of B, then multiplication of A and B is, n, , denoted by AB and it is given by c ik =, , å aij bjk , where, , j=1, , c i k is the (i , k ) th element of matrix C of order m ´ p, where C = AB., Properties of Multiplication of Matrices, (i) Let A, B and C be three matrices of same order., Then, matrix multiplication is associative., i.e., , ( AB) C = A(BC )., , (ii) Existence of multiplicative identity For every, square matrix A, there exists an identity matrix I of, same order such that A × I = A = I × A., (iii) Matrix multiplication is distributive over addition., i.e., A(B + C ) = AB + AC, (iv) Non-commutativity Generally, matrix multiplication, is not commutative i.e. if A and B are two matrices, and AB, BA both exist, then it is not necessary that, AB = BA., , Transpose of a Matrix, The matrix obtained by interchanging the rows and, columns of a given matrix A, is called transpose of a, matrix A. It is denoted by, A¢ or AT or Ac, , ONE DAY REVISION, , Let A and B be two matrices each of same order m ´ n., Then, the sum of matrices A + B is a matrix whose, elements are obtained by adding the corresponding, elements of A and B. If A and B are not of same order,, then A + B is not defined., Properties of Matrix Addition, Let A, B and C are three matrices of same order m ´ n,, then, (i) Matrix addition is commutative,, i.e., A+ B=B+ A, , Multiplication of a Matrix by a Scalar

Page 12 :
04, , CBSE Sample Paper Mathematics Class XII (Term I), , Properties of Transpose of Matrices, (i) ( A¢ ) ¢ = A, (ii) ( A ± B) ¢ = A¢ ± B¢, (iii) (kA) ¢ = kA¢, where k is any constant., (iv) ( AB) ¢ = B¢ A¢, , [reversal law], , Symmetric and Skew-symmetric, Matrices, A square matrix A is called symmetric matrix, if A¢ = A, and a square matrix A is called skew-symmetric, if, A¢ = - A., Properties of Symmetric and Skew-symmetric, Matrices, (i) For a square matrix A with real number entries,, A + A¢ is a symmetric matrix and A - A¢ is a, skew-symmetric matrix., , ●, , (ii) Any square matrix A can be expressed as the sum, of a symmetric and a skew-symmetric matrices., 1, 1, i.e., A = ( A + A¢ ) + ( A - A¢ ), 2, 2, , Inverse of a Matrix, If A and B are two square matrices of same order such, that AB = BA = I, then B is called the inverse matrix of A, and is denoted by A-1,, i.e., B = A-1, Here, A is said to be invertible., ● Inverse of a square matrix, if exists, is unique., ● A rectangular matrix does not possess inverse, matrix., ● If B is the inverse of A, then A is also inverse of B., -1, ● ( AB ), = B - 1 A- 1, , Determinants, , Every square matrix A of order n is associated with a, number, called its determinant and it is denoted by, det(A) or A ., , ●, , Determinant of Matrix of Order 1, Let A = [a] be a square matrix of order 1, then, | A| = | a| = a, i.e. element itself is determinant., Determinant of Matrix of Order 2, a, a, det( A) or |A| = 11 12 = a11 a22 - a12 a21, a21 a22, Determinant of Matrix of Order 3, a11 a12 a13, det( A) = | A| = a21 a22, a31 a32, , a23, , ●, , 1, = |[ x1( y2 - y3 ) + x2( y3 - y1) + x3 ( y1 - y2 )]|, 2, Area is positive quantity. So, we always take the, absolute value of the determinant., If area is given, then use both positive and negative, values of the determinant for calculation., , Condition of Collinearity for Three Points, Three points A( x1, y1), B( x2, y2 ) and C( x3, y3 ) are, collinear if and only if the area of triangle formed by, these three points is zero., x 1 y1 1, i.e., x 2 y2 1 = 0, x 3 y3 1, , Minors, , a33, , = a11 (a22a33 - a32a23 ) - a12(a21a33 - a31a23 ), + a13(a21a32 - a31a22 ), [expanding along R1], We can expand the above determinant corresponding to, any row or column., , Minor of an element aij of a determinant is the, determinant obtained by deleting i th row and j th, column in which element aij lies. It is denoted, by Mij ., a11 a12 a13, e.g., If A = a21 a22 a23 , then, a31 a32, , Important Points, , ONE DAY REVISION, , (i) Multiplying a determinant by k means multiplying, the elements of only one row (or one column) by k., (ii) If A is a square matrix of order n, then| kA| = k n | A|,, where n Î N., (iii) If all the elements of any row or column of a, determinant are zero, then the value of such, determinant is zero., , Area of Triangle, Let A( x1, y1), B( x2, y2 ) and C( x3, y3 ) be the vertices of a, DABC. Then, its area is given by, x 1 y1 1, 1, x 2 y2 1, D=, 2, x 3 y3 1, , a33, , Minors of elements of A are, a22 a23, ,, M11 =, a32 a33, , and, , M12 =, , a21 a23, a31 a33, , M13 =, , a21 a22, , etc., a31 a32, , The minor of an element of a determinant of order, n (n ³ 2 ) is a determinant of order n - 1., , Cofactors, If M ij is the minor of an element aij , then the cofactor of, aij is denoted by Cij or A ij and defined as follows, Aij or Cij = (- 1)i + j Mij

Page 13 :
05, , CBSE Sample Paper Mathematics Class XII (Term I), , Singular and Non-singular Matrices, A square matrix A is said to be a singular matrix, if, A = 0 and if A ¹ 0, then matrix A is said to be, non-singular matrix., , Adjoint of a Matrix, Let A = [ aij ] n ´ n be a square matrix, then adjoint of, A, i.e. adj ( A) = CT , where C = [c ij ] is the cofactor matrix, of A ., Properties of Adjoint of Square Matrix, If A and B are two square matrices of order n, then, (i) adj ( AT ) = (adj A)T, , matrix A. It is denoted by A-1 and is given by, 1, [adj ( A)] ., A- 1 =, | A|, Properties of Inverse of a Matrix, (i) ( A-1)-1 = A, (ii) ( AB)-1 = B -1A-1, (iii) ( AT )-1 = ( A-1)T, (v) AA-1 = A-1A = I, , Solution of System of Linear Equations by, Using Inverse of a Matrix (or by matrix method), Let the system of linear equations be, , (ii) adj (kA) = k n- 1(adj A), k Î R, , a1x + b1y + c 1z = d 1; a2 x + b2 y + c 2 z = d 2;, , (iii) (adj AB) = (adj B) (adj A), , and, , n- 1, , (iv)| adj A| = | A|, , , if| A| ¹ 0, ( n- 1)2, , (v)|adj [adj ( A)]| = | A|, n- 2, , (vi) adj (adj A) = | A|, , , if| A| ¹ 0, , ×A, , Inverse of a Matrix, Suppose A is a non-zero square matrix of order n and, there exists matrix B of same order n such that, AB = BA = In, then such matrix B is called an inverse of, , ●, , (iv) | A-1| =| A|-1, 1, (vi) (kA)-1 = A-1, where k ¹ O, k, , a3 x + b3 y + c 3 z = d 3., , We can write the above system of linear equations in, matrix form as AX = B, where, é xù, éd 1ù, é a1 b1 c 1 ù, A = ê a2 b2 c 2 ú , X = ê y ú and B = êd 2 ú, ê ú, ê ú, ê, ú, êë z úû, êëd 3 úû, êë a3 b3 c 3 úû, If| A| ¹ 0, then solution is given by X = A-1B,, where A-1 =, , adj ( A), ., | A|, , Continuity and Differentiability, , Continuous Function, , (ii) (f - g ) is continuous at x = c ., , A real valued function f is said to be continuous, if it is, continuous at every point in the domain of f., Continuity of a Function at a Point, Suppose, f is a real valued function on a subset of the, real numbers and let c be a point in the domain of f., Then, f is continuous at x = c , if lim f ( x ) = f (c ) ., x® c, , i.e. if f (c ) = lim f ( x ) = lim f ( x ), then f ( x ) is, x® c+, , continuous at x = c ., , x® c-, , Otherwise, f ( x ) is discontinuous at x = c ., Some Basic Continuous Functions, , (vi) All trigonometric functions are continuous in their, domain., , Algebra of Continuous Function, Theorem 1, Let f and g be two real functions continuous at, a real number c, then, (i) (f + g ) is continuous at x = c ., , Theorem 2, Suppose f and g are real valued functions such that, (fog ) is defined at c. If g is continuous at c and if f is, continuous at g(c), then (fog) is continuous at c., , Differentiability or Derivability, A real valued function f is said to be derivable or, differentiable at x = c in its domain, if its left hand and, right hand derivatives at x = c exist and are equal., At x = a, right hand derivative,, f (a + h ) - f (a), Rf ¢ (a) = lim, h®0, h, and left hand derivative,, f (a - h ) - f (a), Lf ¢ (a) = lim, h®0, -h, Thus, f ( x ) is differentiable at x = a, if Rf ¢(a) = Lf ¢(a)., Otherwise, f ( x ) is not differentiable at x = a., , Differentiation, The process of finding derivative of a function is called, the differentiation., , ONE DAY REVISION, , (i) Every constant function is continuous., (ii) Every identity function is continuous., (iii) Every rational functions are always continuous., (iv) Every polynomial function is continuous., (v) Modulus function f ( x ) =| x | is continuous., , (iii) fg is continuous at x = c ., f, (iv) is continuous at x = c provided that, g( x ) ¹ 0., g

Page 14 :
06, , CBSE Sample Paper Mathematics Class XII (Term I), , Derivatives of Some Standard Functions, d, d n, (i), (constant) = 0, (ii), ( x ) = nx n- 1, dx, dx, d, (iii), (cx n ) = cnx n- 1, where c is a constant., dx, d, (iv), (sin x ) = cos x, dx, d, (v), (cos x ) = - sin x, dx, d, (vi), (tan x ) = sec 2 x, dx, d, (vii), (cosec x ) = - cosec x cot x, dx, d, (viii), (sec x ) = sec x tan x, dx, d, (ix), (cot x ) = - cosec 2 x, dx, d, (x), (e x ) = e x, dx, d x, (xi), (a ) = a x loge a, a > 0, dx, 1, d, (xii), (loge x ) = , x > 0, dx, x, d, 1, (xiii), , a > 0, a ¹ 1, (log a x ) =, dx, x loge a, , Algebra of Derivatives, d, du dv, (i), [sum and difference rule], (u ± v ) =, ±, dx, dx dx, d, d, d, (ii), (u × v ) = u, (v ) + v, (u ) [product rule], dx, dx, dx, d, d, (u ) - u, (v ), v, d æu ö, dx, dx, (iii), [quotient rule], ç ÷=, dx è v ø, v2, where, u and v are functions of x., , Derivative of Composite Functions, , ONE DAY REVISION, , Let y be a real valued function which is a composite of, two functions, say y = f (u ) and u = g( x )., dy dy du, Then,, =, ×, = f ¢(u )× g ¢ ( x ), dx du dx, d, i.e., [f { g( x )}] = f ¢ [g( x )]× g ¢ ( x ), dx, , Derivatives of Implicit Functions, Let f ( x, y ) = 0 be an implicit function of x. Then, to find, dy, we first differentiate both sides of equation w.r.t. x, dx, dy, and then take all terms involving, on LHS and, dx, remaining terms on RHS to get required value., , Derivatives of Inverse, Trigonometric Functions, (i), , d, 1, , -1 < x < 1, (sin -1 x ) =, dx, 1 - x2, , (ii), , d, -1, , -1 < x < 1, (cos -1 x ) =, dx, 1 - x2, , (iii), , d, 1, (tan -1 x ) =, dx, 1 + x2, , (iv), , d, -1, (cot -1 x ) =, dx, 1 + x2, , (v), , d, 1, ,| x| > 1, (sec -1 x ) =, dx, x x2 - 1, , (vi), , d, -1, ,| x| > 1, (cosec -1x ) =, dx, x x2 - 1, , Derivative of a Function with, Respect to Another Function, Let y = f ( x ) and z = g( x ) be two given functions, we, firstly differentiate both functions with respect to x, separately and then put these values in the following, formulae, dy dy / dx, dz dz / dx, or, ., =, =, dz dz / dx, dy dy / dx, , Derivative of Logarithmic Function, Suppose, given function is of the form u ( x )v ( x)., In such cases, we take logarithm on both sides and, use properties of logarithm to simplify it and then, differentiate it., , Derivative of Parametric Functions, If x = f(t ) and y = y(t ), then, , dy dy / dt, =, dx dx / dt, , Derivative of Infinite Series, When the value of y is given as an infinite series, then, the process to find the derivatives of such infinite series, is called differentiation of infinite series., In this case, we use the fact that if one term is deleted, from an infinite series, it remains unaffected to replace, all terms except first form by y. Thus, we convert it into, a finite series or function. Then, we differentiate it to find, the required value., , Second Order Derivative, dy, = f ¢( x ) is called, dx, d æ dy ö, the first derivative of y or f ( x ) and, ç ÷ is called the, dx è dx ø, second order derivative of y w.r.t. x and it is denoted by, d 2y, or y ¢¢ or y2., dx 2, Let y = f ( x ) be a given function, then

Page 15 :
07, , CBSE Sample Paper Mathematics Class XII (Term I), , ●, , Applications of Derivative, , Increasing function, Let I be an open interval contained in the domain of, a real valued function f. Then, f is said to be, (a) increasing on I, if x1 < x2, Þ f ( x1) £ f ( x2 ), " x1, x2 Î I, (b) strictly increasing on I, if x1 < x2, Þ f ( x1) < f ( x2 ), "x1, x2 Î I, , Decreasing function, Let I be an open interval contained in the domain of a, real valued function f. Then, f is said to be, (a) decreasing on I, if x1 < x2, Þ f ( x1) ³ f ( x2 ), " x1, x2 Î I, (b) strictly decreasing on I, if x1 < x2, Þ f ( x1) > f ( x2 ), "x1, x2 Î I, Theorem Let f be continuous on [a, b] and, differentiable on (a, b)., ●, , ●, , ●, , If f ¢ ( x ) > 0 for each x Î(a, b), then f ( x ) is said to be, increasing in [a, b] and strictly increasing in (a, b)., If f ¢ ( x ) < 0 for each x Î(a, b), then f ( x ) is said to be, decreasing in [a, b] and strictly decreasing in (a, b)., If f ¢ ( x ) = 0 for each x Î(a, b), then f is said to be a, constant function in [a, b]., A monotonic function f in an interval I, we mean that f, is either increasing in I or decreasing in I., , Tangents and Normals, A tangent is a straight line, which touches the curve, y = f ( x ) at a point., A normal is a straight line perpendicular to a tangent to, the curve y = f ( x ) intersecting at the point of contact., Slope of Tangent and Normal, dy, represents the gradient or slope of a curve y = f ( x )., dx, , Equations of Tangent and Normal, Let y = f ( x )be a curve and P ( x1, y1) be a point on it. Then,, (a) equation of tangent at P( x1, y1) is, é dy ù, ( y - y1 ) = ê ú, ( x - x 1), ë dx û ( x1, y1 ), (b) equation of normal at P ( x1, y1) is, -1, ( y - y1 ) =, ( x - x 1), dy, é ù, êë dx úû, ( x1, y1 ), , Let f be a real valued function and c be an interior point, in the domain of f. Then,, (i) point c is called a local maxima, if there is a h > 0, such that f (c ) > f ( x ), " x in (c - h, c + h )., Here, value f (c ) is called the local maximum value, of f., (ii) point c is called a point of local minima, if there is a, h > 0 such that f (c ) < f ( x ), " x in (c - h, c + h )., Here, value f (c )is called the local minimum value, of f., , Critical Point, A point c in the domain of a function f at which either, f ¢ (c ) = 0 or f is not differentiable, is called a critical, point of f., , First Derivative Test, Let f be a function defined on an open interval I and let f, be continuous at a critical point c in I., Then,, (i) if f ¢( x ) change sign from positive to negative as x, increases through point c, then c is a point of local, maxima., (ii) if f ¢( x ) change sign from negative to positive as x, increases through point c, then c is a point of local, minima., (iii) if f ¢( x ) does not change sign as x increases through, c, then c is neither a point of local maxima nor a, point of local minima. Infact, such a point is called, point of inflection., , Second Derivative Test, Let f be a function defined on an interval I and c Î I,, such that f be twice differentiable at c. Then,, (i) x = c is a point of local maxima, if f ¢ (c ) = 0 and, f ¢¢(c ) < 0. The value f (c ) is local maximum value of f., (ii) x = c is a point of local minima, if f ¢ (c ) = 0, and f ¢¢(c ) > 0. Then, the value f (c ) is local minimum, value of f., (iii) if f ¢ (c ) = 0 and f ¢¢(c ) = 0, then the test fails., , Absolute Maxima and Absolute Minima, Let f be a continuous function on [a, b] and c be a point, in [a, b] such that f ¢ (c ) = 0., Then, find f (a) , f (b) and f (c ) . The maximum of these, values gives a maxima or absolute maxima and, minimum of these values gives a minima or absolute, minima., , ONE DAY REVISION, , If a tangent line to the curve y = f ( x ) makes an angle q, with X-axis in the positive direction, then, dy, Slope of tangent, = tan q, dx, -1, -1, Slope of normal =, =, Slope of tangent dy/ dx, , Maxima and Minima

Page 16 :
08, , CBSE Sample Paper Mathematics Class XII (Term I), , ●, , Linear Programming, , A linear programming problem is one that is concerned, with finding the optimal value (maximum or minimum, value) of a linear function (called objective function) of, several variables (say x and y called decision variable),, subject to the constraints that the variables are, non-negative and satisfy a set of linear inequalities, (called linear constraints)., , Mathematical Form of LPP, The general mathematical form of a linear, programming problem may be written as, Maximise or Minimise Z = c 1x + c 2 y, subject to constraints are a1x + b1y £ d 1, a2 x + b2 y £ d 2,, etc. and non-negative restrictions are x ³ 0, y ³ 0., , Some Terms Related to LPP, (i) Constraints The linear inequations or inequalities, or restrictions on the variables of a linear, programming problem are called constraints. The, conditions x ³ 0, y ³ 0 are called non-negative, restrictions., (ii) Optimisation Problem A problem which seeks to, maximise or minimise a linear function subject to, certain constraints determined by a set of linear, inequalities is called an optimisation problem., Linear programming problems are special type of, optimisation problems., (iii) Objective Function A linear function of two or, more variables which has to be maximised or, minimised under the given restrictions in the form of, linear inequations or linear constraints is called the, objective function.The variables used in the, objective function are called decision variables., (iv) Optimal Values The maximum or minimum value, of an objective function is known as its optimal value., , bounded, if it can be enclosed within a circle., Otherwise, it is said to be unbounded region., (vii) Feasible and Infeasible Solutions Any solution to, the given linear programming problem which also, satisfies the non-negative restrictions of the, problem is called a feasible solution. Any point, outside the feasible region is called an infeasible, solution., (viii) Optimal Solution A feasible solution at which the, objective function has optimal value is called the, optimal solution of the linear programming, problem., (ix) Optimisation Technique The process of obtaining, the optimal solution is called optimisation, technique., , Important Theorems, (i) Theorem 1 Let R be the feasible region (convex, polygon) for a linear programming problem and, Z = ax + by be the objective function., When Z has an optimal value (maximum or, minimum), where the variables x and y are subject, to constraints described by linear inequalities, this, optimal value must occur at a corner point (vertex), of the feasible region., (ii) Theorem 2 Let R be the feasible region for a linear, programming problem and Z = ax + by be the, objective function. If R is bounded, then the, objective function Z has both a maximum and a, minimum value on R and each of these occurs at a, corner point (vertex) of R., , Graphical Method of Solving LPP, The following steps are given below:, ●, , ONE DAY REVISION, , (v) Feasible and Infeasible Regions The common, region determined by all the constraints including, non-negative constraints x, y ³ 0 of a linear, programming problem is called the feasible region, or solution region. Each point in this region, represents a feasible choice. The region other than, feasible region is called an infeasible region., (vi) Bounded and Unbounded Regions A feasible, region of a system of linear inequalities is said to be, , ●, , ●, , Step I Find the feasible region of the linear, programming problem and determine its corner, points (vertices) either by inspection or by solving the, two equations of the lines intersecting at that point., Step II Evaluate the objective function Z = ax + by at, each corner point. Let M and m respectively denote, the largest and smallest values of these points., Step III When the feasible region is bounded, M and, m are the maximum and minimum values of Z.

Page 17 :
9, , CBSE Sample Paper Mathematics Class XII (Term I), , THE, , QUALIFIERS, Chapterwise Set of MCQs to Check Preparation, Level of Each Chapter, , 1. Relations and Functions, Direction (Q. Nos. 1-15) Each of the question has four options out of which only one is correct., Select the correct option as your answer., , 1. Let a relation R defined on the set of real number R by aRb, iff a - b + 3 is an irrational, number. Then, R is, (a) reflexive, (c) transitive and reflexive, , (b) symmetric only, (d) an equivalence relation, , 2. The relation R defined on the set A = {1, 2, 3, 4, 5} by R = {(x , y) =| x 2 - y 2 | < 16} is given, by, (a) {(1, 1), (2, 1), (3, 1), (4, 1), (2, 3)}, (c) {(3, 3), (3, 4), (5, 4), (4, 3), (3, 1)}, , (b) {(2, 2), (3, 2), (4, 2), (2, 4)}, (d) None of these, , 3. Consider the non-empty set of all human beings in a town at a particular time and a, relation R defined as xRy, iff x is wife of y, then R is, (a) reflexive, (b) symmetric, (c) transitive, (d) None of these, , 4. The relation R on the set A = {1, 2, 3} defined by R = {(1, 2), (1, 3)} is, (a) symmetric, (c) transitive, , (b) reflexive, (d) None of these, , 5. Let L be the set of all lines in a plane and R be a relation on L defined by l1 R l2 , iff l1 is, (b) symmetric, (d) equivalence, , 6. The number of equivalence relations on the set {1, 2, 3} containing (1, 2) and (2, 1) is, (a) 0, , (b) 1, , (c) 2, , (d) 3, , 7. Let R be the equivalence relation in the set A = {0, 1, 2, 3, 4, 5} given by R = {(a , b) : 2, divides ( a - b)}. Then [0], the equivalence class containing 0, is, (a) {0, 2, 4, 5}, (b) {0, 3, 5}, (c) f, (d) {0, 2, 4}, , THE QUALIFIER, , perpendicular to l2 , then R is, (a) reflexive, (c) transitive

Page 18 :
10, , CBSE Sample Paper Mathematics Class XII (Term I), , 8. Consider a non-empty set consisting of children in a family and a relation R defined as, aRb, if a is brother of b. Then, R is, (a) symmetric but not transitive, (c) neither symmetric nor transitive, , (b) transitive but not symmetric, (d) both symmetric and transitive, , 9. The maximum number of equivalence relations on the set A = {1, 2, 3} are, (a) 1, , (b) 2, , (c) 3, , (d) 5, , 10. If the set A contains 5 elements and the set B contains 6 elements, then the number of, one-one and onto mappings from A to B is, (b) 120, (c) 0, , (a) 720, , (d) None of these, , 11. The function f : N ® N (N is the set of natural number) defined by f (x) = 2x + 3, is, (a) injective, , (b) surjective, , (c) bijective, , (d) None of these, , 12. The function f : R ® R given by f (x) = cos x for all x Î R, then f is, (a) one-one and onto, (c) onto but not one-one, , (b) one-one but not onto, (d) neither one-one nor onto, , 13. Let A and B be sets, f : A ´ B ® B ´ A such that f (a , b) = (b , a). Then, f is, (a) injective, , (b) surjective, , (c) bijective, x, 14. The function f : [0, ¥) ® R given by f (x) =, is, x +1, (a) one-one and onto, (c) onto but not one-one, , (d) None of these, , (b) one-one but not onto, (d) neither one-one nor onto, , 15. Given a function defined by f (x) = 4 - x 2 ; 0 £ x £ 2, 0 £ f (x) £ 2. Then, function f is, (a) many one, , (b) into, , (c) one-one into, , (d) bijective, , Answers, 1. (a), 6. (c), 11. (a), , 2. (d), 7. (d), 12. (d), , 3. (c), 8. (b), 13. (c), , 4. (c), 9. (d), 14. (b), , For Detailed Solutions, Scan the code, , 5. (b), 10. (c), 15. (d), , 2. Inverse Trigonometric Functions, , THE QUALIFIER, , Direction (Q. Nos. 1-15) Each of the question has four options out of which only one is correct., Select the correct option as your answer., é, , æ, , ë, , è, , 1. The value of cosê cos -1 ç 1, (a), 2, , 3 ö pù, ÷ + ú is, 2 ø 6û, , (b) 1, , æ 3ö, 2, + cos - 1 ç, ÷ is, 3, è 2 ø, 5p, (b), 6, , (c) 0, , (d) - 1, , 2. The value of 2 cosec - 1, (a), , p, 6, , (c), , 7p, 6, , (d), , 2p, 3

Page 19 :
11, , CBSE Sample Paper Mathematics Class XII (Term I), , 1, 3. The principal value of sin - 1 éê cos æç sin - 1 ö÷ ùú is, p, (a), 4, , è, , ë, , p, (b), 3, , 2 øû, , (c), , p, 6, , (d), , p, 2, , 4. The domain of the function cos - 1 (2x - 1) is, (b) [ - 1, 1], , (a) [0, 1], , (c) ( - 1, 1), , (d) [0, p], , (c) 1 - x 2, , (d) x 2 + 1, , 1, 5. If cos - 1 æç ö÷ = q, then tan q will be, èxø, , 1, , (a), , (b) x 2 - 1, , 2, , x -1, , 1, 2, , 1, 2, , 6. The value of cos - 1 + 2 sin - 1 is, (a), , p, 4, , (b), , p, 6, , 1, = - 2, then sin - 1 x is equal to, x, -p, (a) p, (b), 2, , (c), , p, 3, , (d), , (c), , p, 2, , (d), , 2p, 3, , 7. If x +, , -p, 6, , æ 1 + x 2 - 1ö, ÷ , x ¹ 0 is, ç, ÷, x, è, ø, , 8. The simplest form of tan - 1 ç, , 1, (b) tan - 1 x, 2, , (a) tan - 1 x, , (c) 2 tan - 1 x, , - 1ö, æ - p öö, - 1æ 1 ö, - 1æ, ÷ + cot ç, ÷ + tan ç sin ç, ÷ ÷ is, è 3ø, è 3ø, è è 2 øø, -p, p, (b), (c), 12, 12, , (d) sin - 1 x, , 9. The value of tan - 1 æç, (a), , p, 6, , - 1ö, - 1 æ - 1ö, ÷ + sin ç ÷ is, è 2 ø, è 2 ø, p, 3p, (b), (c), 4, 4, , (d), , p, 3, , (d), , 2p, 3, , 10. The value of tan - 1 1 + cos - 1 æç, (a), , p, 2, , 11. The value of sin[cot - 1 {tan(cos - 1 x)}] is, (b) 1 - x 2, , (a) x, , (c), , 1, x, , (d) x 2 - 1, , 12. The value of sec 2 (tan - 1 2) + cosec 2 (cot - 1 3) is, (a) 13, , (b) 5, , (c) 15, , (a) tan - 1, , x, a, , x, , (d) None of these, , 14. The domain of y = cos - 1 (x 2 - 4) is, (a) [3, 5], (c) ( - 5 , - 3 ) È ( 3 , 5 ), , (b) (0, p ), (d) [ - 5 , - 3 ] È [ 3 , 5 ], , THE QUALIFIER, , ö, ÷ , - a < x < a is, ÷, è a2 - x2 ø, x, x, (b) cot - 1, (c) sin - 1, a, a, æ, , 13. The simplest form of tan - 1 çç, , (d) 10

Page 20 :
12, , CBSE Sample Paper Mathematics Class XII (Term I), , 15. The domain of the function defined by f (x) = sin -1 x - 2 is, (a) (2 , 3], , (b) [2 , 3], , (c) (2 , 3), , Answers, 1. (d), 6. (d), 11. (a), , 2. (b), 7. (b), 12. (c), , 3. (b), 8. (b), 13. (c), , 4. (a), 9. (c), 14. (d), , 5. (b), 10. (c), 15. (b), , (d) None of these, , For Detailed Solutions, Scan the code, , 3. Matrices, Direction (Q. Nos. 1-15) Each of the question has four options out of which only one is correct., Select the correct option as your answer., ì 2, i ¹ j, , then A 2 is, 0, ,, =, i, j, î, é0 4ù, (c) ê, ú, ë4 0û, , 1. If matrix A = [a ij ] 2 ´ 2 , where a ij = í, é2 0 ù, (a) ê, ú, ë0 2 û, , é0 2 ù, (b) ê, ú, ë2 0 û, , é4 0ù, (d) ê, ú, ë0 4û, , 2. If a matrix has 28 elements, then the number of possible order of the matrix is, (a) 6, , éa + 4, , 3. If ê, ë 8, (a) 2, , (b) 3, , (c) 4, , 3b ù é 2a + 2 b + 2 ù, , then the value of a - 2b is, =, - 6úû êë 8, a - 8búû, (b) - 1, (c) 0, , (d) 2, , (d) 1, , 4. If A and B are two matrices of order 3 ´ m and n ´ 4 respectively, then the order of, matrix C = ( 5A - 2B) is a ´ b, then, (a) a + b = 6, (b) a + b = 7, , (c) a - b = 1, , (d) ab = 6, , 5. If X m ´ 3 Yp ´ 4 = Z 2 ´ b for three matrices X , Y and Z, then the value of m + p - b is equal to, (a) 1, , (b) 2, , (c) 9, , é 2 - 2ù, 2, ú and A = pA, then p is, 2, 2, ë, û, (b) 4, (c) 1, , (d) 3, , 6. If matrix A = ê, (a) 2, , THE QUALIFIER, , é 0 a - 3ù, 7. If the matrix A = ê 2 0 - 1ú is skew-symmetric, then a - b is, ê, ú, êë b 1 0 úû, (a) - 1, (b) 1, (c) 5, , (d) - 1, , (d) - 5, , 8. If A and B are matrices of same order, then (AB ¢ - BA¢ ) is a, (a) skew-symmetric matrix, (c) symmetric matrix, , (b) null matrix, (d) unit matrix, , 9. If A is a square matrix such that A 2 = I, then (A - I) 3 + (A + I) 3 - 7 A is equal to, (a) A, (c) I + A, , (b) I - A, (d) 13 A

Page 21 :
13, , CBSE Sample Paper Mathematics Class XII (Term I), , 10. If A and B are square matrix of the same order and AB = 3I, then A - 1 is equal to, (a) 3 B, , (b), , é cos a, ë - sin a, , 11. If A = ê, (a), , p, 2, , 1, B, 3, , (c) 3 B- 1, , (d), , 1 -1, B, 3, , sin a ù, p, , then the value of a satisfying 0 < a < , when A + A T = 2 I 2 is, cos aúû, 2, p, p, (b), (c) 0, (d), 3, 4, , é1 0ù, é x 0ù, 2, and B = ê, ú , then value of x for which A = B is, ú, 5, 1, 1, 1, ë, û, ë, û, (a) 1, (b) - 1, (c) 4, , 12. If A = ê, , (d) does not exist, , 13. Total number of possible matrix of order 3 ´ 3 with each entry - 1 or 1 is, (a) 9, , (b) 27, , (c) 81, , (d) 512, , é 3 - 2ù, é1 0ù, 2, and I = ê, ú, ú , then the value of k such that A = kA - 2I, is, ë 4 - 2û, ë 0 1û, (a) - 1, (b) 2, (c) - 2, (d) 1, , 14. If A = ê, , é1 2 0ù é 0ù, 15. The value of x such that [1 2 1] ê 2 0 1ú ê 2ú = O, is, úê ú, ê, êë1 0 2úû êë xúû, (a) 1, , (c) - 1, , (b) 0, , (d) 3, , Answers, 1. (d), 6. (b), 11. (d), , 2. (a), 7. (d), 12. (d), , 3. (c), 8. (a), 13. (d), , 4. (b), 9. (a), 14. (d), , 5. (a), 10. (b), 15. (c), , For Detailed Solutions, Scan the code, , 4. Determinants, Direction (Q. Nos. 1-15) Each of the question has four options out of which only one is correct., Select the correct option as your answer., 1. The value of determinant, (a) p2 - 1, (c) p2 - 2 p + 1, , p, p-1, , p+1, is, p, (b) p, (d) 1, , 3ù, , then the value of| AB | is, 1úû, (c) 14, , (d) 12, , 3. If the area of triangle with vertices (-3, 0), (3, 0) and (0, k) is 9 sq units. Then, the value k, will be, (a) 9, , (b) 3, , (c) - 9, , (d) 6, , THE QUALIFIER, , é1 2 ù, é1, and B = ê, ú, ë 3 - 1û, ë- 1, (a) - 28, (b) 28, , 2. If A = ê

Page 22 :
14, , CBSE Sample Paper Mathematics Class XII (Term I), , 4. If the points (k + 1, 1), (2k + 1, 3) and (2k + 2, 2k) are collinear, then the value of k is, (b) - 2, , (a) 2, , 5. The value of, , (c), , 1, 2, , (d) 1, , cos 15° sin 15°, is, sin 75° cos 75°, (b) - 1, , (a) 1, , (c) 0, , (d), , 1 2 -3, , 6. Cofactor of 4 in the determinant 4 5, 2 0, (b) - 2, , (a) 2, , 0, 1, , 1, 2, , is equal to, (c) - 5, , (d) - 8, , (c) 2, , (d) - 2, , x +1 1 1, 7. If 1 1 - 1 = 4, then x is equal to, -1, , 1, , 1, (b) - 1, , (a) 0, , 8. If A is a non-singular matrix of order 3 and B is its adjoint such that| B| = 64, then| A| is, (b) ± 64, , (a) 64, , é 4 6ù, ú , then A × ( adj A) is, ë7 5û, 0 ù, é - 22, é - 16 - 4 ù, (a) ê, (b) ê, ú, - 22 û, - 29úû, ë 0, ë 5, , (c) ± 8, , (d) 18, , é22 0 ù, (c) ê, ú, ë 0 22 û, , (d) None of these, , 9. If A = ê, , é 1 l 0ù, 10. If ê 3 - 1 2ú is a singular matrix, then the value of l is, ú, ê, êë 4 1 5úû, (a) 1, (b) 0, (c) - 1, , (d) 2, , sin q, 1 ù, é 1, ê, 11. Let A = - sin q, 1, sin qú , where 0 £ q £ 2p, then, ê, ú, 1 úû, - sin q, êë - 1, (a) det A = 0, (b) det A Î (2 , ¥ ), (c) det A Î (2 , 4), , (d) det A Î [2 , 4], , 12. Given, 2x - y + 2z = 2 , x - 2y + z = - 4 and x + y + lz = 4, then the value of l such that the, given system of equations has no solution is, (b) 1, (c) 0, , (d) - 3, , (a) 3, , THE QUALIFIER, , 2 l -3, 13. If A = 0 2 5 , then A - 1 exists, if, 1 1, (a) l = 2, , 3, (b) l ¹ 2, , (c) l ¹ - 2, , (d) None of these, , 1 -2 5, 14. If there are two values of a which makes determinant D = 2 a - 1 = 86, then the, 0, sum of these numbers is, (a) 4, (b) 5, , (c) - 4, , 4, , 2a, (d) 9

Page 23 :
15, , CBSE Sample Paper Mathematics Class XII (Term I), , é 1 4ù, -1, ú , then A is equal to, 2, 1, ë, û, 4ù, é- 1 4 ù, ê, ú, 7ú, (b) ê 7 7 ú, 2 -1, 1 ú, ê, ú, ú, ë 7, 7û, 7 û, , 15. If A = ê, é1, ê, (a) ê 7, 2, ê, ë7, , é- 1 2 ù, ê, ú, (c) ê 7 7 ú, 4, 1, - ú, ê, ë 7, 7û, , (d) None of these, , Answers, 1. (d), 6. (b), 11. (d), , 2. (a), 7. (a), 12. (b), , 3. (b), 8. (c), 13. (d), , 4. (a), 9. (a), 14. (c), , For Detailed Solutions, Scan the code, , 5. (c), 10. (c), 15. (b), , 5. Continuity and Differentiability, Direction (Q. Nos. 1-15) Each of the question has four options out of which only one is correct., Select the correct option as your answer., ì( x + 3) 2 - 36, ï, 1. If the function f (x) = í x - 3, ï, k, î, value of k is, (a) 6, (b) 12, , , x=3, , ìï sin 3x, , , x¹0, , îï k / 2, (b) 6, , , x=0, , 2. If the function f (x) = í x, (a) 3, , , x ¹ 3 is given to be continuous at x = 3 , then the, , ì k cos x, ï, , 3. The function f (x) = í p - 2x, ïî, , (a) 6, , 3, , (c) - 12, , is continuous at x = 0, then the value of k is, (c) 9, , (d) 12, , , x¹p/2, , p, is continuous at x = , then k equals to, 2, , x=p/2, , (b) - 6, , (b) 3, , (d) - 5, , (c) 5, , 4. The number of points at which the function f (x) =, (a) 2, , (d) - 6, , 1, is discontinuous, is, log|x |, , (c) 4, , (d) 1, , ,, x£2, ì 5, ï, 5. If f (x) = íax + b , 2 < x < 10 is continuous function, then the value of a + b is, ï 21, x ³ 10, ,, î, (b) 5, , (c) 4, , (d) 3, , 6. The set of points, where the function f given by f (x) =| 2x - 1|sin x is differentiable, is, (a) R, , 7. If 2x + 3y = sin y, then, (a), , 2, cos y, , 1, (b) R - ìí üý, î2 þ, , dy, dx, (b), , (c) (0, ¥ ), , (d) None of these, , is equal to, 2, cos y + 3, , (c), , 2, cos y - 3, , (d), , 2, 3 - cos y, , THE QUALIFIER, , (a) 2

Page 24 :
16, , CBSE Sample Paper Mathematics Class XII (Term I), , æ1 - x2 ö, dy, ÷ , then, is equal to, 2÷, dx, è1 + x ø, , 8. If y = logçç, (a), , 4x 3, 1 - x4, , (b), , 9. If y = log 2 (log 2 x), then, (a), , log 2 e, log e x, , - 4x, 1 - x4, , dy, , 1, , (d), , 4 - x4, , - 4x 3, 1 - x4, , is equal to, , dx, , (b), , (c), , log 2 e, , (c), , x log x 2, , log 2 e, , (d), , x log e x, , log e x, x log 2 e, , 10. The derivative of cos - 1 (2x 2 - 1) w.r.t cos - 1 x is, (a) 2, , (b), , -1, 2 1- x, , (c), , 2, , 2, x, , (d) 1 - x 2, , p, 11. If f (x) =|cos x - sin x |, then f ¢ æç ö÷ is equal to, è 3ø, , (a), , 3 -1, 2, , (b), , 1- 3, 2, , 12. If x 16 y 9 = (x 2 + y) 17 , then x, (a) y, , dy, dx, , dy, dx, , at q =, , 1, 2, , p, is, 3, (c), , 14. If y = a cos(log x) + b sin(log x), then x 2, (a) 0, , (b) y, , d2y, dx, , 2, , +x, , (b) 1, , 1, 3, , (d), , 3, 2, , dy, , is equal to, dx, (c) - y, , 15. If x = 2at and y = at 2 , where a is constant, then, 1, (a), 2a, , (d) - 2 y, , (c) 3y, , 13. If x = a sec 2 q and y = a tan 3 q, then, (b), , 2, , is equal to, , (b) 2 y, , (a) 1, , æ 3 + 1ö, ÷, (d) - ç, è 2 ø, , 3 +1, , (c), , 2, , d y, dx, , (c) 2 a, , 2, , at x =, , (d) 2 y, , 1, is, 2, (d), , a, 2, , Answers, , THE QUALIFIER, , 1. (b), 6. (b), 11. (c), , 2. (b), 7. (c), 12. (b), , 3. (a), 8. (b), 13. (d), , 4. (b), 9. (c), 14. (c), , 5. (d), 10. (a), 15. (a), , For Detailed Solutions, Scan the code, , 6. Applications of Derivatives, Direction (Q. Nos. 1-15) Each of the question has four options out of which only one is correct., Select the correct option as your answer., p, 1. The slope of the tangent to the curve x = a cos 3 q and y = a sin 3 q at q = is, 4, 1, (a) 1, (b) - 1, (c) 0, (d), 2

Page 25 :
17, , CBSE Sample Paper Mathematics Class XII (Term I), , 2. The equation of normal to the curve 3x 2 - y 2 = 8 which is parallel to the line x + 3y = 8, is, (a) 3x - y = 8, , (b) 3x + y + 8 = 0, , (c) x + 3y ± 8 = 0, , (d) x + 3y = 0, , 3. The point on the curve y = 2x 2 - 6x - 4 at which the tangent is parallel to the X-axis is, 3 13, (a) æç , ö÷, è2 2 ø, , æ - 5 - 17 ö, (b) ç, ,, ÷, è 2, 2 ø, , 3 17, (c) æç , ö÷, è2 2 ø, , æ 3 - 17 ö, (d) ç ,, ÷, è2 2 ø, , 4. The equation of tangent at (2, 3) on the curve y 2 = ax 3 + b is y = 4 x - 5, then the value of, a - b is, (a) 9, , (b) - 5, , (c) - 9, , (d) 5, , 5. The equation of the tangent to the curve 16x 2 + 9y 2 = 145 at the point (x 1 , y1 ), where, x 1 = 2 and y1 > 0, is, , (a) 32 x - 27y = 145, , (b) 32 x + 25y = 140, , (c) 32 x + 27y = 145, , (d) 30x + 20y = 157, , 6. The value of b for which the function f (x) = sin x - bx + c is decreasing in the interval, ( - ¥, ¥) is given by, (a) b < 1, , (b) b ³ 1, , (c) b > 1, , (d) b £ 1, , 7. In the interval [0, 1], the function x 2 - x + 1 is, (a) increasing, (c) neither increasing nor decreasing, , (b) decreasing, (d) None of these, , 8. The function f (x) = 1 - x 3 are, (a) increases everywhere, (c) increases in (0, ¥ ), , (b) decreases in (0, ¥ ), (d) decreases everywhere, , 9. If f (x) = 3x 4 + 4 x 3 - 12x 2 + 12, then f (x) is, (a) increasing in ( - ¥ , - 2 ) È (0, 1), (c) decreasing in ( - 2 , 0) È (0, 1), , (b) increasing in ( - 2 , 0) È (1, ¥ ), (d) decreasing in ( - ¥ , - 2 ) È (1, ¥ ), , 10. The function f (x) = tan x - x, (a) always increases, (c) never increases, , (b) always decreases, (d) None of these, , 11. The function f (x) = tan - 1 (sin x + cos x) is an increasing is, p p, (a) æç , ö÷, è4 2ø, , æ- p pù, (b) ç, ,, è 2 4 úû, , p, (c) æç0, ö÷, è 2ø, , æ- p pö, (d) ç, , ÷, è 2 2ø, , 12. The minimum value of 2x + 3y, when xy = 6 is, (a) 9, , (b) 12, , (c) 18, , (d) 6, , (a) 9, , (b) 0, , (c) 14, , (d), , 5, 4, , 14. A missile is fired from the ground level rises x m vertically upwards in t s, where, 25 2, t . The maximum height reached is, 2, (a) 200 m, (b) 125 m, (c) 190 m, x = 100 t -, , (d) 300 m, , THE QUALIFIER, , 13. The maximum value of f (x) = (x - 1) 1/ 3 (x - 2) in [1, 9] is

Page 26 :
18, , CBSE Sample Paper Mathematics Class XII (Term I), , 15. The maximum slope of curve y = - x 3 + 3x 2 + 9x - 27 is, (a) 0, , (b) 12, , (c) 16, , (d) 32, , Answers, 1. (b), 6. (b), 11. (b), , 2. (c), 7. (c), 12. (b), , 3. (d), 8. (d), 13. (c), , 4. (a), 9. (b), 14. (a), , For Detailed Solutions, Scan the code, , 5. (c), 10. (a), 15. (b), , 7. Linear Programming, Direction (Q. Nos. 1-15) Each of the question has four options out of which only one is correct., Select the correct option as your answer., 1. Variables of the objective function of the linear programming problem are, (a) zero, (c) negative, , (b) zero or positive, (d) zero or negative, , 2. Corner points of the feasible region determined by the system of linear constraints are, (0, 3), (1, 1) and (3, 0). Let Z = px + qy, where p, q > 0. Condition on p and q, so that the, minimum of Z occurs at (3, 0) and (1, 1) is, q, (a) p = 2 q, (b) p =, 2, (c) p = 3q, (d) p = q, , 3. The feasible solution for a LPP is shown in following figure. Let Z = 3x - 4 y be the, objective function. Minimum of Z occurs at, Y, , (4, 10), , (0, 8), , (6, 8), (6, 5), , (0, 0), , X, , (5, 0), , (a) (0, 0), (c) (5, 0), , (b) (0, 8), (d) (4, 10), , THE QUALIFIER, , 4. The region represented by the inequation system x , y ³ 0, y £ 6 and x + y £ 3 is, (a) unbounded in Ist quadrant, (b) unbounded in Ist and IInd quadrants, (c) bounded in Ist quadrant, (d) None of the above

Page 27 :
19, , CBSE Sample Paper Mathematics Class XII (Term I), , 5. The maximum value of Z = 4 x + 3y, if the feasible region for an LPP is as shown below,, is, , Y, (0, 40), , C(0, 24), , X′, , B(16, 16), (48, 0), , O, Y′, , X, , A, (25, 0), , (a) 112, (c) 72, , (b) 100, (d) 110, , 6. Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5)., Let F = 4 x + 6y be the objective function. The minimum value of F occurs at, (a) Only (0, 2), (b) Only (3, 0), (c) the mid-point of the line segment joining the points (0, 2) and (3, 0), (d) any point on the line segment joining the points (0, 2) and (3, 0), , 7. For the LPP, Minimise Z = x + y such that inequalities 5x + 10 y ³ 0, x + y £ 1, y £ 4 and, x, y ³ 0, (a) there is a bounded solution, (c) there are infinite solutions, , (b) there is no solution, (d) None of these, , 8. A furniture dealer deals in only two items namely tables and chairs. He has ` 5000 to, invest and space to store atmost 60 pieces. A table cost him ` 250 and a chair ` 60. He, can sell a table at a profit of ` 15. Assume that, he can sell all the items that he, produced. The number of constraints in the problem are, (a) 2, (b) 3, (c) 4, (d) 5, , 9. The feasible region for the following constraints L1 £ 0, L2 ³ 0, L3 = 0, x ³ 0 and y ³ 0 in, the figure shown is, , Y, F, , L1 = 0, , E, G, , L3 = 0, , D, , I, , H, Y′, , (a) area DHF, (c) line segment EG, , A, , L2 = 0, B, , C, , X, , (b) area AHC, (d) line segment GI, , THE QUALIFIER, , X′

Page 28 :
20, , CBSE Sample Paper Mathematics Class XII (Term I), , 10. The feasible region for an LPP is shown in the following figure. Then, the minimum, value of Z = 11x + 7 y is, , Y, , C(0, 5), (0, 3) B, , A(3, 2), , X′, , x+, 3, , Y′, , (a) 21, , x+y=5, , (b) 47, , X, y=, 9, , (c) 20, , (d) 31, , 11. The corner points of the feasible region determined by the system of linear constraints, are (0, 0), (0, 40), (20, 40), (60, 20) and (60, 0). The objective function is Z = 4 x + 3y., Compare the quantity in column A and column B., Column A, , Column B, , Maximum of Z, , 325, , (a) The quantity in column A is greater, (b) The quantity in column B is greater, (c) The two quantities are equal, (d) The relationship cannot be determined on the basis of the information supplied., , 12. The area of the feasible region for the following constraints 3y + x £ 3, x ³ 0 and y ³ 0, will be, (a) bounded, , (b) unbounded, , (c) convex, , (d) concave, , 13. Shaded region is represented by the constraints, Y, 4x – 2 y = – 3, , (0,3/2) A, X¢, , (–3/4,0)B, , X, , O, , Y¢, , (a) 4x - 2 y £ 3, , (b) 4x - 2 y £ - 3, , (c) 4x - 2 y ³ 3, , (d) 4x - 2 y ³ - 3, , 14. Maximum value of Z = 3x + 4 y subject to constraints x - y ³ - 1, -x + y £ 0, x £ 10, y £ 12, , THE QUALIFIER, , and x , y ³ 0, is given by, (a) 1, (b) 4, , (c) 6, , (d) no feasible region, , 15. The area of the feasible region for the following constraints x + y £ 8, 3x + 5y £ 15, x ³ 0, and y ³ 0 will be, (a) bounded, (c) do not say anything, , (b) unbounded, (d) None of these, , Answers, 1. (b), 6. (d), 11. (b), , 2. (b), 7. (a), 12. (a), , 3. (b), 8. (c), 13. (d), , 4. (c), 9. (c), 14. (d), , 5. (a), 10. (a), 15. (a), , For Detailed Solutions, Scan the code

Page 29 :
21, , CBSE Sample Paper Mathematics Class XII (Term I), , CBSE, QUESTION BANK, Case Study Based Questions, Relations and Functions, 1. A general election of Lok Sabha is a gigantic, exercise. About 911 million people were, eligible to vote and voter turnout was about, 67%, the highest ever., ONE – NATION, , (a) (F1 , F2 ) Î R, (F2 , F3 ) Î R and (F1 , F3 ) Î R, (b) (F1 , F2 ) Î R, (F2 , F3 ) Î R and (F1 , F3 ) Î, / R, (c) (F1 , F2 ) Î R, (F2 , F2 ) Î R but (F3 , F3 ) Î, / R, (F, ), and, (F, ), (d) (F1 , F2 ) Î, R,, ,, F, Î, R, ,, F, /, / R, 2, 3 /, 1, 3 Î, , (iv) The above defined relation R is, (a) Symmetric and transitive but not, reflexive z, (b) Universal relation, (c) Equivalence relation, (d) Reflexive but not symmetric and, transitive, , ONE – ELECTION, FESTIVAL OF, DEMOCRACY, GENERAL, ELECTION – 2019, , Let I be the set of all citizens of India who, were eligible to exercise their voting right in, general election held in 2019. A relation ‘R’, is defined on I as follows, R = {(V1 , V2 ) : V1 , V2 Î I}, and both use their voting right in general, election-2019}, Answer the following questions using the above, information., , (b) (Y , X) Î R, (d) (X, Y) Î, / R, , (ii) Mr. ‘X’ and his wife ‘W’ both exercised, their voting right in general election 2019,, Which of the following is true?, (a) both (X, W) and (W , X) Î R, (b) (X, W) Î R but (W , X) Î, / R, (c) both (X, W) and (W , X) Î, / R, (d) (W , X) Î R but (X, W) Î, / R, , (v) Mr. Shyam exercised his voting right in, General Election-2019, then Mr. Shyam is, related to which of the following?, (a) All those eligible voters who cast their, votes, (b) Family members of Mr.Shyam, (c) All citizens of India, (d) Eligible voters of India, (i) (d) Given, R = {(V1, V2 ) : V1, V2 Î I} and, both use their voting right in general election-2019., Since, X, Y Î I × X exercised his voting right while Y did, not cast her vote in general election-2019., \ Clearly, ( X, Y ) Ï R., (ii) (a) Relation is symmetric., \, ( X, W ) Î R Þ (W , X ) Î R., (iii) (a) Since, (F, F ) Î R, F Î I, and F use their voting right., Þ R is reflexive., (F1, F2 ) Î R Þ (F2 , F1 ) Î R, R is symmetric., and (F1, F2 ) Î R and (F2 , F3 ) Î R Þ (F1, F3 ) Î R, (By transitive property), (iv) (c) Given relation R is reflexive, symmetric and, transitive., \ R is equivalence relation., , CBSE QUESTION BANK, , (i) Two neighbors X and Y Î I . X exercised, his voting right while Y did not cast her, vote in general election-2019. Which of, the following is true?, (a) (X, Y) Î R, (c) (X, X) Î, / R, , (iii) Three friends F1 , F2 and F3 exercised their, voting right in general election-2019, then, which of the following is true?

Page 30 :
22, , CBSE Sample Paper Mathematics Class XII (Term I), , (v) (a) Clearly, Mr. Shyam exercised his voting right in, general election-2019, then Mr. Shyam is related to, all those eligible voters who cast their votes., , 2. Sherlin and Danju are playing Ludo at home, , (i), , during Covid-19. While rolling the dice,, Sherlin’s sister Raji observed and noted the, possible outcomes of the throw every time, belongs to set {1, 2, 3, 4, 5, 6}., Let A be the set of players while B be the set, of all possible outcomes., , (ii), , (iii), , A = {S, D}, B = {1,2,3,4,5,6}, Answer the following questions using the above, information., (i) Let R : B ® B be defined by, R = {( x , y) : y is divisible by x } is, (a) Reflexive and transitive but not, symmetric, (b) Reflexive and symmetric and not, transitive, (c) Not reflexive but symmetric and transitive, (d) Equivalence, , (ii) Raji wants to know the number of, functions from A to B. How many number, of functions are possible?, (a) 62, (c) 6!, , (b) 26, (d) 212, , CBSE QUESTION BANK, , (iii) Let R be a relation on B defined by, R = {(1,2), (2,2), (1,3), (3,4), (3,1), (4,3),, (5,5)}. Then, R is, (a) Symmetric, (c) Transitive, , (b) Reflexive, (d) None of these, , (iv), , Given, A be the set of players i.e. {S, D },, while B be the set of all possible outcomes, i.e. {1, 2, 3, 4, 5, 6}., (a) Clearly, R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5),, (1, 6), (2, 2), (2, 4), (2, 6), (3, 3),, (3, 6), (4, 4), (5, 5), (6, 6)}, Since, x is divisible by x for all x Î B., \ ( x, x) Î R For all x Î R. So, R is reflexive., We observed that, 6 is divisible by 2, Þ (2, 6) ÎR, But (6, 2) ÏR, so R is not symmetric., Again, if y is divisible by x and z is divisible by y, then, z will also be divisible by x., \ ( x, y) Î R, ( y, z) Î R Þ ( x, z) Î R., Þ R is transitive., (a) We know that, if A and B are two sets having m, and n elements, respectively. Then, total number of, functions from A to B is nm , i.e. 62 ., (d) Since, (1, 1) ÏR, So R is not reflexive., (1, 2 ) ÎR but (2, 1) ÏR, So R is not symmetric., and (1, 3) ÎR and (3, 4) ÎR but (1, 4) ÏR., So, R is not transitive., (d) If A and B are finite sets consisting of m and, n elements, respectively, then A ´ B has mn ordered, pairs., \ Total number of relations from A to B is 2 mn ., \ 2 2 ´ 6 = 212, , (v) (b) Since, (a, a) Î R, a Î A Þ R is reflexive., and (a, b ) Î R and (b, c ) Î R, Þ (a, c ) Î R " (a, b )c Î A Þ R is transitive., and (1, 2 ) ÎR but (2, 1) Ï R Þ R is not symmetric., , 3. An organization conducted bike race under, 2 different categories-boys and girls. Totally, there were 250 participants. Among all of, them finally three from Category 1 and two, from Category 2 were selected for the final, race. Ravi forms two sets B and G with these, participants for his college project., Let B = {b1 , b 2 , b 3 } G = {g1 , g 2 } where B, represents the set of boys selected and G the, set of girls who were selected for the final, race., , (iv) Raji wants to know the number of, relations possible from A to B. How many, numbers of relations are possible?, (a) 62, (c) 6!, , (b) 26, (d) 212, , (v) Let R : B ® B be defined by R={(1,1),(1,2),, (2,2), (3,3), (4,4), (5,5),(6,6)}, then R is, (a) Symmetric, (b) Reflexive and Transitive, (c) Transitive and symmetric, (d) Equivalence, , Ravi decides to explore these sets for various, types of relations and functions.

Page 31 :
23, , CBSE Sample Paper Mathematics Class XII (Term I), , Answer the following questions using the above, information., (i) Ravi wishes to form all the relations, possible from B to G. How many such, relations are possible?, (b) 25, (d) 23, , (a) 26, (c) 0, , (ii) Let R : B ® B be defined by R = {( x , y) : x, and y are students of same sex}, Then this, relation R is, , 4. Students of Grade 9, planned to plant, saplings along straight lines, parallel to each, other to one side of the playground ensuring, that they had enough play area. Let us, assume that they planted one of the rows of, the saplings along the line y = x - 4. Let L be, the set of all lines which are parallel on the, ground and R be a relation on L., , (a) Equivalence, (b) Reflexive only, (c) Reflexive and symmetric but not, transitive, (d) Reflexive and transitive but not, symmetric, , (iii) Ravi wants to know among those, relations, how many functions can be, formed from B to G ?, 2, , 12, , (a) 2, (c) 3 2, , (b) 2, (d) 23, , (iv) Let R : B ® G be defined by, R = {(b1 , g1 ), (b2 , g2 ), (b 3 , g1 )}, then R is, , (v) Ravi wants to find the number of injective, functions from B to G. How many, numbers of injective functions are, possible?, (b) 2!, , (c) 3!, , (i) Let relation R be defined by R = {, ( L1 , L 2) : L1|| L 2 where L1 , L2 ÎL}, then R, is ……… relation ., (a) Equivalence, (b) Only reflexive, (c) Not reflexive, (d) Symmetric but not transitive, , (a) Injective, (b) Surjective, (c) Neither Surjective nor Injective, (d) Surjective and Injective, , (a) 0, , Answer the following questions using the above, information., , (d) 0!, , (i) (a) Since, n(B) = 3, n(G ) = 2, \ Total possible relations are 2 3 ´ 2 = 2 6, , (a) R is Symmetric but neither reflexive nor, transitive, (b) R is Reflexive and transitive but not, symmetric, (c) R is Reflexive but neither symmetric nor, transitive, (d) R is an Equivalence relation, , (iii) The function, f ( x) = x - 4 is, , f:R ® R, , defined, , by, , (a) Bijective, (b) Surjective but not injective, (c) Injective but not Surjective, (d) Neither Surjective nor Injective, , (iv) Let f : R ® R be defined by f ( x) = x - 4., Then, the range of f ( x) is, (a) R, , (b) Z, , (c) W, , (d) Q, , (v) Let R = {(L 1 , L 2 ) : L 1 is parallel to L 2 and, L 1 : y = x - 4}, then which of the, following can be taken as L2 ?, (a) 2x - 2y + 5 = 0, (c) 2x + 2y + 7 = 0, , (b) 2x + y = 5, (d) x + y = 7, , (i) (a) Given, R = {(L1L2 ) : L1 is parallel to L2 }., R is reflexive as any line L1 is parallel to itself, Þ (L1, L1 ) Î R., , CBSE QUESTION BANK, , (ii) (a) Clearly, ( x, x) Î R, " x Î B, Þ R is reflexive., ( x, y) Î R Þ( y, x) Î R, x, y Î B, Þ R is symmetric., and ( x, y) Î R and ( y, z) Î R "x, y, z Î B, Þ ( x, z) Î R, Þ R is transitive., Hence, R is equivalence relation., (iii) (d) Since, n(B) = 3, n(G ) = 2, \ Number of functions from B to G are 2 3 ., (iv) (b) Since, range of R = G, Þ Range = codomain, Þ R is surjective., (v) (a) If A and B are finite sets having m and n elements,, respectively then the number of injective function, ìn p , n ³ m, from A to B is í m, ., n<m, î 0,, Here, n(B) = 3, n(G ) = 2, \ Total number of injective relation from B to G is 0., , (ii) Let R = {( L1 , L 2) : L1 ^ L 2 where, L1, L 2 Î L}, which of the following is true?

Page 37 :
29, , CBSE Sample Paper Mathematics Class XII (Term I), , 5. Two farmers Ramakishan and Gurucharan, Singh cultivate only three varieties of rice, namely Basmati, Permal and Naura. The, sale (in rupees) of these varieties of rice by, both the farmers in the month of September, and October are given by the following, matrices A and B., , September sales (in `), é10000 20000 30000ù Ramakishan, A=ê, ú, ë50000 30000 10000 û Gurucharan, October sales (in `), é 5000 10000 6000 ù Ramakishan, B=ê, ú, ë 20000 10000 10000û Gurucharan, Answer the following questions using the above, information., (i) The total sales in September and October, for each farmer in each variety can be, represented as, (a) A + B, (c) A > B, , (ii) What is the value of A23 ?, (a) 10000, (c) 30000, , (b) A - B, (d) A < B, , (iv) (a) 2% of B in October, 2, é 5000 10000 6000 ù, =, ´ B = 0.02 ´ ê, ú, 100, ë20000 10000 10000û, é100 200 120ù, =ê, ú, ë 400 200 200û, Hence, the profit received by Ramakishan in sales of, each variety of rice are ` 100, ` 200 and ` 120, respectively., 2, (v) (b) 2% of A in September =, ´ A, 100, 2 é10000 20000 30000ù, =, 100 êë 50000 30000 10000 úû, é 200 400 600ù, =ê, ú, ë1000 600 200 û, Hence, the profit received by Gurucharan in the sale, of each variety of rice are ` 1000, ` 600, ` 200, respectively., , Determinants, 1. Manjit wants to donate a rectangular plot of, land for a school in his village. When he was, asked to give dimensions of the plot, he told, that if its length is decreased by 50 m and, breadth is increased by 50m, then its area, will remain same, but if length is decreased, by 10m and breadth is decreased by 20m,, then its area will decrease by 5300 m 2 ., , (b) 20000, (d) 40000, , y, , (iii) The decrease in sales from September to, October is given by, (a) A + B, (c) A > B, , (b) A - B, (d) A < B, , (iv) If Ramakishan receives 2% profit on gross, sales, compute his profit for each variety, sold in October., , (v) If Gurucharan receives 2% profit on gross, sales, compute his profit for each variety, sold in September., (a) ` 100, ` 200, ` 120, (b) ` 1000 , ` 600, ` 200, (c) ` 400, ` 200, ` 120, (d) ` 1200, ` 200, ` 120, (i) (a) Combined sales in September and October for, each farmer in each variety is represented by A + B., (ii) (a) Clearly, A23 = 10000, (iii) (b) The decrease in sales from September to October, is given by A - B., , Based on the information given above, answer, the following questions., (i) The equations in terms of x and y are, (a) x - y = 50, 2x - y = 550, (b) x - y = 50, 2x + y = 550, (c) x + y = 50, 2x + y = 550, (d) x + y = 50, 2x - y = 550, , (ii) Which of the following matrix equation, represent the information given above., é 1 - 1ù é x ù é 50 ù, (a) ê, ú, úê ú=ê, ë 2 1 û ë y û ë550û, é1, (b) ê, ë2, é1, (c) ê, ë2, é1, (d) ê, ë2, , 1ù é x ù é 50 ù, =, 1úû êë y úû êë550úû, 1 ù é x ù é 50 ù, =, - 1úû êë y úû êë550úû, 1ù é x ù é - 50 ù, =, 1úû êë y úû êë - 550úû, , CBSE QUESTION BANK, , (a) ` 100, ` 200 and ` 120, (b) ` 100, ` 200 and ` 130, (c) ` 100, ` 220 and ` 120, (d) ` 110, ` 200 and ` 120, , x

Page 38 :
30, , CBSE Sample Paper Mathematics Class XII (Term I), , (iii) The value of x (length of rectangular, field) is, (a) 150 m, (c) 200 m, , (b) 400 m, (d) 320 m, , (iv) The value of y (breadth of rectangular, field) is, (a) 150 m, (c) 430 m, , (b) 200 m, (d) 350 m, , (v) How much is the area of rectangular, field?, (a) 60000 sq m, (c) 30000 m, , (b) 30000 sq m, (d) 3000 m, , According to the question, when length is decreased, by 50 m and breadth is increased by 50 m., \, ( x - 50)( y + 50) = xy, …(i), Þ, x - y = 50, and when length is decreased by 10 m and breadth, is decreased by 20 m., \, ( x - 10)( y - 20) = xy - 5300, …(ii), Þ, 2 x + y = 550, (i) (b) x - y = 50, 2 x + y = 550, (ii) (a) Eqs. (i) and (ii) can be written in matrix form as, é 1 -1ù é xù é 50 ù, ê2 1 ú ê yú = ê 550ú, û, ûë û ë, ë, (iii) (c) We have,, é 1 -1ù, ê2 1 ú, û, ë, , é xù é 50 ù, ê yú = ê 550ú, û, ë û ë, -1, , é xù é 1 -1ù é 50 ù, ê yú = ê2 1 ú ê 550ú, ë û ë, û ë, û, 1, é 1 1ù é 50 ù, =, 1 - (2 )(-1) êë -2 1úû êë 550úû, é é a b ù -1, 1, é d - bù ù, =, êQ ê, ú, ú, ad - bc êë -c a úû ú, êë ëc d û, û, 1 é 50 + 550 ù 1 é 600ù, = ê, =, 3 ë -100 + 550úû 3 êë 450úû, é200ù, =ê, ú, ë150û, \ x = 200, y = 150, Lenght of rectangular field Þ x = 200 m, (iv) (a) Breadth of rectangular field, y = 150 m, (v) (b) Area of rectangular field = 200 ´ 150, = 30000 sq m, , CBSE QUESTION BANK, , Þ, , Continuity and, Differentiability, 1. The Relation between the height of the plant, (y in cm) with respect to exposure to, sunlight is governed by the following, 1, equation y = 4x - x 2 where x is the number, 2, of days exposed to sunlight., , Answer the following questions using the above, information., (i) The rate of growth of the plant with, respect to sunlight is, (a) 4x -, , 1 2, x, 2, , (b) 4 - x, (d) x -, , (c) x - 4, , 1 2, x, 2, , (ii) What is the number of days it will take for, the plant to grow to the maximum, height?, (a) 4, (c) 7, , (b) 6, (d) 10, , (iii) What is the maximum height of the, plant?, (a) 12 cm, (c) 8 cm, , (b) 10 cm, (d) 6 cm, , (iv) What will be the height of the plant after 2, days?, (a) 4 cm, (c) 8 cm, , (b) 6 cm, (d) 10 cm, , (v) If the height of the plant is 7/2 cm, the, number of days it has been exposed to the, sunlight is, (a) 2, (c) 4, Given equation y = 4 x -, , (b) 3, (d) 1, 1 2, x, 2, , (i) (b) The ratio of growth of the plant with respect to, 1, dy, sunlight, i.e., = 4 - (2 x) = 4 - x, 2, dx, (ii) (a) For maximum height,, dy, put, = 0Þ4 - x = 0Þ x = 4, dx, d2y, Now, 2 = - 1 < 0, dx, \ Number of days for plant to grow to the maximum, height = 4 days, 1, (iii) (c) Maximum height of plant, y = 4 x - x2, 2, Put x = 4, we get, 1, y = 4(4) - (4)2 = 16 - 8 = 8 cm, 2, \ Maximum height of plant = 8 cm

Page 39 :
31, , CBSE Sample Paper Mathematics Class XII (Term I), 1 2, x, 2, Height of plant after 2 days,, we put x = 2, we get, 1, y = 4(2 ) - (2 )2 = 8 - 2 = 6 cm, 2, 1, (v) (d) We have, y = 4 x - x2, 2, 7, Putting y = , then, 2, 7, 1, = 4 x - x2 Þ x2 - 8 x + 7 = 0, 2, 2, Þ ( x - 7 )( x - 1) = 0 Þ x = 1, 7, \ Number of days to exposed sunlight = 1day, , (iv) (b) We have, y = 4 x -, , 2. P( x) = - 5x 2 + 125x + 37500 is the total profit, function of a company, where x is the, production of the company., , \ Profit is maximum when P¢( x) = 0, 125, = 12.5, 10, The production of company is 12.5 unit, when profit, is maximum., (ii) (b) Profit is maximum at 12.5., \ P(12.5) = - 5(12.5)2 + 125 ´ (12.5) + 37500, - 10 x + 125 = 0 Þ x =, , Þ, , = - 781.25 + 1562.5 + 37500 = ` 38281.25, (iii) (d) Profit is strictly increasing,, when,, P ¢ ( x) > 0, \, - 10 x + 125 > 0 Þ 10 x < 125 Þ x <12.5, [Q x > 0], \, x Î(0, 12.5), (iv) (b) We have, P( x) = - 5 x2 + 125 x + 37500, P(2 ) = - 5(2 )2 + 125 ´ 2 + 37500, = - 20 + 250 + 37500 = 37730, (v) (a) We have, P( x) = - 5 x2 + 125 x + 37500, and, Þ, , P( x) = 38250, 38250 = - 5 x2 + 125 x + 37500, , Þ, , 5 x2 - 125 x + 750 = 0, , Þ, , x2 - 25 x + 150 = 0, , Þ, ( x - 15)( x - 10) = 0 Þ x = 10, 15, The production of company be 15 units., , 3. A potter made a mud vessel, where the, Answer the following questions using the above, information., (i) What will be the production when the, profit is maximum?, (a) 37500, (c) - 12.5, , shape of the pot is based on, f ( x) =| x - 3| +| x - 2|, where f ( x), represents the height of the pot., , (b) 12.5, (d) 37500, , (ii) What will be the maximum profit?, (a) ` 3828125, (c) ` 39000, , (b) ` 38281.25, (d) None of these, , (iii) Check in which interval the profit is, strictly increasing., (a) (12.5, ¥), (b) for all real numbers, (c) for all positive real numbers, (d) (0, 12.5), , (iv) When the production is 2units what will, be the profit of the company?, (b) 37730, (d) None of these, , (v) What will be production of the company, when the profit is ` 38250?, (a) 15, (b) 30, (c) 2, (d) data is not sufficient to find, Given total profit function, P( x) = - 5 x2 + 125 x + 37500, (i) (b) We have, P( x) = - 5 x2 + 125 x + 37500, P¢( x) = - 10 x + 125, p¢¢( x) = - 10 < 0, , (i) When x > 4 What will be the height in, terms of x?, (a) x - 2, (c) 2x - 5, , (b) x - 3, (d) 5 - 2x, , (ii) Will the slope vary with x value?, (a) Yes, (b) No, (c) Slope is not defined for any value of x., (d) Insufficient data for the slope., , (iii) What is, , dy, at x = 3 ?, dx, , (a) 2, (b) - 2, (c) Function is not differentiable, (d) 1, , (iv) When the x value lies between (2, 3) then, the function f ( x) is, (a) 2x - 5, , (b) 5 - 2x, , (c) 1, , (d) 5, , CBSE QUESTION BANK, , (a) 37500, (c) 37770, , Answer the following questions using the above, information.

Page 40 :
32, , CBSE Sample Paper Mathematics Class XII (Term I), , (v) If the potter is trying to make a pot using, the function f ( x) = [x] , will he get a pot or, not? Why?, , (iii) What will be the equation of the tangent, at the critical point if it passes through (2,, 3)?, (a) x + 360y = 1082, (b) y = 360x - 717, (c) x = 717 y + 360, (d) None of the above, , (a) Yes, because it is a continuous function, (b) Yes, because it is not continuous, (c) No , because it is a continuous function, (d) No , because it is not continuous, Given, f( x) = | x - 3| + | x - 2 |, x<2, ì- x + 3 - x + 2, ï, f ( x) = í - x + 3 + x - 2 2 £ x < 3, ï x- 3+ x-2, x³3, î, , (i), (ii), , (iii), (iv), (v), , x<2, ì5 - 2 x, ï, Þ, f ( x) = í 1, 2£ x<3, ï2 x - 5, x³3, î, (c) When x > 4, f( x) = 2 x - 5, x<2, ì- 2, ï, (a) (yes) f ¢( x) = í 1 2 < x < 3, ï2, x>3, î, Clearly, slope vary with x value., (c) f¢(3- ) = 1and f¢(3+ ) = 2, \ f( x) is not differentiable at x = 3, (c) When x Î(2, 3), f( x) = 1, (d) When f( x) = [ x] ,, f( x) is not continuous at integral value., \ Potter will not get a pot., , 4. The shape of a toy is given as, , f ( x) = 6 ( 2x 4 - x 2 ). To make the toy, beautiful 2 sticks which are perpendicular, to each other were placed at a point (2, 3),, above the toy., , (iv) Find the second order derivative of the, function at x = 5., (a) 598, (c) 3588, , (b) 1176, (d) 3312, , (v) At which of the following intervals will, f ( x) be increasing?, (a) ( - ¥ , - 1 / 2) È (1 / 2, ¥), (b) ( - 1 / 2, 0) È (1 / 2, ¥), (c) ( 0, 1 / 2) È (1 / 2, ¥), (d) ( - ¥ , - 1 / 2) È ( 0, 1 / 2), Given function, f( x) = 6(2 x4 - x2 ), (i) (b) We have, f( x) = 6(2 x4 - x2 ), f ¢( x) = 6(8 x3 - 2 x), For critical point, put f ¢( x) = 0, we get, 8 x3 - 2 x = 0, Þ, , 2 x(4 x2 - 1) = 0, , x = 0, 4 x2 - 1 = 0, 1, 1, Þ, x = 0, x2 = Þ x = ±, 4, 2, 1, \ Critical point is ±, 2, (ii) (d) We have, f ¢( x) = 6(8 x3 - 2 x), Þ, , f¢(2 ) = 6(8(2 )3 - 2(2 )) = 6(64 - 4) = 360, -1, 1, Slope of normal at (2, 3) is =, f ¢(2 ) 360, (iii) (b) Slope of tangent at x = 2 is 360, Equation of tangent passes through (2, 3) and slope, 360 is, y - 3 = 360( x - 2 ), Þ, y - 3 = 360 x - 720 Þ y = 360 x - 717, (iv) (c) We have, f( x) = 6(2 x4 - x2 ), , CBSE QUESTION BANK, , f ¢( x) = 6(8 x3 - 2 x), , Answer the following questions using the above, information., (i) Which value from the following may be, abscissa of critical point?, 1, 4, (c) ± 1, , (a) ±, , 1, 2, (d) None of these, , (b) ±, , (ii) Find the slope of the normal based on the, position of the stick., (a) 360, 1, (c), 360, , (b) - 360, -1, (d), 360, , f ¢¢( x) = 6(24 x2 - 2 ), f ¢¢(5) = 6(24(5)2 - 2 ) = 6(600 - 2 ), = 6 ´ 598 = 3588, (v) (b) For increasing f ¢( x) > 0, \, 6(8 x3 - 2 x) > 0, Þ, , 2 x(2 x - 1) (2 x + 1) > 0, +, , –, –1/2, , +, , –, 0, , 1/2, , Þ f( x) is increasing when,, 1, 1, x Î æç - , 0ö÷ È æç , ¥ö÷, è 2 ø è2 ø

Page 41 :
33, , CBSE Sample Paper Mathematics Class XII (Term I), , Latest CBSE, , SAMPLE PAPER, Latest Sample Question Paper for Class XII (Term I), Issued by CBSE on 2 Sept 2021, , Mathematics Class 12 (Term I), Instructions, 1. This question paper contains three sections - A, B and C. Each part is compulsory., 2. Section - A has 20 MCQs, attempt any 16 out of 20., 3. Section - B has 20 MCQs, attempt any 16 out of 20., 4. Section - C has 10 MCQs, attempt any 8 out of 10., 5. There is no negative marking., 6. All questions carry equal marks., Maximum Marks : 40, , Time : 90 Minutes, , Section A, In this section, attempt any 16 questions out of Questions 1-20. Each question is of 1 mark weightage., , p, , 1. sin éê - sin -1 æç - ö÷ ùú is equal to, è 2 øû, ë3, (a), , 1, 2, , 1, , (b), , 1, 3, , (c) -1, , (d) 1, , ì1, when i ¹ j, , , then A 2 is, 3. If A = [a ij ] is a square matrix of order 2 such that a ij = í, 0, ,, when, i, =, j, î, é1 0ù, (a) ê, ú, ë1 0û, , é 1 1ù, (b) ê, ú, ë0 0û, , é1 1ù, (c) ê, ú, ë1 0û, , ék 8 ù, ú is a singular matrix, is, ë 4 2kû, (b) -4, (c) ± 4, , é1 0ù, (d) ê, ú, ë0 1û, , 4. Value of k, for which A = ê, (a) 4, , (d) 0, , Latest CBSE SAMPLE PAPER, , ì1 - cos kx, , x¹0, ï, is, 2. The value of k(k < 0) for which the function f defined as f (x) = í x sin x, 1, ï, x=0, ,, 2, î, continuous at x = 0, is, 1, 1, (d), (a) ±1, (b) -1, (c) ±, 2, 2

Page 42 :
34, , CBSE Sample Paper Mathematics Class XII (Term I), , 5. Find the intervals in which the function f given by f (x) = x 2 - 4 x + 6 is strictly, increasing, (a) ( -¥ , 2 ) È (2 , ¥ ), , (b) (2 , ¥ ), , (c) ( -¥ , 2 ), , (d) ( -¥ , 2 ] È (2 , ¥ ), , 6. Given that A is a square matrix of order 3 and|A| = - 4, then|adj A| is equal to, (a) -4, , (c) -16, , (b) 4, , (d) 16, , 7. A relation R in set A = {1, 2, 3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which of the, following ordered pair in R shall be removed to make it an equivalence relation in A?, (a) (1, 1), (b) (1, 2), (c) (2, 2), (d) (3, 3), é 2a + b, , a - 2b ù, , é4, , -3ù, , 8. If ê, ú =ê, ú , then the value of a + b - c + 2d is, ë 5c - d 4 c + 3dû ë11 24û, (d) -8, 1, 9. The point at which the normal to the curve y = x + , x > 0 is perpendicular to the line, x, 3x - 4 y - 7 = 0 is, 5, 5, -1 5, 1 5, (c) æç , ö÷, (b) æç ±2 , ö÷, (a) æç2 , ö÷, (d) æç , ö÷, è 2ø, è, ø, è, è, ø, 2, 2 2ø, 2 2, (a) 8, , (b) 10, , (c) 4, , 10. sin(tan -1 x), where|x| < 1 is equal to, (a), , x, , (b), , 1 - x2, , 1, 1 - x2, , (c), , 1, 1 + x2, , (d), , x, 1 + x2, , 11. Let the relation R in the set A = {x Î Z: 0 £ x £ 12}, given by R = {(a , b):|a - b|is a multiple, of 4}. Then [1], the equivalence class containing 1, is, (a) {1, 5, 9}, (b) {0, 1, 2, 5}, (c) f, dy, 12. If e x + e y = e x + y , then is, dx, (a) e y - x, (b) e x + y, (c) - e y - x, , (d) A, , (d) 2 e x - y, , 13. Given that matrices A and B are of order 3 ´ n and m ´ 5 respectively, then the order of, , Latest CBSE SAMPLE PAPER, , matrix C = 5A + 3B is, (a) 3 ´ 5 and m = n, (b) 3 ´ 5, , 14. If y = 5 cos x - 3 sin x, then, (a) -y, , (b) y, , (c) 3 ´ 3, , (d) 5 ´ 5, , (c) 25y, , (d) 9y, , 2, , d y, dx 2, , is equal to, , é 2 5ù, ú , then ( adjA ) ¢ is equal to, ë -11 7û, é 7 5ù, é 7 11ù, (b) ê, (c) ê, ú, ú, ë11 2 û, ë -5 2 û, , 15. For matrix A = ê, é -2, (a) ê, ë 11, , -5 ù, -7úû, , 2, x2 y, +, = 1 at which the tangents are parallel to Y-axis are, 9, 16, (b) ( ±4, 0), (c) ( ±3, 0), (d) (0, ± 3), , 16. The points on the curve, (a) (0, ± 4), , é 7 -5 ù, (d) ê, ú, ë11 2 û

Page 43 :
35, , CBSE Sample Paper Mathematics Class XII (Term I), , 17. Given that A = [a ij ] is a square matrix of order 3 ´ 3 and|A| = -7, then the value of, 3, , å a i2 Ai2 where Aij denotes the cofactor of element a ij is, , i=1, , (b) -7, , (a) 7, , 18. If y = log(cos e x ), then, (a) cos e x - 1, , (c) 0, , (d) 49, , (c) e x sin e x, , (d) -e x tan e x, , dy, , is, dx, (b) e - x cos e x, , 19. Based on the given shaded region as the feasible region in the graph, at which point (s), is the objective function Z = 3x + 9y maximum., Y, x=y, 25, , D (0, 20), C (15, 15), , 15, (0, 10), X′, , A, B (5, 5), , 5, O, , 5, Y′, , (60, 0), , 20, , (10, 0), , 35, , X, , 50, , x+3y=60, x+y=10, , (a) Point B, (c) Point D, , (b) Point C, (d) Every point on the line segment CD, , p, 20. The least value of the function f (x) = 2 cos x + x in the closed interval éê 0, ùú is, p, (b) + 3, 6, , (a) 2, (c), , p, 2, , ë, , 2û, , (d) The least value does not exist, , Section B, In this section, attempt any 16 questions out of Questions 21-40. Each question is of 1 mark weightage., , (a) one-one but not onto, (c) Neither one-one nor onto, , 22. If x = a secq, y = b tan q then, (a), , -3 3b, 2, , a, -3 3b, (c), a, , (b) Not one-one but onto, (d) Both one-one and onto, , d2y, dx, , 2, , at q =, , p, is, 6, (b), (d), , -2 3b, a, -b, 3 3a 2, , Latest CBSE SAMPLE PAPER, , 21. The function f : R ® R defined as f (x) = x 3 is

Page 44 :
36, , CBSE Sample Paper Mathematics Class XII (Term I), , 23. In the given graph, the feasible region for a LPP is shaded. The objective function, Z = 2x - 3y will be minimum at, Y, (4, 10), , (6, 8), , (0, 8), , (6, 5), , (0, 0), , (a) (4, 10), , X, , (5, 0), , (b) (6, 8), , (c) (0, 8), , 24. The derivative of sin -1 (2x 1 - x 2 ) w.r.t. sin -1 x,, (a) 2, , (b), , p, -2, 2, , (c), , p, 2, , (d) (6, 5), , 1, < x < 1 is, 2, (d) -2, , 2 - 4ù, é2, é1 -1 0ù, ê, ú, ê, 25. If A = 2 3 4 and B = -4 2 -4ú , then, ê, ú, ú, ê, êë 2 -1 5 úû, êë 0 1 2úû, (a) A-1 = B, , (b) A-1 = 6 B, , (c) B-1 = B, , (d) B-1 =, , 1, A, 6, , 26. The real function f (x) = 2x 3 - 3x 2 - 36x + 7 is, , Latest CBSE SAMPLE PAPER, , (a) Strictly increasing in ( -¥ , - 2 ) and strictly decreasing in ( -2 , ¥ ), (b) Strictly decreasing in ( -2 , 3), (c) Strictly decreasing in ( -¥ , 3) and strictly increasing in (3, ¥ ), (d) Strictly decreasing in ( -¥ , - 2 ) È (3, ¥ ), , æ 1 + cos x + 1 - cos x ö, 3p, ÷, p < x <, is, ÷, 2, è 1 + cos x - 1 - cos x ø, x, 3p x, (b), (c) 2, 2 2, , 27. Simplest form of tan -1 çç, (a), , p x, 4 2, , (d) p -, , x, 2, , 28. Given that A is a non-singular matrix of order 3 such that A 2 = 2A, then the value of|2A|, is, (a) 4, , (b) 8, , (c) 64, , (c) 16, , 29. The value of b for which the function f (x) = x + cos x + b is strictly decreasing over R is, (a) b < 1, (c) b £ 1, , (b) No value of b exists, (d) b ³ 1, , 30. Let R be the relation in the set N given byR = {(a , b): a = b - 2, b > 6} , then, (a) (2 , 4) Î R, (c) (6, 8) Î R, , (b) (3, 8) Î R, (d) (8, 7) Î R

Page 45 :
37, , CBSE Sample Paper Mathematics Class XII (Term I), , ì, 31. The points, at which the function f given by f (x) = ïí|x| x < 0 is continuous, is/are, x, , ïî -1 x ³ 0, (b) x = 0, (d) x = - 1 and 1, , (a) x Î R, (c) x Î R - {0}, , é 0 3a ù, é0 2 ù, and kA = ê, ú , then the value of k , a and b respectively are, ú, ë 2b 24û, ë 3 - 4û, (a) -6, - 12 , - 18, (b) -6, - 4, - 9, (c) -6, 4, 9, (d) -6, 12 , 18, , 32. If A = ê, , 33. A linear programming problem is as follows, Minimise Z = 30x + 50y, Subject to the constraints, 3x + 5y ³ 15, 2x + 3y £ 18, x ³ 0, y ³ 0, In the feasible region, the minimum value of Z occurs at, (a) a unique point, (b) no point, (c) infinitely many points, (d) two points only, , 34. The area of a trapezium is defined by function f and given by f (x) = (10 + x) 100 - x 2 ,, then the area when it is maximised is, (b) 7 3 cm 2, (a) 75 cm 2, , (c) 75 3 cm 2, , (d) 5 cm 2, , 35. If A is square matrix such that A 2 = A, then (I + A) 3 - 7 A is equal to, (b) I + A, , (a) A, , (c) I - A, , (d) I, , 36. If tan -1 x = y, then, (a) -1 < y < 1, , (b), , -p, p, £y£, 2, 2, , (c), , -p, p, <y<, 2, 2, , -p p ü, (d) y Î ìí, , ý, î 2 2þ, , 37. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B., Based on the given information f is best defined as, (a) Surjective function, (b) Injective function, (c) Bijective function, (d) None of these, 1ù, , then 14 A -1 is given by, 2úû, é 4 -2 ù, (b) ê, ú, ë2 6 û, , é2 -1ù, (c) 2 ê, ú, ë 1 -3 û, , é3 -1ù, (d) 2 ê, ú, ë1 2 û, , 39. The points on the curve y = x 3 - 11x + 5 at which the tangent is y = x - 11, is/are, (a) ( -2 , 19), (c) ( ±2 , 19), , (b) (2 , - 9), (d) ( -2 , 19) and (2 , - 9), , éa, ëg, , 40. Given that A = ê, , (a) 1 + a 2 + bg = 0, (c) 3 - a 2 - bg = 0, , bù, and A 2 = 3I then, -aúû, (b) 1 - a 2 - bg = 0, (d) 3 + a 2 + bg = 0, , Latest CBSE SAMPLE PAPER, , é3, ë -1, é2 -1ù, (a) 14ê, ú, ë1 3 û, , 38. For A = ê

Page 46 :
38, , CBSE Sample Paper Mathematics Class XII (Term I), , Section C, In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , 41. For an objective function Z = ax + by, where, a , b > 0; the corner points of the feasible, region determined by a set of constraints (linear inequalities) are (0, 20), (10, 10),, (30, 30) and (0, 40). The condition on a and b such that the maximum Z occurs at both, the points (30, 30) and (0, 40) is, (a) b - 3a = 0, (b) a = 3b, (c) a + 2 b = 0, (d) 2 a - b = 0, , 42. For which value of m is the line y = mx + 1 a tangent to the curve y 2 = 4 x?, (a), , 1, 2, , (b) 1, , (c) 2, , (d) 3, , 1, , 43. The maximum value of [x(x - 1) + 1] 3 , 0 £ x £ 1 is, (a) 0, , (b), , 1, 2, , (c) 1, , (d) 3, , 1, 3, , 44. In a linear programming problem, the constraints on the decision variables x and y are, x - 3y ³ 0, y ³ 0, 0 £ x £ 3. The feasible region, (a) is not in the first quadrant, (b) is bounded in the first quadrant, (c) is unbounded in the first quadrant, (d) does not exist, é 1, 45. Let A = ê - sin a, ê, êë -1, , sin a, 1, - sin a, , (a) |A|= 0, , 1 ù, sin aú , where 0 £ a £ 2p, then, ú, 1 úû, , (b) |A|Î (2 , ¥ ), , (c) |A|Î (2 , 4), , (d) |A|Î [2 , 4], , CASE STUDY, , Latest CBSE SAMPLE PAPER, , The fuel cost per hour for running a train is proportional to the square of the speed it generates, in km per hour. If the fuel costs ` 48 per hour at speed 16 km per hour and the fixed charges to, run the train amount to ` 1200 per hour., , Based on the given information, answer the following questions., , 46. Given that the fuel cost per hour is k times the square of the speed the train generates, in km/h the value of k is, 16, 1, (b), (a), 3, 3, , (c) 3, , (d), , 3, 16, , 47. If the train has travelled a distance of 500 km, then the total cost of running the train is, given by function, 15, 600000, (a), v+, 16, v, , (b), , 375, 600000, v+, 4, v, , (c), , 5 2 150000, v +, 16, v, , (d), , 3, 6000, v+, 16, v

Page 51 :
43, , CBSE Sample Paper Mathematics Class XII (Term I), , 30. Given, R = {( a, b) : a = b - 2, b> 6}, , 34. We have, f ( x ) = (10 + x ) 100 - x 2, , R = {(5, 7), (6, 8), (7, 9), (8, 10), (9, 11) …}, (6, 8) Î R, ìï x , x < 0, 31. We have, f ( x ) = í|x|, ïî - 1, x ³ 0, ìï x , x < 0 ì -1, x < 0, =í, = í -x, ïî -1, x ³ 0 î -1, x ³ 0, \, , Þ f ( x ) = -1, "x Î R, As we know that, constant function is, continuous, "x Î R., Therefore, f ( x ) is continuous "x Î R., é 0 2k ù, é0 2 ù, 32. We have, A = ê, ú Þ kA = ê3 k -4 k ú, 3, 4, û, ë, û, ë, , f ¢ (x) =, =, , 100 - x 2 - 10 x - x 2, 100 - x 2, 100 - 10 x - 2 x 2, 100 - x 2, , For maxima or minima put f ¢ ( x ) = 0., 100 - 10 x - 2 x 2, =0, \, 100 - x 2, , é 0 3aù, kA = ê, ú, ë2 b 24 û, \ 2 k = 3 a, 2 b = 3 k and - 4 k = 24, Þ k = - 6, a = - 4 and b = - 9, , Þ, 2 x 2 + 10 x - 100 = 0, [divide by 2], Þ, x 2 + 5 x - 50 = 0, Þ, ( x + 10 ) ( x - 5 ) = 0, Þ, x = - 10, 5, At x = - 10, f ( x ) = 0, Now, area of trapezium at x = 5 is, f ( 5 ) = (10 + 5 ) 100 - 25, = 15 75, = 15 ´ 5 3 = 75 3 cm 2, , Given,, , 33. Minimize Z = 30 x + 50 y, Subject to constraints, 3 x + 5 y ³ 15 , 2 x + 3 y £ 18, and, x ³ 0, y ³ 0, Graph of inequalities are, , 35. Given, A2 = A, , Y, , Now, ( I + A)3 - 7 A, = I 3 + A3 + 3 I 2 A + 3 IA2 - 7 A, = I + A × A2 + 3 IA + 3 IA - 7 A, [Q A2 = A and I 2 = I 3 = I ], = I + ( A × A) + 3 A + 3 A - 7 A, [Q IA = A], = I + A+3A+3A-7A, [Q A2 = A], = I +7A-7A= I, , C(0, 6), , (0, 3) D, B(9, 0), O, , A, (5, 0), , X, , 3x+5y=15 2x+3y=18, , Y′, , Corner points, , Z = 30x + 50y, , A (5, 0), , 30( 5) + 0 = 150, , B (9, 0), , 30( 9) + 0 = 270, , C (0, 6), , 0 + 50 ´ 6 = 300, , D (0, 3), , 0 + 50 ´ 3 = 150, , The minimum value of Z lies on line segment, AD., \Minimum value of Z occurs at infinitely, many points., , 36. Given, tan -1 x = y, p p, Range of tan -1 x is æç - , ö÷, è 2 2ø, p, p, - <y<, \, 2, 2, 37. We have, A = {1, 2, 3}, B = {4, 5, 6, 7}, Function f = {(1, 4), (2, 5), (3, 6)}, Clearly, f is injective function., Now, range of f = {4, 5, 6}, Codomain = {4, 5, 6, 7}, Here, Range ¹ Codomain, \ f is not surjective function., , Latest CBSE SAMPLE PAPER, , X′, , On differentiating w.r.t. x, we get, (10 + x ) (2 x ), f ¢ ( x ) = 100 - x 2 2 100 - x 2, (10 + x )x, f ¢ ( x ) = 100 - x 2 100 - x 2

Page 53 :
45, , CBSE Sample Paper Mathematics Class XII (Term I), , é 1, 45. We have, A = ê - sin a, ê, êë - 1, Þ, Þ, Q, Þ, Þ, \, , sin a, 1, - sin a, , 1 ù, sin a ú, ú, 1 úû, , |A| = 1(1 + sin 2 a ), - sin a ( - sin a + sin a ) + 1(sin 2 a + 1), |A| = 2 + 2 sin 2 a, 0 £ sin 2 a £ 1, 0 £ 2 sin 2 a £ 2, 0 + 2 £ 2 + 2 sin 2 a £ 2 + 2 Þ 2 £ |A| £ 4, |A|Î [2 , 4 ], , 46. Let the fuel cost per hour be x., Speed of train = v km/h, Given, x is proportional to square of speed, in km/h., \, x = kv 2, Given, x = 48 and v = 16, 48, 48, 3, \, k=, =, =, 2, 16, 16, 16, ´, (16 ), 47. Given, the fixed charges to run the train, amount to ` 1200 per hour., Cost of running the train per hour,, 3, c1 = 1200 + x Þ c1 = 1200 + v 2, 16, \Cost of running the train per kilometre., 1200 3, é Here, c = c1 ù, i.e. c =, +, v, 16, v úû, v, ëê, Cost of running the train 500 km, 600000 1500, +, v, Þ 500 c =, v, 16, , 600000 375, v, +, v, 4, 1200 3, 48. Here, c =, +, v, 16, v, On differentiating w.r.t. v, we get, =, , 1200 3, dc, =- 2 +, 16, dv, v, For maxima or minima, put, -1200, v, Þ, , 2, , +, , dc, =0, dv, , 3, =0, 16, 1200 ´ 16, v2 =, 3, , Þ, , v 2 = 6400, , Þ, , v = ± 80, , Þ, , v = - 80 (rejected), , \The most economical speed to run the train is, 80 km/h., 49. The fuel cost for the train to travel 500 km of, the most economical speed is, 3, 3, 500 æç ö÷v = 500 ´, ´ 80, è 16 ø, 16, = `7500, 50. The total cost of the train to travel 500 km at, the most economical speed is, 1200 3, +, ´ 80 ö÷ = 500(15 + 15 ), 500 æç, ø, è 80, 16, = ` 15000, , Latest CBSE SAMPLE PAPER

Page 57 :
49, , CBSE Sample Paper Mathematics Class XII (Term I), , SAMPLE PAPER 1, MATHEMATICS, A Highly Simulated Practice Questions Paper, for CBSE Class XII (Term I) Examination, , Instructions, 1. This question paper contains three sections - A, B and C. Each section is compulsory., 2. Section - A has 20 MCQs, attempt any 16 out of 20., 3. Section - B has 20 MCQs, attempt any 16 out of 20., 4. Section - C has 10 MCQs, attempt any 8 out of 10., 5. There is no negative marking., 6. All questions carry equal marks., Maximum Marks : 40, Time allowed : 90 min, , Roll No., , Section A, In this section, attempt any 16 questions out of Questions 1-20. Each question is of 1 mark weightage., , é1 - 1ù, 1. If matrix A given by A = ê 0 3 ú , then the order of the matrix A is, ê, ú, êë 2 5 úû, (a) 1 ´ 2, (b) 2 ´ 3, (c) 3 ´ 2, (d) 2 ´ 2, , 2. The minimum value of the function f (x) = x - 4 exists at, (a) x = 0, (c) x = 4, , (b) x = 2, (d) x = - 4, , 3. The domain of sec - 1 x is, (b) R, (d) None of these, , 4. The region represent by the inequation system x, y ³ 0, y £ 5 and x + y £ 2 is, (a) unbounded in Ist quadrant, (c) bounded in Ist quadrant, , 5. The value of, (a) - 1, , x, , -7, , x 5x + 1, , (b) unbounded in Ist and IInd quadrants, (d) None of these, , at x = - 1 is, , (b) - 3, , (c) 2, , (d) - 5, , SAMPLE PAPER 1, , (a) R - [ -1, 1], (c) [ - 1, 1]

Page 58 :
50, , CBSE Sample Paper Mathematics Class XII (Term I), , 6. The function f given by f (x) = 3x + 17, is, (a) strictly increasing on R, (c) decreasing on R, , (b) strictly decreasing on R, (d) Both (b) and (c) are correct, , 7. Maximise Z = 10x + 5y subject to constraints x £ 4, y £ 6 and x, y ³ 0., (a) 40 at (0, 0), , 8. The value of, (a) 0, , (b) 40 at ( 4, 0), , (c) 70 at ( 4, 6), , (d) 30 at (0, 6), , (c) 2, , (d) 3, , cos q sin q, is, - sin q cos q, (b) 1, , 9. A vertex of a feasible region by the linear constraints 2x + y £ 30, x + 2y £ 24, and x, y ³ 0 is, (a) (0, 10), , 10. If, , (b) (12 , 6), , (c) (0, 2 ), , (d) (14, 0), , (c) ± 6, , (d) 0, , (c) 64, , (d) None of these, , 2x 5, 6 -2, x, , then value of is, =, 8 x, 7 3, 6, , (a) - 5, , (b) ± 1, , 1 4 3, 11. The value of determinant 9 - 1 4 is, 5, (a) 21, , (b) 166, , 0, , 2, , 12. The feasible solution for a LPP is shown in following figure. Let Z = 2x - 3y be the, objective function. (Maximum Value of Z + Minimum Value of Z) is equal to, Y, , (5, 6), (8, 5), , (0, 5), (7, 4), , (0, 0), , X, , SAMPLE PAPER 1, , (6, 0), , (a) 0, , (b) - 2, , (c) - 1, , 13. The function f (x) = x 2 - 4 x, x Î[0, 4] attains minimum value at, (a) x = 0, (c) x = 2, , (b) x = 1, (d) x = 4, , 14. If A is 3 ´ 3 matrix such that A = 10 , then 3A equals, (a) 270, (c) 72, , (b) 240, (d) 216, , (d) - 3

Page 59 :
51, , CBSE Sample Paper Mathematics Class XII (Term I), , 15. The range of cosec - 1 x is, p p, (a) é - , ù, êë 2 2 úû, , p p, (b) ù - , é, úû 2 2 êë, , p p, (c) é - , ù - {0}, êë 2 2 úû, , (d) None of these, , 16. The domain in which sine function will be one-one, is, -p p ù, (a) é, ,, ëê 2 2 ûú, , p 3p, (b) é , ù, ëê 2 2 ûú, , (c) [0, p ], , (d) Both (a) and (b), , 17. If Radha has 15 notebooks and 6 pens, Fauzia has 10 notebooks and 2 pens and Simran, has 13 notebooks and 5 pens, then the above information is expressed as, é15 6ù, I. ê10 2ú, ê, ú, êë13 5úû, (a) Only by I, (c) Both I and II, , é15 10 13ù, II. ê, ú, ë 6 2 5û, (b) Only by II, (d) None of these, , 6, 3y - 2ù, é x + 3 z + 4 2y - 7ù é 0, ê, ú, ê, 18. If - 6 a - 1, 0, = -6, - 3 2c + 2ú , then the values of x , y, z , a , b and c are, ê, ú ê, ú, 0 úû êë 2b + 4 - 21, 0 úû, êë b - 3 - 21, (a) x = - 3, y = - 5, z = 2, a = - 2 , b = - 7 and c = - 1, (b) x = - 2 , y = - 7, z = - 1, a = - 3, b = - 5 and c = 2, (c) x = - 3, y = - 5, z = 2, a = 2 , b = 7 and c = 1, (d) x = 3, y = 5, z = 2 , a = 2 , b = 7 and c = 1, , 19. If A is a matrix defined by A = [a ij ] = [sin j x i ]; 1 £ i £ 3, 1 £ j £ 3, and B is a matrix defined by B = [ b ij ] = [cos i x j ]; 1 £ i £ 3, 1 £ j £ 3., a, Then, the value of 22 is, b 12, (a) 2 cos x 2, (c) 2 sin x 2, , (b) 2 sin 2 x, (d) None of these, , 5 3 8, 20. The value of D = 2 0 1 using cofactors of elements of second row is, 1 2 3, (a) - 14, , (c) - 7, , (b) 14, , (d) 7, , Section B, In this section, attempt any 16 questions out of Questions 21-40. Each question is of 1 mark weightage., , (a), , 4+x -2, , 1, 2, , x, , , x ¹ 0 be continuous at x = 0, then 4 f ( 0) is equal to, (b), , 1, 4, , (c) 1, , (d), , 3, 2, , 1 2 k, 22. If k 4 5 = 0, then the possible values of k are, 5 6 7, (a) -3 and 3, , (b) -3 and, , -8, 3, , (c) -3 and, , -1, 3, , (d) 3 and, , 8, 3, , SAMPLE PAPER 1, , 21. If f (x) =

Page 60 :
52, , CBSE Sample Paper Mathematics Class XII (Term I), , 23. The least value of f (x) = e x + e - x is, (a) -2, (c) 0, , (b) 2, (d) 1, , 24. The points on the curve x 2 + y 2 - 2x - 3 = 0 at which the tangents are parallel to the, X-axis, are, (a) (1, ± 2 ), (c) (2 , ± 2 ), , (b) (1, ± 3), (d) None of these, , 25. Which of the following statements is true for f (x) = 4 x 3 - 6x 2 - 72x + 30 ?, I. f is strictly increasing in the interval (- ¥, - 2)., II. f is strictly increasing in the interval (3, ¥)., III. f is strictly decreasing in the interval (- 2, 3)., IV. f is neither increasing nor decreasing in R., (a) I and Il are true, (b) II and III are true, (c) II and IV are true, (d) All are true, ln (1 + ax) - ln(1 - bx), , x ¹ 0. If f ( x) is continuous at x = 0, then f ( 0) - a - b is, 26. Let f (x) =, x, equal to, (a) 0, (b) 1, (c) 2, (d) 3, , 27. All the points of discontinuity of f defined by f (x) =|x| -| x + 1| is/are, (a) 0, 1, (c) no point of discontinuity, , (b) 1, 0, 2, (d) None of these, , 28. Secant function is bijective when its domain and range are ..A.. and R - ( -1, 1), respectively. Here, A refers to, (a) [0, p ], p, (c) [0, p ] - ìí üý, 2, î þ, , p, (b) é0, ù, êë 2 úû, (d) None of these, , 29. The domain of the function y = sin -1 (-x 2 ) is, (a) [0, 1], (c) [ -1, 1], , 30. Which of the following is true?, , SAMPLE PAPER 1, , (a) tan -1 1 = tan 1, (c) tan 1 < tan -1 1, , (b) [0,1], (d) f, (b) tan 1 > tan -1 1, (d) None of these, , 31. The function f (x) = 4 sin 3 x - 6 sin 2 x + 12 sin x + 100 is strictly, 3p, (a) increasing in æç p, ö÷, è 2ø, p p, (c) decreasing in é - , ù, êë 2 2 úû, , p, (b) decreasing in æç , p ö÷, è2 ø, p, (d) decreasing in é0, ù, êë 2 úû, , 32. The interval in which y = x 2 e - x is increasing, is, (a) ( - ¥ , ¥ ), (c) (2 , ¥ ), , (b) ( - 2 , 0), (d) (0, 2 )

Page 61 :
53, , CBSE Sample Paper Mathematics Class XII (Term I), , 33. Derivative of log [log (log x 5 )] w.r.t. x is, (a), (c), , 1, , (b), , 5, , x log x × log (log x ), 5, , 1, x log (log x 5 ), , (d) None of these, , x log (log x 5 ), , æ x2 ö, dy, ÷ , then, is equal to, 2÷, dx, è1 + x ø, 1, 2x, (b), (a), 2, x (1 + x ), 1 + x2, , 34. If y = log çç, , (c), , 2, 2, , x (1 + x ), , (d), , x, (1 + x 2 ), , 35. If area of triangle is 4 sq units with vertices (-2, 0), (0, 4) and (0, k), then k is equal to, (a) 0 and - 8, , (c) -8, , (b) 8, , (d) 0 and 8, , 36. The equation of the tangent to the curve y = 4 x + 5, which is parallel to the line, 2x - y + 3 = 0, is, (a) 2 x - y + 3 = 0, , (b) x - y + 3 = 0, , (c) 2 x + y + 3 = 0, , (d) x - 2 y + 3 = 0, , 37. The equations of the tangent and normal to the parabola y 2 = 4 ax at the point (at 2 , 2at), are respectively, (a) ty = x + at 2 and y = - tx + 2 at + at 3, (c) y = tx + 2 at + at 3 and ty = x + at 2, , (b) ty = x - at 2 and y = tx - 2 at + at 3, (d) y = - tx + 2 at + at 3 and ty = x + at 2, , 38. If x = a (cos q + q sin q) and y = a(sin q - q cos q), then, , dy, , is equal to, dx, (a) tan q, (b) cot q, (c) sin q, (d) cos q, 5ù é 3 - 4ù é 7 6ù, éx, is, 39. The value of y - x from the equation 2 ê, =, ú +ê, 2úû êë15 14úû, ë 7 y - 3û ë 1, (a) - 5, (b) 5, (c) - 7, (d) 7, , 40. If Z = 2x + 3y, subject to constraints x + 2y £ 10, 2x + y £ 14 and x , y ³ 0, then one of the, corner point of feasible region is, (a) (0, 7), (b) (6, 2), , (c) (5, 0), , (d) (8, 1), , Section C, In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , 41. The maximum value of the objective function, Z = 34 x + 45y, , 42. If y =, , ( x - 1)( x - 2), ( x - 3), , , then, , dy, dx, , is equal to, , é 1, 1, 1 ù, +, ú, ê, ë x - 1 x - 2 x - 3û, ( x - 1) ( x - 2 ) é 1, 1, 1 ù, (c), +, ú, ê, ( x - 3), ë x - 1 x - 2 x - 3û, , (a), , 1 ( x - 1) ( x - 2 ), 2, ( x - 3), , (d) 1290, , (b), , 1 ( x - 1) ( x - 2 ), 2, ( x - 3), , (d) None of these, , é 1, 1, 1 ù, +, ú, ê, ë x - 1 x - 2 x - 3û, , SAMPLE PAPER 1, , Subject to constraints x + y £ 300, 2x + 3y £ 70 and x , y ³ 0, is, (b) 1100, (c) 1150, , (a) 1190

Page 63 :
OMR SHEET, , SP 1, , Roll No., Sub Code., , Student Name, , Instructions, Use black or blue ball point pens and avoid Gel & Fountain pens for filling the OMR sheet., Darken the bubbles completely. Don’t put a tick mark or a cross mark, half-filled or over-filled bubbles will not be read, by the software., Correct, , ✔, , ✗, , Incorrect, , Incorrect, , Incorrect, , Do not write anything on the OMR Sheet., Multiple markings are invalid., , 1, , 18, , 35, , 2, , 19, , 36, , 3, , 20, , 37, , 4, , 21, , 38, , 5, , 22, , 39, , 6, , 23, , 40, , 7, , 24, , 41, , 8, , 25, , 42, , 9, , 26, , 43, , 10, , 27, , 44, , 11, , 28, , 45, , 12, , 29, , 46, , 13, , 30, , 47, , 14, , 31, , 48, , 15, , 32, , 49, , 16, , 33, , 50, , 17, , 34, , Check Your Performance, Total Questions:, Total Correct Questions:, , If Your Score is, , Score Percentage =, , Total Correct Questions, Total Questions, , × 100, , Less than 60%, > Average (Revise the concepts again), Greater than 60% but less than 75% > Good (Do more practice), Above 75%, > Excellent (Keep it on)

Page 64 :
56, , CBSE Sample Paper Mathematics Class XII (Term I), , Answers, 1. (c), 11. (a), 21. (c), , 2. (c), 12. (d), 22. (d), , 3. (d), 13. (c), 23. (b), , 4. (c), 14. (a), 24. (a), , 5. (b), 15. (c), 25. (d), , 6. (a), 16. (d), 26. (a), , 7. (c), 17. (c), 27. (c), , 8. (b), 18. (a), 28. (c), , 9. (b), 19. (c), 29. (c), , 10. (b), 20. (d), 30. (b), , 31. (b), 41. (a), , 32. (d), 42. (a), , 33. (a), 43. (b), , 34. (c), 44. (a), , 35. (d), 45. (d), , 36. (a), 46. (d), , 37. (a), 47. (c), , 38. (a), 48. (d), , 39. (d), 49. (a), , 40. (b), 50. (b), , SOLUTIONS, é 1 - 1ù, 1. Given matrix, A= ê 0 3 ú has 3 rows and, ê, ú, êë2 5 úû, 2 columns., \ Order of matrix A is 3 ´ 2 ., 2. Given function, f ( x ) = x - 4, , 5. Given,, , -7, , x, , = x( 5 x + 1) + 7( x ), , x 5x + 1, , = 5x2 + x + 7x = 5x2 + 8x, = x( 5 x + 8 ), , Now, at x = - 1,, Required result = ( - 1)( - 5 + 8 )= ( - 1) (3 ) = - 3, 6. Given, f ( x )= 3 x + 17, On differentiating w.r.t. x, we get, f ¢ ( x ) = 3 > 0, in every interval of R., Thus, the function is strictly increasing on R., , Graph of f ( x ),, Y, , |x – 4|, , 7. Given, objective function Z = 10 x + 5 y, and x £ 4, y £ 6 and x, y ³ 0., , X′, , X, , 4, , Let draw the inequations,, Y, B (4, 6), , C (0, 6), 6, Y′, , 5, , From graph, we observe that f ( x ) has, minimum value at x = 4., , 4, 3, , 3. The domain of sec- 1 x is, , Feasible region, , 2, , ( - ¥ , - 1] È [1, ¥ ) or R - ( - 1, 1), , 1, , 4. Given, inequation system,, x , y ³ 0, y £ 5 and x + y £ 2, , A (4, 0), , X′, O, (0, 0), , Y, , 1, , 2, , 3, , X, , 4, x=4, , y=5, , 5, , Y′, , 4, , Corner Points, , Value of Z = 10 x + 5 y, , 2, , O(0, 0 ), , Z =0 +0=0, , 1, , A(4 , 0 ), , Z = 40 + 0 = 40, , B(4 , 6 ), , Z = 40 +30 = 70, (Maximum), , C (0, 6 ), , Z = 0 + 30 = 30, , 3, , X, O, (0, 0), , 1, , 2, , 3, , 4, , 5, , x+, y=, 2, , SAMPLE PAPER 1, , y=6, , Hence, the inequation system gives bounded, region in Ist quadrant., , The maximum of Z is 70 at ( 4 ,6 ).

Page 65 :
57, , CBSE Sample Paper Mathematics Class XII (Term I), , 8. Given,, , cos q, , sin q, , - sin q, , cos q, , 13. Given function, f ( x ) = x 2 - 4 x, x Î[0 , 4 ], , = (cos q) (cos q) - (sin q) ( - sin q), = cos 2 q + sin 2q, =1, 9. Given, linear constraints,, 2 x + y £ 30, x + 2 y £ 24 and x, y ³ 0, Y, , On differentiating w.r.t. x, we get, f ¢ (x) = 2 x - 4, For maximum or minimum of f ( x ) , f ¢ ( x ) = 0, Þ, 2x - 4 = 0 Þ x = 2, Again, on differentiating w.r.t. x, we get, [minimum], f ¢¢( x ) = 2 > 0, Hence, f ( x ) is minimum at x = 2., 14. Given, A =10 and A is a 3 ´ 3 matrix., \, 3 A = (3 )3 A = 27 ´ 10 = 270, [Q if matrix A is a order of n, then |kA|= k n|A|], p p, 15. The range of cosec- 1x is é - , ù - {0 }, êë 2 2 úû, , 2x+y=30, (0, 12), (12, 6), , x+2y=24, (0, 0), , (15, 0), , X, , Hence, (12 , 6 ) is one of the vertex of a feasible, region., 2x 5, 6 -2, 10. Given,, =, 8 x, 7 3, 2 x 2- 40 = 18 + 14, 2 x 2 = 72, x 2 = 36 Þ x = ± 6, x, Now,, = ±1, 6, 1 4 3, 11. We have, 9 - 1 4, Þ, Þ, Þ, , 5, , 0, , 16. Since, the domain of sine function is the set of, all real numbers and range is the closed, interval [-1, 1]. If we restrict its domain to, é - p , p ù, then it becomes one-one and onto, ëê 2 2 ûú, with range [-1, 1]., Actually, sine function restricted to any of the, p, -3 p, - p p ù é p 3p ù, intervals é, etc., is, , ,, , - ù, é, ,, ú, êë 2, ê, 2 û ë 2 2 úû êë 2 2 úû, one-one and its range is [-1, 1]., 17. If Radha has 15 notebooks and 6 pens, Fauzia, has 10 notebooks and 2 pens and Simran has, 13 notebooks and 5 pens, then this could be, arranged in tabular form as, , 2, , Now expanding along R3, we get, 5(16 + 3 ) + 2 ( - 1 - 36 ) = 95 - 74 = 21, 12. Given objective function, Z = 2 x - 3 y, , Name, , Notebooks Pens, , Radha, , 15, , 6, , Fauzia, , 10, , 2, , Simran, , 13, , 5, , Value of Z = 2 x - 3 y, , (0, 0 ), , Z =0 -0=0, , (6, 0 ), , Z = 12 - 0 = 12, (Maximum), , é15 6 ù, This can be expressed as ê10 2 ú., ê, ú, êë13 5 úû, , (7, 4 ), , Z = 14 - 12 = 2, , The above information can also be arranged in, tabular form as, , (8, 5 ), , Z = 16 - 15 = 1, , ( 5, 6 ), , Z = 10 - 18 = - 8, , (0, 5 ), , Z = 0 - 15 = - 15, (Minimum), , Maximum Z + Minimum Z = 12 + ( - 15 ), = 12 - 15 = - 3, , Name, , Radha, , Fauzia, , Simran, , Notebooks, , 15, , 10, , 13, , Pens, , 6, , 2, , 5, , é15 10 13 ù, This can be expressed as ê, ú., ë6 2 5û, , SAMPLE PAPER 1, , Corner Points

Page 66 :
58, , CBSE Sample Paper Mathematics Class XII (Term I), , 18. As the given matrices are equal, therefore their, corresponding elements must be equal., Comparing the corresponding elements,, we get, x + 3 = 0,, z + 4 = 6,, 2 y - 7 = 3y - 2,, a - 1 = - 3,, 0 = 2 c + 2,, b - 3 = 2 b + 4,, On simplifying, we get, a = - 2 , b = - 7 , c = - 1, x = - 3 , y = - 5 and z = 2, 19. The given matrix is, A = [ aij ] = [sin jx i ] ; 1 £ i , j £ 3, \, a22 = sin 2 x 2, Also, given that matrix, B = [ bij ] = [cos i x j ] ; 1 £ i , j £ 3, \, b12 = cos x 2, sin 2 x 2 2 sin x 2 cos x 2, a, Now, consider 22 =, =, b12, cos x 2, cos x 2, = 2 sin x 2, 5 3 8, 20. Given, D = 2 0 1, 1 2 3, Cofactors of the elements of second row are, 3 8, A21 = ( - 1)2 + 1, = - (9 - 16 ) = 7, 2 3, A22 = ( - 1)2 + 2, , SAMPLE PAPER 1, , and, , A23 = ( - 1)2 + 3, , [Q Aij = ( - 1)i + j Mij], 5 8, = 15 - 8 = 7, 1 3, 5 3, 1 2, , = - (10 - 3 ) = - 7, Now, expansion of D using cofactors of, elements of second row is given by, D = a21 A21 + a22 A22 + a23 A23, = 2 ´ 7 + 0 ´ 7 + 1 (- 7), = 14 - 7 = 7, 4+ x -2, 21. Given, f ( x )=, x, Q f ( x ) is continuous at x = 0., \ lim f ( x ) = f (0 ), x®0, 4+ x -2, Þ, f (0 ) = lim f ( x ) = lim, x®0, x®0, x, æ 4+ x-2, 4 + x +2ö, = lim ç, ´, ÷, x ® 0è, x, 4 + x +2ø, , 4+x-4, x, = lim, x( 4 + x + 2 ) x ® 0 x( 4 + x + 2 ), 1, 1, 1, = lim, =, =, x®0 4 + x +2, 2 +2 4, = lim, , x®0, , \ 4 f (0 ) = 1, 1 2 k, 22. Given, k 4 5 = 0, 5 6 7, Expanding along R1, we get, 1(28 - 30 ) - 2 (7 k - 25 ) + k (6 k - 20 ) = 0, Þ, -2 - 14 k + 50 + 6 k 2 - 20 k = 0, Þ, 6 k 2 - 34 k + 48 = 0, Þ, 3 k 2 - 17 k + 24 = 0, [divide by 2], Þ, 3 k 2 - 9 k - 8 k + 24 = 0, Þ, 3k (k - 3) - 8(k - 3) = 0, Þ, (3 k - 8 )( k - 3 ) = 0, 8, Þ, k = 3,, 3, 23. Given, f ( x ) = ex + e- x, On differentiating w.r.t. x, we get, f ¢( x ) = ex - e- x, For least value of f ( x ), put f ¢( x ) = 0, \, ex - e- x = 0 Þ ex = e- x, Þ, e2 x = 1 Þ 2 x = 0 Þ x = 0, Now, f ¢¢( x ) = ex + e- x, At x = 0, f ¢¢(0 ) = e0 + e-0 = 1 + 1 = 2 > 0, \ f ( x ) is least (or minima) at x = 0., So, the least value of f ( x ) at x = 0 is, f (0 ) = e0 + e-0 = 1 + 1 = 2, 24. The equation of the given curve is, ...(i), x2 + y2 - 2 x - 3 = 0, On differentiating w.r.t. x, we get, dy, dy, 2x + 2y, - 2 = 0 Þ 2y, = 2 - 2x, dx, dx, dy 2 (1 - x ) 1 - x, =, =, Þ, 2y, dx, y, dy, For tangent parallel to X-axis, we have, =0, dx, 1- x, = 0 Þ1 - x = 0 Þ x = 1, Þ, y, On substituting x = 1 in Eq.(i), we get, 12 + y 2 - 2 ´ 1 - 3 = 0, Þ, y2 - 4 = 0 Þ y = ± 2, Hence, the point at which the tangents are, parallel to the X-axis are (1, 2 ) and (1, - 2 ).

Page 67 :
59, , CBSE Sample Paper Mathematics Class XII (Term I), , 25. We have, f ( x ) = 4 x 3 - 6 x 2 - 72 x + 30, On differentiating w.r.t. x, we get, f ¢ ( x ) = 12 x 2 - 12 x - 72, = 12 ( x 2 - x - 6 ) = 12 ( x - 3 ) ( x + 2 ), Therefore, f ¢( x ) = 0 gives x = - 2, 3., The points x = - 2 and x = 3 divides the real, line into three disjoint intervals, namely,, (- ¥, – 2), (– 2, 3) and (3, ¥)., –, , +, , –∞, , –2, , +, , +∞, , 29. Given, y = sin -1( - x 2 ), , 3, , In the intervals (- ¥, – 2) and (3, ¥), f ¢( x ) is, positive while in the interval (- 2, 3), f ¢( x ) is, negative. Consequently, the function f is, strictly increasing in the intervals (- ¥, - 2), and (3, ¥) while the function is strictly, decreasing in the interval (- 2, 3). However, f is, neither increasing nor decreasing in R., Interval, , Sign of, f ¢ (x ), , Nature of, function f, , ( - ¥, - 2 ), , (-)(-) > 0, , f is strictly, increasing, , ( - 2, 3 ), , (-) (+) < 0, , f is strictly, decreasing, , ( 3, ¥ ), , (+) (+) > 0, , 28. If we restrict the domain of secant function to, p, [0 , p ] - ìí üý, then it is one-one and onto with its, î2 þ, range as the set R - ( -1, 1)., Actually, secant function restricted to any of, -p, p, the intervals [ - p , 0 ] - ìí üý, [0 , p ] - ìí üý and, î 2 þ, î2 þ, 3p ü, ì, [ p , 2 p ] - í ý etc., is bijective and its range is, î2 þ, R - ( -1, 1)., Þ sin y = - x 2, We know that, - 1 £ sin y £ 1, Þ, - 1 £ - x2 £ 1, Þ, -1 £ x 2 £ 1 Þ 0 £ x 2 £ 1, Þ, |x| £ 1 Þ - 1 £ x £ 1, 30. From figure, we note that tan x is an increasing, p p, p, function in the interval æç - , ö÷, since 1 > ., è 2 2ø, 4, This gives, p, tan1 > 1 Þ tan1 > 1 >, 4, Þ, tan 1 > 1 > tan -1(1), Y, , tan x, , f is strictly, increasing, , 26. Given, f ( x ) is continuous at x = 0., \ f (0 ) = lim f ( x ), , X′, , O π/4 π/2, , –π/2, , X, , x®0, , ln(1 + ax ) - ln(1 - bx ), x, ln(1 + ax ), ln(1 - bx ), = lim, - lim, x®0, x, ®, 0, x, x, ln(1 ± x ), é, ù, = a - ( - b), = ±1ú, êQ xlim, ®, 0, x, ë, û, =a+b, f (0 ) - a - b = 0, = lim, , Y′, , x®0, , Þ, , 27. Let g( x ) = |x|and h( x ) = |x + 1 |, , f ( x ) = 4 sin 3 x - 6 sin 2 x + 12 sin x + 100, , On differentiating w.r.t. x, we get, f ¢ ( x ) = 12 sin 2 x × cos x - 12 sin x × cos x, + 12 cos x + 0, = 12 cos x(sin 2 x - sin x + 1), Since, in IInd quadrant sin x is +ve and the, cos x is -ve., p, So, f ¢ ( x ) < 0 for all x Î æç , p ö÷., è2 ø, 32. Given,, , y = x 2 e- x, , On differentiating w.r.t. x, we get, dy, = x 2 e- x ( -1) + e- x (2 x ), dx, = x e- x ( - x + 2 ) = x (2 - x ) e- x, dy, For increasing function,, >0, dx, Þ, xe- x (2 - x ) > 0, , SAMPLE PAPER 1, , Now, g( x ) = |x|is the absolute value function,, so it is a continuous function for all x Î R., h( x ) = |x + 1 |is the absolute value function, so, it is a continuous function for all x Î R., Since, g( x ) and h( x ) are both continuous, functions for all x Î R, so difference of two, continuous function is a continuous function, for all x Î R., Thus, f ( x ) = |x| - |x + 1 |is a continuous, function at all points., Hence, there is no point at which f ( x ) is, discontinuous., , 31. Q

Page 68 :
60, , CBSE Sample Paper Mathematics Class XII (Term I), , Case I, Þ, x > 0 and 2 - x > 0, Þ, x > 0 and x < 2, Þ, 0< x<2, Case II, Þ, x < 0 and 2 - x < 0, Þ, x < 0 and x > 2, Hence, there is no value of x exist., Clearly, it is increasing in (0, 2). So, correct, answer is (d)., 5, , 33. Let y = log[log (log x )], On differentiating w.r.t. x, we get, dy, 1, d, =, × [log(log x 5 )], dx log(log x 5 ) dx, 1, , ×, , 1, , 5, , ×, , 1, , 5, , d, (log x 5 ), log(log x ) log x dx, , =, , 1, , =, , 5, , 5, , ×, , 1, , log(log x ) (log x ) x, , 5, , ×, , d 5, (x ), dx, , 5x4, , =, , x 5 log x 5 log(log x 5 ), 5, , =, , x × 5 log x × log(log x 5 ), 1, , =, , x log x × log(log x 5 ), , æ x2 ö, 34. y = log ç, ÷, 2, è1 + x ø, On differentiating w.r.t. x, we get, dy, =, dx, , d æ x ö, ç, ÷, dx è 1 + x 2 ø, 2, , 1, x2, 1 + x2, , dy 1 + x 2 é (1 + x 2 ) (2 x ) - ( x 2 ) (2 x ) ù, =, ´ê, ú, dx, x2, (1 + x 2 )2, ë, û, , SAMPLE PAPER 1, , =, =, , 1 + x2, x, , 2, , 1 + x2, x, , 1, 35. Given,, 2, , 2, , -2, , ´, ´, , 2 x(1 + x 2 - x 2 ), (1 + x 2 )2, 2x, 2 2, , (1 + x ), , =, , 2, x (1 + x 2 ), , 0 1, , 0 4 1 = ±4, 0 k 1, , Þ, , - 2 ( 4 - k ) + 1(0 - 0 ) = ± 8, , Þ, , - 2 ( 4 - k ) + 1(0 - 0 ) = ± 8, , Þ, , (- 8 + 2 k ) = ± 8, , Taking positive sign,, 2 k - 8 = 8 Þ 2 k = 16 Þ k = 8, Taking negative sign,, 2k - 8 = - 8 Þ 2k = 0 Þ k = 0, \, k = 0, 8, 36. The equation of the given curve is, y = 4x + 5, The slope of the tangent at any point ( x , y ) is, given by, 1, , -1, dy 1, 2, = ( 4x + 5)2 × 4 =, 4x + 5, dx 2, The equation of the given line is, 2 x - y + 3 = 0 Þ y = 2 x + 3,, which is of the form y = mx + c., \ Slope of this line is 2., Now, tangent to the curve is parallel to the line, 2 x - y + 3 = 0 i.e. the slope of the tangent is, equal to the slope of the line., 2, = 2 Þ 4x + 5 = 1, 4x + 5, , Þ, 4 x + 5 = 1 Þ x = -1, Put x = -1 in y = 4 x + 5, we get, y = -4 + 5 = 1 = 1, \ Equation of the tangent passing through the, point ( -1, 1) having slope 2 is given by, y - 1 = 2 ( x - ( -1)), Þ, y -1 = 2x +2 Þ 2x - y +3 = 0, Hence, the equation of the required tangent is, 2 x - y + 3 = 0., 37. The equation of the given parabola is, y 2 = 4 ax, On differentiating w.r.t. x, we get, dy 2 a, dy, =, = 4a Þ, 2y, dx, dx, y, , …(i), , \ Slope of the tangent at ( at 2 , 2 at ) is, 2a 1, æ dy ö, =, =, ç ÷, è dx ø( at 2 , 2at ) 2 at t, Hence, the equation of the tangent at ( at 2 , 2 at ), is, 1, y - 2 at = ( x - at 2 ), t, Þ, yt - 2 at 2 = x - at 2 Þ x - ty + at 2 = 0, Also, slope of the normal at ( at 2 , 2 at ), -1, =-t, =, Slope of tangent at ( at 2 , 2 at ), \ The equation of the normal at ( at 2 , 2 at ) is, y - 2at = - t( x - at 2 ), Þ t x + y - 2 at - at 3 = 0

Page 69 :
61, , CBSE Sample Paper Mathematics Class XII (Term I), , 38. Given, x = a (cos q + q sin q),, y = a(sin q - q cos q), On differentiating w.r.t. q, we get, d, dx, =a, (cos q + q sin q ), dq, dq, d, d, (cos q ) +, (q sin q )üý, = a ìí, dq, þ, î dq, = a{- sin q + (q cos q + sin q × 1)} = aq cos q, d, [using product rule in, (q sin q )], dq, dy, d, and, =a, (sin q - q cos q), dq, dq, d, d, (sin q ) (q cos q )üý, = a ìí, dq, þ, î dq, , \, , = a[cos q - {q ( - sin q) + cos q × 1}], = aq sin q, d, [using product rule in, (q cos q )], dq, dy, dy dq, aq sin q, =, =, = tan q, dx dx aq cos q, dq, , 39. The given equation is, 5 ù é3 -4 ù é 7 6 ù, éx, 2ê, +, =, 7, y, - 3 úû êë 1 2 úû êë15 14 úû, ë, 10 ù é3 -4 ù é 7 6 ù, é2 x, Þ ê, +, =, 14, 2, y, - 6 úû êë 1 2 úû êë15 14 úû, ë, 6 ù é7 6ù, é2 x + 3, Þ, =, ê 15, 2 y - 4 úû êë15 14 úû, ë, On equating the corresponding elements, we, have, 2 x + 3 = 7 and 2 y - 4 = 14, Þ, 2 x = 4 and 2 y = 18, Þ, x = 2 and y = 9, \, y- x=9-2 =7, 40. Given, maximise Z = 2 x + 3 y, , x, , 0, , 4, , 10, , y, , 5, , 3, , 0, , So, the line is passing through the points (0 , 5 ),, ( 4 , 3 ) and (10 , 0 )., , x, , 4, , 6, , 7, , y, , 6, , 2, , 0, , So, the line is passing through the points ( 4 , 6 ),, (6 , 2 ) and (7 , 0 )., On putting (0 , 0 ) in the inequality 2 x + y £ 14,, we get 0 + 0 £ 14, which is true., So, the half plane is towards the origin., The intersection point of lines corresponding to, Eqs. (i) and (ii) is B(6 , 2 )., On shading the common region, we get the, feasible region OABD., x+, , 2y, , Y, , x=0, 0, 2x + y =14, D (0, 5), 5, (4, 6), 4, C(4, 3), 3, B(6, 2), 2, 1, , =1, , X′, (0, 0)O, Y′, , (10, 0), , 1 2 3 4 5 6 7 8 9 10, A(7, 0), , X, , The corner points are, O(0 , 0 ), A(7 , 0 ), B(6 , 2 ) and D(0 , 5 ), 41. We have the following LPP, Maximise Z = 34 x + 45 y, Subject to the constraints, x + y £300, 2 x + 3 y £ 70 and x, y ³0, Now, considering the inequations as, equations, we get, x + y =300, and, 2 x + 3 y = 70, Table for line x + y =300 is, x, , 0, , 300, , y, , 300, , 0, , …(i), …(ii), , So, the line passes through the points (0 ,300 ), and (300 ,0 )., On putting (0 ,0 ) in the inequality x + y £ 300, we, get, 0 + 0 £ 300, which is true., So, the half plane is towards the origin., Table for line 2 x + 3 y = 70 is, x, , 35, , 0, , y, , 0, , 70/3, , SAMPLE PAPER 1, , Subject to constraints, ...(i), x + 2 y £ 10, ...(ii), 2 x + y £ 14, and, ...(iii), x, y ³ 0, Shade the region to the right of Y-axis to show, x ³ 0 and above X-axis to show y ³ 0., Table for line x + 2 y = 10 is, , On putting (0 , 0 ) in the inequality x + 2 y £ 10,, we get 0 + 0 £ 10, which is true., So, the half plane is towards the origin., Table for line 2 x + y = 14 is

Page 72 :
64, , CBSE Sample Paper Mathematics Class XII (Term I), , SAMPLE PAPER 2, MATHEMATICS, A Highly Simulated Practice Questions Paper, for CBSE Class XII (Term I) Examination, , Instructions, 1. This question paper contains three sections - A, B and C. Each section is compulsory., 2. Section - A has 20 MCQs, attempt any 16 out of 20., 3. Section - B has 20 MCQs, attempt any 16 out of 20., 4. Section - C has 10 MCQs, attempt any 8 out of 10., 5. There is no negative marking., 6. All questions carry equal marks., Maximum Marks : 40, Time allowed : 90 min, , Roll No., , Section A, In this section, attempt any 16 questions out of Questions 1-20. Each question is of 1 mark weightage., , 1. If D =, , 1, log b a, , then D is equal to, log a b, 1, , (a) 1, , (b) -1, , (c) 0, , (d) 2, , 2. If A = [a ij ] 2 ´ 2 , where a ij = e 2 ix sin j x. Then, A is equal to, é e 2x, (a) ê 4x, êë e, é e 2x, (c) ê 4x, êë e, , sin x e 2x sin 2 xù, ú, sin x e 4x sin x úû, , é e x sin x e 2x sin xù, (b) ê 2x, ú, x, êë e sin x e sin x ûú, , sin x e 2x sin 2 xù, ú, sin x e 4x sin 2 x úû, , (d) None of these, , SAMPLE PAPER 2, , 3. If A and B are square matrices of the same order 3, such that A = 3 and AB = 3I, then, the value of B is equal to, (a) 2, (b) 3, (c) 9, (d) 1, sin( 8x), 4. If a function f (x) =, , x ¹ 0 is continuous at x = 0, then f ( 0) is equal to, x, (a) 4, (b) 8, (c) 10, (d) 5, , 5. The function given by f (x) = x 3 - 3x 2 + 3x - 100 is, (a) increasing on R, (c) strictly decreasing on R, , (b) decreasing on R, (d) None of these

Page 73 :
65, , CBSE Sample Paper Mathematics Class XII (Term I), , 6. The point at which the tangent to the curve y = 2x 2 - x + 1 is parallel to y = 3x + 9 will be, (a) (2, 1), , (b) (1, 2), 2, , (c) (3, 9), , (d) (- 2, 1), , 2, , 7. If A = [a ij ] 2 ´ 2 , where a ij = i + 2 j , then A is equal to, é3 9 ù, (a) ê, ú, ë6 12 û, , é3 6 ù, (b) ê, ú, ë9 12 û, , é6 12 ù, (c) ê, ú, ë3 9 û, , é9 3ù, (d) ê, ú, ë6 1û, , 8. Feasible region (shaded) for a LPP is shown in following figure., B (2, 3), , C (0, 1), O (0, 0), , Maximum of Z = 10x - 2y is, (a) 60, (b) 40, , A (6, 0), , (c) 80, , (d) 70, , 9. The function f : N ® N, N being the set of natural numbers, defined by f (x) = 2x + 3 is, (a) injective and surjective, (c) not injective but surjective, , (b) injective but not surjective, (d) neither injective nor surjective, , 10. If A is a square matrix satisfying A¢ A = I, then the value of A, (a) - 1, , (b) 1, , (c) 2, , 2, , is, (d) - 2, , 11. If the domain and range of cosine function are [0, p] and [-1, 1] respectively, then, it is, (a) one-one, (c) both one-one and onto, , (b) onto, (d) None of these, , 12. Find the values of a , b , c and d from the equation, é a - b 2a + cù é - 1 5ù, ê 2a - b 3c + dú = ê 0 13ú, û, û ë, ë, (a) a = 1, b = 3, c = 4 and d = 2, (b) a = 2 , b = 1, c = 4 and d = 3, (c) a = 1, b = 2 , c = 3 and d = 4, (d) a = 1, b = 3 , c = 2 and d = 4, , 13. Suppose there is a relation R between the positive numbers x and y given by xRy if and, , (a) - 3, , (b) 0, , (c) 3, , (d) - 1, , SAMPLE PAPER 2, , only if x £ y 2 . Then, which one of the following is correct?, (a) R is reflexive but not symmetric, (b) R is symmetric but not reflexive, (c) R is neither reflexive nor symmetric, (d) None of the above, ì kx , if x <0, is continuous, 14. The value of the constant ‘k’ so that the function f (x) = ïí x, ïî 3,, if x ³ 0, ï, at x = 0 is

Page 74 :
66, , CBSE Sample Paper Mathematics Class XII (Term I), , ìx 2 - 1, 15. If the function f (x) = ïí x - 1 , when x ¹ 1 is given to be continuous at x = 1, then the, ï 2k ,, when x = 1, îï, value of k is equal to, (a) 1, (b) 2, (c) 0, (d) - 1, , 16. If C is a matrix having 2 rows and 3 columns, then number of elements in matrix C is, (a) 6, , (b) 3, , é 4 0 0ù, 17. If A = ê 0 4 0ú , then|adj A| is equal to, ú, ê, êë 0 0 4ûú, (a) 4 3, (b) 4 6, , (c) 2, , (d) 5, , (c) 4 2, , (d) 4 5, , 18. If matrix A = [a ij ] 3 ´ 1 , where a ij = i 3 + j 3 , then A is equal to, é2 ù, ê ú, (a) ê 9 ú, êë28úû, , (b) [2 9 8], , é28ù, ê ú, (c) ê 1 ú, êë 2 úû, , (d) None of these, , 19. If R and R¢ are symmetric relations (not disjoint) on a set A, then the relation R Ç R ¢ is, (a) not symmetric, , (b) symmetric, , (c) cannot determine, , (d) None of these, , 20. Let R be a relation on the set N of natural numbers defined by ‘nRm Û n is a factor of, m’. Then, which one of the following is correct?, (a) R is reflexive, symmetric but not transitive, (b) R is transitive, symmetric but not reflexive, (c) R is reflexive, transitive but not symmetric, (d) R is an equivalence relation, , Section B, In this section, attempt any 16 questions out of Questions 21-40. Each question is of 1 mark weightage., , 21. The value of a for which the following is continuous at x = 3:, ì( x + 3) 2 - 36, ï, , x¹3, f ( x) = í, x-3, ïî, x=3, a, ï, (b) 12, , SAMPLE PAPER 2, , (a) 9, , (c) 3, , (d) 15, , 22. The function f : R ® R is defined by f (x) = 3 - x, I. f is one-one function., III. f is a decreasing function., , II. f is onto function., , Which of the above statement(s) is/are correct?, (a) I and II, (b) II and III, (c) I and III, (d) All of these, , 23. The curve y = xe x has minimum value equal to, (a) -, , 1, e, , (b), , 1, e, , (c) - e, , (d) e

Page 75 :
67, , CBSE Sample Paper Mathematics Class XII (Term I), , 24. The maximum value of Z = 11x + 7 y, Subject to the constraints 2x + y £ 6, x £ 2 and x ³ 0, y ³ 0 is, (a) 36, (b) 38, (c) 40, , (d) 42, , 25. A trust fund has ` 30000 that must be invested in two different types of bonds. The first, bond pays 5% interest per year, and the second bond pays 7% interest per year. If the, trust must obtain an annual total interest of ` 2000, then using matrix multiplication,, the amounts invested in two types of bonds are respectively, (a) ` 5000 and ` 25000, (b) ` 25000 and ` 5000, (c) ` 500 and ` 25000, (d) ` 5000 and ` 2500, , 26. If Z = 2x + 3y subject to the constraints x + y £ 1, x ³ 0 and y ³ 0, then maximum of Z is, (a) 1, , (b) 2, , (c) 3, , (d) 4, , 27. Minimum of Z = 10x - 3y subject to the constraints x + y £ 7, 2x - 3y + 6 ³ 0 and x ³ 0, y ³ 0, is, (a) -5, , (b) -6, , (d) -8, , (c) 18, , 28. Maximum value of Z = 5x + 2y, if the feasible region (shaded) for a LPP is shown in, following figure., , Y, B, , x+, D, , 2 y=, , 76, E (44, 16), , O, , X, , A C, (52, 0), , 2x + y=104, , (a) 260, , é2, 29. If A = ê 2, ê, êë 1, é0 1, ê, (a) ê 1 0, êë0 1, , (b) 252, , (c) 280, , 0 1ù, 1 3ú , then A 2 - 5A + 6I is equal to, ú, - 1 0úû, 0ù, é0 0 0ù, ú, ê, ú, (c), (b) ê0 0 0ú, 0ú, ú, ê, ú, 0û, ë0 0 0û, , é 1 - 1 - 3ù, ú, ê, ê - 1 - 1 - 10ú, êë - 5, 4, 4úû, , (d) 290, , é1 0 0ù, ú, ê, (d) ê 1 0 1ú, êë0 1 0úû, , p, I. strictly increasing in é 0, ù, êë 2 úû, , p, II. strictly decreasing in æç , p ö÷, è2 ø, , III. neither increasing nor decreasing in [ 0, p ], (a) I and Il are true, (b) II and III are true, (c) Only II is true, (d) Only III is true, , SAMPLE PAPER 2, , 30. The function f (x) = sin x is

Page 76 :
68, , CBSE Sample Paper Mathematics Class XII (Term I), , 31. The function f (x) = tan - 1 (sin x + cos x) is an increasing function in, æp pö, (a) ç , ÷, è4 2ø, , æ p pö, (b) ç - , ÷, è 2 4ø, æ p pö, (d) ç - , ÷, è 2 2ø, , æ pö, (c) ç0, ÷, è 2ø, , 32. The distance between the origin and the normal to the curve y = e 2 x + x 2 at x = 0 is, 3, units, 2, 2, (d), units, 5, , (a) 2 units, (c), , (b), , 5, units, 2, , 33. If y = log 2 [log 2 (x)], then, (a), (c), , 34. If, , dy, dx, , is equal to, , log 2 e, , (b), , log e x, log 2 x, , log 2 e, x log x 2, log 2 e, , (d), , log e 2, , x log e x, , y 3, 2 4, 2x 4, 2 3, and, , then the value of x + y is, =, =, 5 1, 6 x, 4 5, 2y 5, , (a), (c), , 3 -2, 3 +2, , (b) 2 3, (d) None of these, , 35. Let f (x) = e x , g (x) = sin - 1 x and h (x) = f [g (x)], then, (a) esin, , -1, , x, , (d), , h ( x), , is equal to, , 1, , (b), , (c) sin -1 x, , h' ( x), , 1 - x2, 1, 1 - x2, , 45 2, -3 4, x - 8x 3 x + 105, then which of the following holds?, 2, 4, I. f ( x) has local maxima at x = 2., II. f ( x) has local maxima at x = 5., III. f ( x) has local minima at x = - 3., (a) Only I is true, (b) Only III is true, (c) Both I and II are true, (d) All I, II and III are true, , SAMPLE PAPER 2, , 36. If f (x) =, , p dy, is equal to, ,, 2 dx, (b) -1, (c) 5, , 37. If y = cos(sin x 2 ), then at x =, (a) 2, , 2 -3 5, 38. If D = 6 0 4 , then, 1, (a) 1, , 5, , 7, , M 21, M 32 - 1, (b) 2, , (d) 0, , is equal to, (c) 3, , (d) 4

Page 77 :
69, , CBSE Sample Paper Mathematics Class XII (Term I), , 39. If the graphical representation of an LPP is shown below, x+, , 2y, , Y, , x=0, 0, 2x + y =14, D (0, 5), 5, (4, 6), 4, C(4, 3), 3, B(6, 2), 2, 1, , =1, , X′, (0, 0)O, Y′, , (10, 0), , 1 2 3 4 5 6 7 8 9 10, A(7, 0), , X, , Then, sum of values of Z = 2x + 3y occurs at all the corner points is, (a) 32, (b) 33, (c) 29, (d) 47, é 1 4 4ù, 40. If the adjoint of a 3 ´ 3 non-singular matrix P is ê 2 1 7 ú , then the possible values of, ê, ú, êë1 1 3úû, the determinant of P are, (a) ± 2, (b) ± 1, (c) ± 3, (d) ± 4, , Section C, In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , ìæ 1 - cos 4 x ö, ÷ , x ¹ 0 is continuous at x = 0, then the value of k is, 8x 2 ø, ïî, k,, x=0, ï, (b) -1, (c) 0, (d) 1, dy, p, at x = is, (tan x), then, dx, 4, -4, 1, (c), (b), (d) None of these, log 2, log 2, , 41. If the function f (x) = ïíçè, (a) 2, , 42. If y = logsin x, (a), , 4, log 2, , 43. If f (x) = log x (log e x ), then f ' (x) at x = e is equal to, (a) 1, , (b) 2, , (c) 0, , (d), , 1, e, , 44. Equation of tangent at the curve y = be - x / a , where it crosses the Y-axis is, x y, - =1, a b, 2, x2 y, (d) 2 - 2 = 1, a, b, , (b), , 45. The points at which the tangent passes through the origin for the curve y = 4 x 3 - 2x 5, are, (a) (0, 0), (2 , 1) and ( - 1, - 2 ), (c) (2 , 0), (2 , 1) and ( - 3 , 1), , (b) (0, 0), (2 , 1) and ( - 2 , - 1), (d) (0, 0), (1, 2 ) and ( - 1, - 2 ), , SAMPLE PAPER 2, , x y, + =1, a b, 2, x2 y, (c) 2 + 2 = 1, a, b, , (a)

Page 79 :
OMR SHEET, , SP 2, , Roll No., Sub Code., , Student Name, , Instructions, Use black or blue ball point pens and avoid Gel & Fountain pens for filling the OMR sheet., Darken the bubbles completely. Don’t put a tick mark or a cross mark, half-filled or over-filled bubbles will not be read, by the software., Correct, , ✔, , ✗, , Incorrect, , Incorrect, , Incorrect, , Do not write anything on the OMR Sheet., Multiple markings are invalid., , 1, , 18, , 35, , 2, , 19, , 36, , 3, , 20, , 37, , 4, , 21, , 38, , 5, , 22, , 39, , 6, , 23, , 40, , 7, , 24, , 41, , 8, , 25, , 42, , 9, , 26, , 43, , 10, , 27, , 44, , 11, , 28, , 45, , 12, , 29, , 46, , 13, , 30, , 47, , 14, , 31, , 48, , 15, , 32, , 49, , 16, , 33, , 50, , 17, , 34, , Check Your Performance, Total Questions:, Total Correct Questions:, , If Your Score is, , Score Percentage =, , Total Correct Questions, Total Questions, , × 100, , Less than 60%, > Average (Revise the concepts again), Greater than 60% but less than 75% > Good (Do more practice), Above 75%, > Excellent (Keep it on)

Page 80 :
72, , CBSE Sample Paper Mathematics Class XII (Term I), , Answers, 1. (c), 11. (c), , 2. (c), 12. (c), , 3. (c), 13. (a), , 4. (b), 14. (a), , 5. (a), 15. (a), , 6. (b), 16. (a), , 7. (a), 17. (b), , 8. (a), 18. (a), , 9. (b), 19. (b), , 10. (b), 20. (c), , 21. (b), 31. (b), 41. (d), , 22. (c), 32. (d), 42. (b), , 23. (a), 33. (d), 43. (d), , 24. (d), 34. (d), 44. (a), , 25. (a), 35. (b), 45. (d), , 26. (c), 36. (b), 46. (c), , 27. (b), 37. (d), 47. (b), , 28. (a), 38. (b), 48. (a), , 29. (c), 39. (d), 49. (d), , 30. (c), 40. (a), 50. (b), , SOLUTIONS, 1. Consider, D =, , 1, , logb a, , loga b, , 1, , 5. Given, f ( x ) = x 3 - 3 x 2 + 3 x - 100, Þ, , f ¢ (x)= 3x2 - 6x + 3, , = 1 - logb a × loga b, é, 1 ù, ú, êQ logb a =, loga b û, ë, , =1-1, =0, , 2. Given that, A = [ aij ]2 ´ 2, where aij = e2ix sin j x, é a11, In general, matrix A = ê, ë a21, Now,, , a12 ù, a22 úû, , a11 = e2x sin x ; a12 = e2x sin 2 x, a21 = e4x sin x ; a22 = e4x sin 2 x, , \, , é e sin x e sin 2 x ù, A = ê 4x, ú, 4x, ë e sin x e sin 2 x û, 2x, , 3. We know that,, , 2x, , AB = A × B, , AB= 3 I, A =3, 3 0 0, AB = 0 3 0 = 27, 0 0 3, , Given,, and, \, Þ, Þ, , AB = 27, A × B = 27, , Þ, , 3 B = 27, , B =9, sin(8 x ), 4. Given that, f ( x ) =, is continuous at x = 0, x, \, lim f ( x ) = f (0 ), , SAMPLE PAPER 2, , \, , x®0, , sin(8 x ), = f (0 ), x®0, x, sin(8 x ), Þ 8 lim, = f (0 ), 8x ® 0, 8x, , \, , Þ, \, , [differentiate w.r.t. x], = 3 ( x - 2 x + 1), 2, , = 3 ( x - 1)2, For any x Î R, ( x - 1)2 ³ 0 since, a perfect square, cannot be negative., \ f ¢ ( x ) ³ 0., Hence, the given function f is an increasing, function on R., 6. Given equation is y = 2 x 2 - x + 1, On differentiating w.r.t. x, we get, dy, = 4x - 1, dx, Since, this is parallel to the given line y = 3 x + 9, dy, Slope of second line =, =3, dx, Therefore, these slopes are equal., Þ, Þ, At x =1,, Þ, , f (0 ) = 8, , y = 2 (1)2 - 1 + 1, y=2, , Thus, the point is (1, 2)., 7. Given, A is 2 ´ 2 order matrix., é a11 a12 ù, \, A= ê, ú, ë a21 a22 û, Now, aij = i 2 + 2 j 2, \, , lim, , 8 ´ 1 = f (0 ), , 4x - 1 = 3, x=1, , a11= 1 + 2 × 1 = 1 + 2 = 3, a12 = 1 + 2 × 4 = 1 + 8 = 9, a21 = 4 + 2 ×1 = 4 + 2 = 6, , \, , a22 = 4 + 2 × 4 = 4 + 8 = 12, é3 9 ù, A= ê, ú, ë6 12 û

Page 81 :
73, , CBSE Sample Paper Mathematics Class XII (Term I), , 8. The shaded region is bounded and has, coordinates of corner points as (0, 0), (6, 0),, (2, 3) and (0, 1). Also, Z = 10 x - 2 y, Corner points, , Corresponding value of, Z, , (0, 0), , 0, , (6, 0), , 60 ¬ Maximum, , (2, 3), , 14, , (0, 1), , -2, , Hence, the maximum value of Z is 60 at (6, 0)., 9. Given, f : N ® N defined by f ( x ) = 2 x + 3, , Þ, , x ® 0+, , Now, f (0 ) = 3, LHL = lim f ( x ) = lim f (0 - h ), = lim, , h®0, , h®0, , k(0 - h ), - kh, = lim, =-k, h®0 h, 0-h, , RHL = lim f ( x ) = lim f (0 + h ) = lim 3 = 3, x ® 0+, , h®0, , h®0, , Q LHL = RHL = f (0 ) Þ - k = 3 or k = - 3, ì x2 - 1, ï, , when x ¹ 1, 15. Given, f ( x ) = í x - 1, ï 2k,, when x = 1, îï, Also, given that f ( x ) is continuous at x = 1, , 10. Given, A is a square matrix and A¢ A = I, A¢, , x ® 0-, , x ® 0-, , Let f ( x1 ) = f ( x2 ), Þ 2 x1 + 3 = 2 x2 + 3 Þ x1 = x2, Hence, f ( x ) is injective., Let f ( x )= y, Þ, y= 2x + 3, y-3, x=, Þ, 2, 1, Let y = 4 Þ x =, 2, i.e. y Î N but xÏ N, Hence, f ( x ) is not surjective., Þ, , Symmetric xRy is not equivalent to yRx, because, 1 R2 Þ 1 is less than 2 2., 2 R1 Þ 2 is less than 12., Thus, it is not symmetric., ì kx, , x<0, ï, 14. Given, f ( x ) = í x, ïî 3 , x ³ 0, ï, Q f ( x ) is continuous at x = 0, \ lim f ( x ) = lim f ( x ) = f (0 ), , A =1, 2, , A =1, , Þ, , lim f ( x ) = f (1), , x®1, , x2 - 1, x®1 x -1, ( x - 1) ( x + 1), = lim, x®1, x-1, , Now, lim f ( x ) = lim, x®1, , [Q|A¢| = |A|], 11. The cosine function is a function whose, domain is the set of all real numbers and range, is the set [-1, 1]. If we restrict the domain of, cosine function to [0 p], then it becomes, one-one and onto with range [-1, 1]., 12. The given equation is, é a - b 2 a + cù é - 1 5 ù, ê2 a - b 3 c + d ú = ê 0 13 ú, ë, û ë, û, , 13. Reflexive Given, xRy Þ x is less than y 2., \xRx Þ x is less than x 2, which is true., Hence, R is reflexive., , x®1, , \, Þ, Þ, , f (1) = 2, 2 k =2, k =1, , 16. If a matrix has m rows and n columns, then, number of elements in the matrix is given by, m ´ n., \Number of elements in matrix C = 2 ´ 3 = 6, é4, 17. Given, A = ê 0, ê, êë 0, 4, \, |A| = 0, , 0 0ù, 4 0ú, ú, 0 4 úû, 0 0, , 4 0 = 43, 0 0 4, , Q |adj A| = |A|n -1,, where n is order of the matrix., \ |adj A| = |A|3- 1 = |A|2 = ( 4 3 )2 = 4 6, , SAMPLE PAPER 2, , On equating the corresponding elements, we, get, a- b= -1, 2a + c = 5, 2a - b = 0, 3 c + d = 13, On simplifying, we get, a = 1, b = 2 , c = 3 and d = 4, , = lim ( x + 1) = 2

Page 82 :
74, , CBSE Sample Paper Mathematics Class XII (Term I), , ( x + 3 )2 - (6 )2, x®3, x-3, (x + 3 + 6) (x + 3 - 6), = lim, x®3, x-3, ( x + 9 )( x - 3 ), = lim, x®3, (x - 3), , 18. Given, A = [ aij ]3 ´ 1, \, , = lim, , é a11 ù, A = ê a21 ú, ê ú, êë a31 úû, , Now, aij = i 3+ j 3, \, , a21 = 8 + 1 = 9, a31 = 27 + 1 = 28, \, , é2 ù, A= ê 9 ú, ê ú, êë28 úû, , 19. Since, R Ç R ¢ are not disjoint, there is at least, one ordered pair, say, (a, b) in R Ç R ¢., But, ( a, b) Î R Ç R ¢, Þ ( a, b)Î R and ( a, b)Î R ¢, Since, R and R¢ are symmetric relations, we get, ( a, b)Î R and ( b, a) Î R, and, ( a, b)Î R ¢ and ( b, a) Î R ¢, and consequently ( b, a)Î R Ç R ¢., Hence, R Ç R ¢ is symmetric., 20. Given, R is a relation on the set N of natural, numbers defined by nRm Û n is a factor of m., Reflexive Since, n is a factor of n for each nÎ N,, therefore, nRn, " nÎ N, i.e. R is reflexive., Symmetric Note that 2 is a factor of 4 but 4 is, not a factor of 2, i.e. 2 R 4 but 4 R, / 2 . Thus, R is, not symmetric., Transitive Let l , m, n Î N, , SAMPLE PAPER 2, , = lim ( x + 9 ) = 3 + 9 = 12, , a11 = 1 + 1 = 2, , \ lRm Þ l is a factor of m, Þ m = lk (for some, k Î N ), mRn Þ m is a factor of n, Þ n = mk ¢ (for some, k ¢ Î N), Þ n = ( lk ) k ¢, Þ n = lkk ¢, Þ n = lk ¢¢ (for some k ¢¢ Î N), Þ l is a factor of n, \ lRm, mRn Þ lRn, Thus, R is transitive., ì ( x + 3 )2 - 36, ï, , x¹3, 21. Given, f ( x ) = í, x -3, îïï, a, x= 3, ( x + 3 )2 - 36, Now, lim f ( x ) = lim, x®3, x®3, x-3, , x®3, , Now, it is given that, f ( x ) is continuous at x = 3, \, lim f ( x ) = f (3 ), x®3, , Þ, , 12 = a or a = 12, , 22. Since, f : R ® R such that f ( x )= 3 - x, Let y1 and y2 be two elements of f ( x ) such that, y1 = y2, - x1, = 3- x 2, Þ, 3, Þ, x1 = x 2, Since, if two images are equal, then their, elements are equal, therefore it is one-one, function., Since, f ( x ) is positive for every value of x,, therefore f ( x ) in into., On differentiating w.r.t. x, we get, dy, = - 3 - x log 3 < 0,, dx, for every value of x., \It is decreasing function., Q Statements I and III are true., 23. Given curve, y = xex, On differentiating w.r.t. x, we get, dy, = x × ex + ex × 1 = xex + ex, dx, dy, For max and min of y,, =0, dx, Þ, , ex ( x + 1) = 0 Þ x = - 1, , [Q ex ¹ 0], , Again, on differentiating w.r.t. x, we get, d 2y, = x × ex + ex ×1 + ex, dx 2, = xex + 2 ex, æ d 2y ö, 1, ç 2÷, = ( - 1)e- 1 + 2 e- 1 = > 0, e, è dx ø, at x = - 1, [minimum], \ f ( x ) has minimum value at x = - 1., Hence, its minimum value is, -1, ., y( - 1) = ( - 1)e- 1 =, e

Page 83 :
75, , CBSE Sample Paper Mathematics Class XII (Term I), , 24. We have, maximise Z = 11x + 7 y, , ...(i), , Subject to the constraints, …(ii), 2x + y £ 6, …(iii), x£2, and, …(iv), x ³ 0, y ³ 0, We see that, the feasible region as shaded, determined by the system of constraints (ii) to, (iv) is OABC and is bounded., So, now we shall use corner point method to, determine the maximum value of Z., Y, , (0, 6) C, , Þ, 5 x+ 210000 - 7 x = 200000, Þ, 210000 - 200000 = 2 x, \, x = 5000, Hence, the amounts invested in two types, of bonds are respectively ` 5000 and, ` (30000 - 5000 ) = ` 25000., 26. Maximise Z = 2 x + 3 y,, Subject to the constraints, x + y £ 1, x ³ 0 and y ³ 0., The shaded region shown in the figure as, OABO is bounded and the coordinates of, corner points O, A and B are (0, 0), (1, 0) and, (0, 1), respectively., Y, , (0, 1) B, A, O, (0, 0) (1, 0), , X′, B (2, 2), , X, , (x, +, , Y′, , (3, 0), , 1), , O, (0, 0), , y=, , X′, , X, , A (2, 0), x=2, , Corner points, , 2x + y = 6, , Corresponding value of, Z = 11 x + 7 y, , Y′, , Corner points, , Corresponding value of, Z = 2x + 3 y, , (0, 0), , 0, , (0, 0), , 0, , (1, 0), , 2, , (2, 0), , 22, , (0, 1), , 3 ¬ Maximum, , (2, 2), , 36, , (0, 6), , 42 ¬ Maximum, , Hence, the maximum value of Z is 42 at (0, 6)., , Subject to the constraints, x + y £ 7, 2 x - 3 y + 6 ³ 0 and x ³ 0, y ³ 0., Y, , (0, 7), B, (3, 4), C, (0, 2), X′, , (–3, 0) (0, 0), , 2x – 3y+6=0, , Y′, , A, X, (7, 0), x+y=7, , SAMPLE PAPER 2, , 25. Let the amount invested in first type of bond, be ` x. Then, the amount invested in second, type of bond will be ` (30000 - x )., According to the given condition,, é 5 ù, ê, ú, [ x 30000 – x ] ê 100 ú = [2000 ], 7, ê, ú, ë 100 û, é 5 x (30000 - x ) 7 ù, +, Þê, úû = [2000 ], 100, ë 100, 5 x+ (30000 - x ) 7, Þ, = 2000, 100, , Hence, the maximum value of Z is 3 at (0, 1)., 27. Minimise Z = 10 x - 3 y

Page 85 :
77, , CBSE Sample Paper Mathematics Class XII (Term I), , 1, 1, =2 +0, 2, 1, \Equation of normal is y - 1 = - ( x - 0 ), 2, Þ, 2y - 2 = - x, Slope of normal at (0 , 1) = -, , Þ, , x + 2y - 2 = 0, 0+0-2, 2, units, =, 5, 1+ 4, , \ Required distance =, 33. Given, y = log2 [log2( x )], , Þ, , é loge x ù, logê, ú, loge (log2 x ), ë loge 2 û, =, =, loge 2, loge 2, loge (loge x ) - loge(loge 2 ), y=, loge 2, , On differentiating w.r.t. x, we get, ù, log2 e, dy, 1 é 1, =, - 0ú =, ê, dx loge 2 ë x loge x, û x loge x, 34. Given,, , 2 4, 2x 4, =, 5 1, 6 x, , On expanding both determinants, we get, 2 ´ 1 - 5 ´ 4 = 2x ´ x - 6 ´ 4, Þ, 2 - 20 = 2 x 2 - 24, Þ, 2 x 2 = - 18 + 24, 6, Þ, x2 = = 3, 2, x=± 3, Þ, 2 3, y 3, Given,, =, 4 5, 2y 5, On expanding both determinants, we get, 2 ´ 5 - 4 ´ 3 = 5 ´ y -3 ´ 2 y, Þ, 10 - 12 = 5 y - 6 y, Þ, -2= -y, Þ, y=2, Hence, x + y = ± 3 + 2, Þ, x+y =2 ± 3, 35. f ( x ) = ex and g( x ) = sin -1 x, Þ, , h( x ) = f [ g( x )], h( x ) = f (sin, sin, , -1, , -1, , sin -1 x, , x) = e, , h( x ) = e, , Þ, , h ¢ ( x ) = esin, , Þ, , 1, h¢ (x), =, h( x ), 1 - x2, , -1, , x, , \ f ¢( x ) = – 3 x 3 – 24 x 2 – 45 x, = – 3 x ( x 2 + 8 x + 15 ) = – 3 x ( x + 5 ) ( x + 3 ), f ¢ (x) = 0, Þ, x = – 5 , x = - 3 and x = 0, f ¢ ¢ ( x ) = – 9 x 2 – 48 x – 45, = – 3 (3 x 2 + 16 x + 15 ), f ¢ ¢ (0 ) = – 45 < 0, Therefore, x =0 is a point of local maxima., f ¢ ¢ (–3 ) = 18 > 0, Therefore, x = – 3 is a point of local minima., f ¢ ¢ (–5 ) = – 30 < 0, Therefore x = – 5 is a point of local maxima., 37. Given, y = cos(sin x 2 ), dy, d, \, = - sin(sin x 2 ) (sin x 2 ), dx, dx, d, = - sin(sin x 2 ) cos x 2 ( x 2 ), dx, = - sin(sin x 2 ) cos x 2 (2 x ), pö, pæ pö, æ dy ö, æ, Now, ç ÷, = - sin ç sin ÷ cos ç2, ÷, p, è, ø, è dx ø x =, 2, 2è 2ø, 2, , =0, -3 5, 0 4, , 2, 38. We have, D = 6, 1, \, , M21 =, , and M32 =, Now,, , 5, , -3 5, 5, , 7, , 2, , 5, , 6, , 4, , 7, , = -21 - 25 = -46, , = 8 - 30 = -22, , -46, -46, M21, =2, =, =, M32 - 1 -22 - 1 -23, , 39. Given, objective function, Z = 2 x + 3 y, Corner points are (0, 0), (7, 0), (6, 2) and (0, 5), respectively., The values of Z at corner points are given, below, Corner points, , Z = 2x + 3 y, , O(0, 0), , Z = 2´0+ 3´0=0, , A(7, 0), , Z = 2 ´ 7 + 3 ´ 0 = 14, , 1, , B(6, 2), , Z = 2 ´ 6 + 3 ´ 2 = 18, , 1 - x2, , D(0, 5), , Z = 2 ´ 0 + 3 ´ 5 = 15, , x, , \, , 45 2, -3 4, x - 8x3 x + 105, 2, 4, , Values of Z are 0, 14, 18 and 15 at the, respective corner points., \ Required sum = 0 + 14 + 18 + 15 = 47, , SAMPLE PAPER 2, , and, , 36. Given, f ( x ) =

Page 86 :
78, , CBSE Sample Paper Mathematics Class XII (Term I), , 40. Given that, adjoint of 3 ´ 3 matrix P is, é 1 4 4ù, ê2 1 7 ú., ê, ú, êë 1 1 3 úû, 1 4 4, 2 1 7 = | P |2, \, , ( 2 )2, æ 1 ö, log ç, ÷, è 2ø, -2 ´ 2, -4, =, =, log2, log2, =, , 43. Given, f ( x ) = logx (loge x ) =, , 1 1 3, , Þ, , [Q| adj A | = | A |2 for n non-singular square, matrix A of order 3], 1 7, 2 7, 2 1, 2, | P| = 1, -4, +4, 1 3, 1 3, 1 1, , = (3 - 7 ) - 4 (6 - 7 ) + 4 (2 - 1), = -4 + 4 + 4, =4, Þ, | P |= ± 2, ì 1 - cos 4 x, , x¹0, ï, 41. Given, f ( x ) = í 8 x 2, ïî, k,, x=0, ï, Q The function f ( x ) is continuous at x = 0, …(i), \ lim f ( x ) = f (0 ), x ®0, 1 - cos 4 x, Now, lim f ( x ) = lim, x ®0, x ®0, 8x2, 2 sin 2 2 x, = lim, x ®0, 8x2, sin 2 2 x, = lim, 2x ® 0 4 x 2, 2, æ sin 2 x ö, = lim ç, ÷, 2x ® 0 è 2 x ø, = (1)2 = 1, f (0 ) = k, \ From Eq. (i),, 1= k Þ k =1, 42. We have, y = log sin x (tan x ) =, , 1, × sec2 x, tan x, 1, - logtan x ´, cos x, sin x, , SAMPLE PAPER 2, , logsin x ×, dy, =, \, dx, p, At x = ,, 4, , log tan x, logsin x, , (logsin x )2, , 1 ö, æ, ç log, ÷ ( 2 )2, è, dy, æ ö, 2ø, =, ç ÷, 2, è dx ø x = p, é, æ 1 öù, log, 4, ÷, ç, ê, è 2 ø úû, ë, , loge (loge x ), loge x, , On differentiating w.r.t. x, we get, 1, 1, 1, loge x ×, × - loge (loge x ) ×, loge x x, x, f ¢ (x) =, (loge x )2, 1 - loge(loge x ), Þ f ¢ (x) =, x (loge x )2, Þ, , f ¢ ( e) =, , 1 - loge(loge e), e (loge e)2, , 44. Given curve is y = be, , =, , 1 - loge 1 1, =, e, e, , -x, a, , ...(i), , At the point where curve crosses the Y-axis,, x=0, \ From Eq. (i), y = be0 = b, Thus, curve crosses the Y-axis at point P(0 , b)., On differentiating Eq. (i), we get, dy, = be, dx, , -x, a, , æ 1ö, ç- ÷, è aø, -x, , b a, e, a, dy, b, At P(0 , b),, =- ., dx, a, Now, equation of tangent at P(0 , b) is, b, y - b = - (x - 0), a, Þ, ay - ab = - bx, Þ, bx + ay = ab, x y, Þ, + =1, a b, x y, Hence, line + = 1 touches the curve, a b, =-, , ...(ii), , -x, , y = be, , a, , at the point where it crosses the, , Y-axis., 45. The equation of the given curve is, y = 4x3 - 2 x5, dy, = 12 x 2 - 10 x 4, dx, Therefore, the slope of the tangent at point, (x, y) is 12 x 2 - 10 x 4.

Page 88 :
80, , CBSE Sample Paper Mathematics Class XII (Term I), , SAMPLE PAPER 3, MATHEMATICS, A Highly Simulated Practice Questions Paper, for CBSE Class XII (Term I) Examination, , Instructions, 1. This question paper contains three sections - A, B and C. Each section is compulsory., 2. Section - A has 20 MCQs, attempt any 16 out of 20., 3. Section - B has 20 MCQs, attempt any 16 out of 20., 4. Section - C has 10 MCQs, attempt any 8 out of 10., 5. There is no negative marking., 6. All questions carry equal marks., Maximum Marks : 40, Time allowed : 90 min, , Roll No., , Section A, In this section, attempt any 16 questions out of Questions 1-20. Each question is of 1 mark weightage., , 1. The principal value branch of cosec –1 is ...A... Here, A refers to, æ -p p ö, (a) ç, , ÷, è 2 2ø, , é -p p ù, (b) ê, ,, ë 2 2 úû, æ -p p ö, (d) ç, , ÷ - {0}, è 2 2ø, , é -p p ù, (c) ê, ,, - {0}, ë 2 2 ûú, , ì1, , æ, , îï 2, , è, , 2. The value of sin í cot -1 ç tan cos -1, , SAMPLE PAPER 3, , (a), , 3, 2, , (b), , 1, 2, , 3 öü, ÷ ý is, 2 øþ, (c), , 1, 2, , (d) None of these, , 3. If A is an invertible matrix of order 3 and|A| = 2, then the value of det (A -1 ) is, (a) -, , 1, 2, , (b), , 1, 4, , (c), , 2, 3, , 1, 2, , 4. If a ij = (3i$ + 2 $j ) and A = [a ij ] 2 ´ 2 , then a 21 + a 22 is equal to, (a) 1, (c) 9, , (b) 8, (d) -1, , (d), , 1, 2

Page 89 :
81, , CBSE Sample Paper Mathematics Class XII (Term I), , 5. The points on the curve y = x 3 at which the slope of the tangent is equal to the, y-coordinate of the points, are, (a) (0, 0) and (27, 3), (c) (2 , 3) and (27, 14), , (b) (0, 0) and (3, 27), (d) (3 , 2 ) and (14, 27), , 6. The region represented by the system of inequation x , y ³ 0, x + 2y £ 2 and x + 2y £ 8 is, (a) unbounded in Ist quadrant, (b) unbounded in Ist and IInd quadrant, (c) bounded in Ist quadrant, (d) None of the above, , é1 0ù, é cos x sin x ù, and A ( adjA) = k ê, ú , then k is equal to, ú, ë 0 1û, ë - sin x cos xû, (a) -1, (b) 0, (c) 2, (d) 1, , 7. If A = ê, , 2ù, é1, é 1 3ù, and B = ê, ú, ú , then the value of det (AB ) is, ë 3 - 1û, ë - 1 1û, (a) 28, (b) 7, (c) - 28, (d) 4, , 8. If A = ê, , 7p ö, ÷ is, 6 ø, p, (b) 6, , 9. The value of cos -1 æç cos, è, , (a), , p, 6, , (c), , 7p, 6, , (d), , 5p, 6, , 10. Let R be the relation in the set {1, 2, 3, 4 } given by, R = {(1, 2), ( 2, 2), (1, 1), ( 4 , 4), (1, 3), ( 3, 3), ( 3, 2)}, Choose the correct answer., (a) R is reflexive and symmetric but not transitive, (b) R is reflexive and transitive but not symmetric, (c) R is symmetric and transitive but not reflexive, (d) R is an equivalence relation, , 11. The minimum value of Z, where Z = 2x + 3y,, subject to constraints 2x + y ³ 23, x + 3y £ 24 and x , y ³ 0, is, (a) 10, (b) 23, (c) 33, (d) 48, , 12. The slope of the tangent to the curve y = x 3 - x at x = 2 is, (b) 6, (d) 11, , 13. If A is any square matrix of order 3 ´ 3 such that|A| = 9, then the value of|adj A| is, (a) 3, (c) 9, , (b) 81, (d) 27, , 14. Let f : Z ® Z be a function given by f (x) = x + 2. Then, f (x) is, (a) one-one, (c) neither one-one nor onto, , (b) one-one and onto, (d) None of these, , SAMPLE PAPER 3, , (a) 5, (c) 7

Page 90 :
82, , CBSE Sample Paper Mathematics Class XII (Term I), , 15. The feasible region of a LPP is shown in following figure. Let Z = 3x - 2y be the, objective function. Minimum of Z occurs at, Y, , (3, 6), , (0, 5), , (5, 4), , (5, 3), , (0, 0), , X, , (4, 0), , (a) ( 4, 0), (c) ( 5, 4), , (b) (0, 5), (d) (0, 0), , 16. If A¢ is the transpose of a square matrix A, then, (a) |A|¹ |A¢ |, (b) |A|= |A¢ |, (c) |A|+ |A¢ |= 0, (d) |A|= |A¢ |only, when A is symmetric, , 17. If y = log (tan x), then, (a) 1, , é1 3ù, , é y 0ù, , dy, p, at x = is equal to, 4, dx, (b) 2, (c) 3, , (d) 4, , é 5 6ù, , 18. If 2 ê, ú , then ( x - y) is equal to, ú =ê, ú +ê, ë 0 xû ë1 2û ë1 8û, (a) 2, , (b) -1, , (c) 1, , (d) 0, , 19. The feasible region for an LPP is shown in the following figure. Minimum of Z = 2x + y, is, , Y, , D (0, 8), C (2, 5), , SAMPLE PAPER 3, , B (4, 3), , A (9, 0), O, , (a) 11, , (b) 6, , X, , (c) 3, , 20. The equation of normal to the curve y = (x - 1) 2 at (2, 1) is given by, (a) x + 2 y = -4, (c) x - 2 y = 4, , (b) x + 2 y = 4, (d) None of these, , (d) 8

Page 91 :
83, , CBSE Sample Paper Mathematics Class XII (Term I), , Section B, In this section, attempt any 16 questions out of Questions 21-40. Each question is of 1 mark weightage., , 21. If the relation R be defined on the set A = {1, 2, 3, 4 , 5} by R = {(a , b) :|a 2 - b 2 | < 8}, then, number of elements in R is, (a) 8, (c) 10, , (b) 9, (d) 11, , 22. The value of k for which the following function is continuous at x = 3, ì( x + 3) 2 - 36, , x¹3, ï, f ( x) = í, x-3, ï, 12k , x = 3, îï, (a) 1, (c) 4, , (b) 3, (d) 12, , 23. If the following function f (x) is continuous at x = 0, then the value of k is, , 5, 2, 3, (c), 2, , ì sin 5x, ï, 2 , x¹0, f ( x) = í, x, ï, k,, x=0, ïî, , (a), , (b), , 1, 2, , (d) 0, , 24. The principal value of sec -1 2 is ..A.. Here, A refers to, p, 3, -p, (c), 3, , p, 6, 2p, (d), 3, (b), , (a), , 25. The function given by f (x) =, (a) x = e, (c) x = 2, , log x, x, , has maximum at, (b) x = 1, (d) None of these, , 26. If the function f be given by f (x) = (x + 2) e - x , then, , 27. The equation of the normal to the curve y = x (2 - x) at the point (2, 0) is, (a) x + 2 y = 2, (b) x - 2 y = 2, (c) 2 x + y = 4, (d) None of the above, , SAMPLE PAPER 3, , (a) f is increasing in ( - ¥ , - 1], (b) f is decreasing in [ - 1, ¥ ), (c) Both (a) and (b) are true, (d) Both (a) and (b) are false

Page 92 :
84, , CBSE Sample Paper Mathematics Class XII (Term I), , 28. If the graphical form of an LPP is as follows, Y, 100, 90, 80, 75, 70, , C, , 60, 50, 40, 30, 23, 20, , A, , B, , x +5 y =, , 10, X′, (0,0)O, , 11 5, , (115, 0), 30, , 40, , 60, , 70, , 80, , 90 100 110 120, , X, , y, +2, =1, , y = 80, , Y′, , 50, , 3x, , 4x+, , 10 20, , 50, , The coordinate of the corner point A of the feasible region of the LPP is, (a) (40, 15), (b) (15, 15), (c) (2, 70), (d) None of these, , 29. The point on the curve x 2 + y 2 = a 2 and y ³ 0 at which the tangent is parallel to X-axis,, is, (a) (0, a), , æa 3 ö, (c) ç ,, a÷, è2 2 ø, , (b) ( a , 0), , (d) ( - a, 0), , 30. The feasible region of an LPP is given below, Y, 200, , D(0, 200), , C(50, 100), , 100, 80, 60, A(0, 50), 40, , B(20, 40), (10, 20), , SAMPLE PAPER 3, , 20, X′, (0, 0) O, 2x–y=0, , 20 40, Y′, , (100, 0), 60, , 80, , 100 120 140 160, , 2 x + y = 200, , X, , x + 2y = 100, , The square root of maximum of Z, where Z = x + 2y, is, (a) 20, (b) 21, (c) 24, é 1 2 2ù, 31. If A = ê 2 1 2ú , then A 2 - 4 A is equal to, ú, ê, êë 2 2 1úû, (a) 2 I 3, (b) 3 I 3, (c) 4 I 3, , (d) 25, , (d) 5 I 3

Page 93 :
85, , CBSE Sample Paper Mathematics Class XII (Term I), , 32. The line y = x + 1 is a tangent to the curve y 2 = 4 x, then the point of contact is, (a) (1, 2 ), , (c) (1, - 2 ), , (b) (2 , 1), , (d) ( - 1, 2 ), , é 1 2 3ù é xù, , 33. If [1 x 1] ê 0 5 1ú ê 1ú = 0, then the value of x is, úê ú, ê, êë 0 3 2úû ëê - 2úû, 2, (b), 3, , (a) 0, , (c), , 5, 4, , (d) -, , 4, 5, , 34. The area of a triangle with vertices (-3, 0), (3, 0) and (0, k) is 9 sq units. Then, the value, of k will be, (a) 9, , (c) -9, , (b) 3, , (d) 6, , 35. Let us define a relation R in R as aRb, if a ³ b. Then, R is, (a) an equivalence relation, (b) reflexive, transitive but not symmetric, (c) symmetric, transitive but not reflexive, (d) neither transitive nor reflexive but symmetric, , 36. Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? If g is described by g(x) = a x + b, then the, value which should be assigned to a and b is, (a) g is a function and a = 2 and b = -1, (b) g is a function and a = -1 and b = 2, (c) g is a function and a = 1 and b = -1, (d) g is not a function, 37. If a and b are positive numbers such that a > b , then the minimum value of, p, a sec q - b tan q, æç 0 < q < ö÷ is, è, 2ø, 1, 1, (a) a 2 - b 2, (d), (b) a 2 + b 2, (c), a 2 - b2, a 2 + b2, , 38. If 12 is divided into two parts such that the product of the square of one part and the, fourth power of the second part is maximum, then its parts are, (a) 5 and 7, (b) 6 and 6, (c) 3 and 9, dy, 39. If y = sec (tan -1 x), then is equal to, dx, xy, x, (c), (b) xy 1 + x 2, (a), 2, 1+ x, 1 + x2, , (d) 4 and 8, , (d) None of these, , 40. If A = {1, 2, 3, K , n} and B = {a , b }. Then, the number of surjections from A into B is, (a), , n, , P2, , (b) 2 n - 2, , (c) 2 n - 1, , (d) None of these, , In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , 41. If f : (-1, 1) ® R be a differentiable function with f (0) = - 1 and f ¢ (0) = 1. Let, g( x) = [ f ( 2 f ( x) + 2)] 2 . Then, g¢ ( 0) is equal to, (a) 4, (b) -4, (c) 0, (d) -2, , SAMPLE PAPER 3, , Section C

Page 95 :
OMR SHEET, , SP 3, , Roll No., Sub Code., , Student Name, , Instructions, Use black or blue ball point pens and avoid Gel & Fountain pens for filling the OMR sheet., Darken the bubbles completely. Don’t put a tick mark or a cross mark, half-filled or over-filled bubbles will not be read, by the software., Correct, , ✔, , ✗, , Incorrect, , Incorrect, , Incorrect, , Do not write anything on the OMR Sheet., Multiple markings are invalid., , 1, , 18, , 35, , 2, , 19, , 36, , 3, , 20, , 37, , 4, , 21, , 38, , 5, , 22, , 39, , 6, , 23, , 40, , 7, , 24, , 41, , 8, , 25, , 42, , 9, , 26, , 43, , 10, , 27, , 44, , 11, , 28, , 45, , 12, , 29, , 46, , 13, , 30, , 47, , 14, , 31, , 48, , 15, , 32, , 49, , 16, , 33, , 50, , 17, , 34, , Check Your Performance, Total Questions:, Total Correct Questions:, , If Your Score is, , Score Percentage =, , Total Correct Questions, Total Questions, , × 100, , Less than 60%, > Average (Revise the concepts again), Greater than 60% but less than 75% > Good (Do more practice), Above 75%, > Excellent (Keep it on)

Page 96 :
88, , CBSE Sample Paper Mathematics Class XII (Term I), , Answers, 1. (c), 11. (b), 21. (d), , 2. (c), 12. (d), 22. (a), , 3. (d), 13. (b), 23. (a), , 4. (c), 14. (b), 24. (a), , 5. (b), 15. (b), 25. (a), , 6. (c), 16. (b), 26. (c), , 7. (d), 17. (b), 27. (b), , 8. (c), 18. (d), 28. (a), , 9. (d), 19. (d), 29. (a), , 10. (b), 20. (b), 30. (a), , 31. (d), 41. (b), , 32. (a), 42. (a), , 33. (c), 43. (a), , 34. (b), 44. (c), , 35. (b), 45. (c), , 36. (a), 46. (a), , 37. (a), 47. (a), , 38. (d), 48. (b), , 39. (a), 49. (c), , 40. (b), 50. (c), , SOLUTIONS, 1. cosec-1 is a function whose domain is, R - ( -1, 1) and range could be any of the, pù, é p pù, é 3p, intervals - ,, êë 2 2 úû - {0 }, êë - 2 , - 2 úû - {- p },, é p 3p ù, - {p } etc., corresponding to each such, ,, ëê 2 2 ûú, interval, we get a branch of the function, cosec-1., é p pù, The branch with range ê - , ú - {0 } is called, ë 2 2û, the principal value branch of cosec-1., ì1, æ, 3 öü, 2. Given, sin í cot -1 ç tan cos -1, ÷ý, 2 øþ, è, îï 2, ì1, Þ sin í cot -1, îï 2, ì1, Þ sin í cot -1, îï 2, , æ 1 öü, ç ÷ý, è 3 øþ, , Y, 5, 4, 3, , 3. Q A is invertible matrix of order 3., |A-1| = |A|-1 =, , 1, |A|, , 2, 1, , 1, 2, 1, 4. We have, aij = (3 i$ + 2 j$ ), 2, , SAMPLE PAPER 3, , |A-1| =, , \, , a21 =, , and a22 =, , When, the slope of the tangent is equal to the, y-coordinate of the point, then y = 3 x 2., [Q y = x 3 given], Þ, 3x2 = x3, 2, Þ, x (3 - x ) = 0, Þ, x = 0 or x = 3, When, x = 0, then from Eq. (i), we get y = 0 3 = 0, When, x = 3, then from Eq. (i), we get, y = 3 3 = 27, Hence, the required points are (0, 0) and (3, 27)., 6. Given inequation system, x , y ³ 0, x + 2 y £ 2 and x + 2 y £ 8, Let l1 : x + 2 y = 2 ,, l2 : x + 2 y = 8, l3 : x = 0 and l4 : y = 0, , æ, æ p öü, ç tan çè ÷øý, è, 6 þ, , p 1, ì 1 pü, Þ sin í ´ ý Þ sin =, 2, 3, 6 2, þ, îï, , \, , The slope of the tangent at the point (x, y) is, given by, æ dy ö, = 3x2, ç ÷, è dx ø( x , y ), , O, , 1, , 2 3 4 5 6 7, l1, , 8, , X, l2, , Hence, inequation system gives bounded, region in first quadrant., é cos x sin x ù, 7. Given, A = ê, ú, ë - sin x cos x û, , 8, 1, (6 + 2 ) = = 4, 2, 2, 10, 1, (6 + 4 ) =, =5, 2, 2, , Now, a21 + a22 = 4 + 5 = 9, 3, , 5. The equation of the given curve is y = x …(i), dy, \, = 3x2, dx, , \, |A|= cos 2 x + sin 2 x = 1, Q ( adjA) A =|A|I, Þ ( adjA) A = 1 I = I, Also, given A( adjA) = kI, From Eqs. (i) and (ii),, kI = I Þ k = 1, , …(i), …(ii)

Page 98 :
90, , CBSE Sample Paper Mathematics Class XII (Term I), , 19. Given, objective function Z = 2 x + y, , 5x, ì, ï sin 2, ,, 23. Given, f ( x ) = í, x, ï, k,, îï, , Value of Z = 2x + y, , A ( 9, 0), , Z = 18 + 0 = 18, , Q f ( x ) is continuous at x = 0., , B ( 4, 3), , Z = 8 + 3 = 11, , Þ, , C ( 2, 5), , Z =4+ 5=9, , D ( 0, 8), , Z =0+ 8=8, (Minimum), , The minimum value of Z is 8., , \, , Þ, , 2, , 20. Given curve is y = ( x - 1) ., dy, = 2 ( x - 1), \, dx, æ dy ö, = 2 (2 - 1) = 2, ç ÷, è dx ø( 2, 1), 1, -1, Slope of normal =, =æ dy ö, 2, ç ÷, è dx ø( 2, 1), Equation of normal is given by, 1, y - 1 = - (x - 2 ), 2, Þ, 2 y - 2 = -x + 2 Þ x + 2 y = 4, 21. Given, A = {1, 2 , 3 , 4 , 5 },, R = {( a, b) :|a2 - b2| < 8 }, R = { (1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2),, (3, 3), (4, 3), (3, 4), (4, 4), (5, 5)}, The number of elements in R is 11., ì ( x + 3 )2 - 36, ,x¹3, ï, 22. Given, f ( x ) = í, x-3, ï, ,x=3, 12 k, îï, Also, given that f ( x ) is continuous at x = 3, Þ, lim f ( x ) = 12 k, x ®3, , \, , SAMPLE PAPER 3, , Þ, Þ, Þ, Þ, Þ, \, , x¹0, , Corner points, , ( x + 3 )2 - 36, = 12 k, x ®3, x-3, lim, , ( x + 3 )2 - 6 2, = 12 k, x ®3, x-3, (x + 3 - 6) (x + 3 + 6), = 12 k, lim, x ®3, x-3, lim, , lim ( x + 3 + 6 ) = 12 k, , x ®3, , 3 + 3 + 6 = 12 k, 12 k = 12, k =1, , Þ, , x=0, , lim f ( x ) = f (0 ), 5x, sin, 2 =k, lim, x ®0, x, 5x, sin, 5, 2 =k, lim, 2 x ®0 5x, 2, 5, 5, ´1= k Þ k =, 2, 2, x ®0, , 24. Let sec-1 2 = q, Þ, , sec q = 2, , We know that, the range of principal value, ì pü, branch of sec-1 q is [0 , p ] - í ý., îï 2 þ, p, p, Q sec q = 2 = sec Þ q =, 3, 3, ì pü, where, q Î [0 , p ] - í ý, îï 2 þ, p, Þ sec-1 2 =, 3, p, Hence, the principal value of sec-1 2 is ., 3, log x, 25. Let f ( x ) =, x, On differentiating w.r.t. x, we get, æ1ö, x ç ÷ - (log x ) × 1, èxø, 1 - log x, f ¢ (x) =, =, 2, x, x2, Again, differentiating w.r.t. x, we get, æ 1ö, x 2 ç - ÷ - (1 - log x ) × 2 x, è xø, f ¢¢ ( x ) =, ( x 2 )2, - x - 2 x + 2 x log x, =, x4, x (2 log x - 3 ) 2 log x - 3, =, =, x4, x3, 1 - log x, For maximum put f ¢ ( x ) = 0 Þ, =0, x2, Þ, log x = 1 Þ x = e, 2 log e - 3 2 × 1 - 3 -1, At x = e, f ¢¢ ( e) =, =, = 3 <0, e3, e3, e, Therefore, by second derivative test, f is the, maximum at x = e.

Page 99 :
91, , CBSE Sample Paper Mathematics Class XII (Term I), , 26. Given, f ( x ) = ( x + 2 ) e- x, -x, , Þ, , f ¢ (x) = - (x + 2 ) e + e, = - ( x + 1) e- x, , -x, , - ( x + 1) e- x ³ 0, , Þ, , ( x + 1) £ 0, , \, , On differentiating w.r.t. x, we get, dy, \, 2x + 2y, =0, dx, dy, x, =Þ, dx, y, dy, Given,, =0, dx, [Q tangent is parallel to X-axis], x, - =0, Þ, y, , -1 1, =, -2 2, , 1, (x - 2 ), 2, 2y = x -2, x -2y = 2, y -0 =, , Þ, x=0, \, y=±a, But, y³0, \, y=a, \ Required point is (0 , a)., , 28., Y, 100, 90, , The corner points of the feasible region are, A(0, 50), B(20, 40), C(50, 100) and D(0, 200)., The values of Z at corner points are given, below, , 50, 40, , Corner points, , A, , B, , 115, , (115, 0), 10 20, , 30, , 40, , 50, , 60, , 70, , 80, , 90 100 110 120, , y, +2, =1, 50, , y = 80, , A(0, 50), , Z = 0 + 2 ´ 50 = 100, , B(20, 40), , Z = 20 + 2 ´ 40 = 100, , C (50, 100), , Z = 50 + 2 ´ 100 = 250, , D(0, 200), , Z = 0 + 2 ´ 200 = 400, , 3x, , 4x+, , Y′, , X, , From the graph, we can see that corner point A, of the feasible region can be obtained from the, intersection points of the given equation of, lines., , Value of Z = x + 2y, , The maximum value of Z is 400 at D(0, 200)., The square root of maximum of Z = Z, = 400 = 20, , SAMPLE PAPER 3, , x +5 y =, , 10, X′, (0,0)O, , (given), , 30. Given objective function, Z = x + 2 y, , C, , 60, , 30, 25, 20, , x + 5 ´ 15 = 115, , 29. Given curve is x 2 + y 2 = a2, , \ Equation of normal,, , 80, 75, 70, , …(ii), , Þ, x = 40, Hence, the corner point of the feasible region is, A (40, 15)., , 27. Given curve is y = x (2 - x ) = 2 x - x 2, dy, \, = 2 -2x, dx, -1, -1, Now, slope of normal =, =, dy, æ ö, 2 - 2 (2 ), ç ÷, è dx ø( 2, 0), , Þ, Þ, , 3 x + 2 y = 150, , 3 x + 2 y = 150, - _____________, 13 y = 195, y = 15, Put y = 15 in Eq. (i),, , [Q e- x > 0], , Þ, x£ -1, Þ, x Î (- ¥ , - 1], (b) For decreasing, f ¢ ( x ) £ 0, Þ, - ( x + 1) e- x £ 0, Þ, ( x + 1) e- x ³ 0, [Q e- x > 0], Þ, x³ -1, \ f is decreasing on [ - 1, ¥ )., , =, , …(i), , Multiply Eq. (i) by 3 and subtracting Eq. (ii), from Eq. (i), we get, 3 x + 15 y = 345, , (a) For increasing, f ¢( x ) ³ 0, Þ, , x + 5 y = 115

Page 101 :
93, , CBSE Sample Paper Mathematics Class XII (Term I), , dy, =0, dq, , Put, Þ, Þ, , =, , 2, , sec q ( a sin q - b) = 0, b, sin q =, a, a 2 - b2, , secq =, a, , Þ cos q =, and tanq =, , (Q sec q ¹ 0 ), , a 2 - b2, , b, 2, , \Minimum value is, a, b, y = a., – b., 2, 2, 2, a –b, a – b2, = a 2 – b2, 38. Let x and y be the two parts., x + y = 12, , Let, , P = x 2y 4, , Þ, , P = xy 2 = L (say), , Þ (12 – x ) (12 – 3 x ) = 0, x = 12 and x = 4, , At x = 4, it is maximum, 2 4, , [Q x ¹ 12 ], , Hence, x y is maximum, when the parts are, 4 and 8., 39. Given, y = sec (tan -1 x ), dy, d, \, sec (tan -1 x ), =, dx dx, , Clearly, function will not be onto, if all, elements of A map to either a or b., \ Total number of surjections from A to B, = 2n -2, 41. Given, g( x ) = [ f (2 f ( x ) + 2 )]2, \g ¢ ( x ) = 2 f (2 f ( x ) + 2 ) × f ¢ (2 f ( x ) + 2 ) × 2 f ¢ ( x ), = 4 f (2 f ( x ) + 2 ) f ¢ (2 f ( x ) + 2 ) f ¢ ( x ), g ¢ (0 ) = 4 f (2 f (0 ) + 2 ) f ¢ (2 f (0 ) + 2 ) f ¢ (0 ), = 4 f (0 ) f ¢ (0 ) f ¢ (0 ), = -4, 42. Let x = a cos 3 q and y = a sin 3 q ., 2, , 2, , 2, , 2, , Then, x 3 + y 3 = ( a cos 3 q ) 3 + ( a sin 3 q ) 3, 2, , 2, , = a 3 [(cos 2 q + (sin 2 q )] = a 3, Hence, x = a cos 3 q and y = a sin 3 q is parametric, 2, , 2, , 2, , equation of x 3 + y 3 = a 3, dx, Now,, = - 3 a cos 2 q sin q, dq, dy, and, =3 a sin 2 q cos q, dq, dy, 3 a sin 2 q cos q, dy dq, =, =, = - tan q = dx dx, -3 a cos 2 q sin q, dq, é 1 2 -2 ù, 43. Given, B = ê -1 3, 0 ú, ú, ê, êë 0 -2 1 úû, 1, Here, |B| = - 1, 0, , 2, , -2, , 3, , 0, , -2, , 1, , 3, , = 1 (3 - 0 ) - 2 ( -1 - 0 ) - 2 (2 - 0 ), =3 +2 - 4 =1¹0, -1, \ B exists., Cofactors of|B|are, B11 = (3 - 0 ) = 3, B12 = - ( - 1 - 0 ) = 1,, B13 = (2 - 0 ) = 2 ,, B21 = - (2 - 4 ) = 2 , B22 = (1 - 0 ) = 1,, B23 = - ( - 2 - 0 ) = 2 ,, , y, x, , SAMPLE PAPER 3, , L = x (12 – x )2, dL, Þ, = (12 – x )2 – 2 x (12 – x ), dx, dL, For maxima, put, =0, dx, Þ, , Þ, , (1 + x 2 ), dy, x, x, =, .y, = sec (tan -1 x ) ., dx, (1 + x 2 ) (1 + x 2 ), 40. Total number of functions = ( n( B))n ( A ) = 2 n ., , a, , a - b2, Now, differentiating Eq. (i) w.r.t. q, we get, d 2y, = 2 secq × secq tan q ( a sin q - b), dq 2, + sec2 q × ( a cos q - 0 ), 2, d y, = 2 sec2 q × tan q ( a sin q - b) + a secq, Þ, dq 2, b, At sinq = ,, a, d 2y, a2, b, æ b, ö, =, 2, ×, ×, ç a × - b÷, 2, 2, 2, ø, 2, 2, dq, (a - b ) a - b è a, a, +a ×, 2, a - b2, a2, a2, =0+, =, >0, a 2 - b2, a 2 - b2, , Then,, , sec (tan -1 x ) .tan (tan -1 x )

Page 103 :
95, , CBSE Sample Paper Mathematics Class XII (Term I), , SAMPLE PAPER 4, MATHEMATICS, A Highly Simulated Practice Questions Paper, for CBSE Class XII (Term I) Examination, , Instructions, 1. This question paper contains three sections - A, B and C. Each section is compulsory., 2. Section - A has 20 MCQs, attempt any 16 out of 20., 3. Section - B has 20 MCQs, attempt any 16 out of 20., 4. Section - C has 10 MCQs, attempt any 8 out of 10., 5. There is no negative marking., 6. All questions carry equal marks., Maximum Marks : 40, Time allowed : 90 min, , Roll No., , Section A, In this section, attempt any 16 questions out of Questions 1-20. Each question is of 1 mark weightage., , 1. Let Z = x + y be the objective function and maximum Z = 30. The maximum value occurs, at point, æ 50 40 ö, (a) ç , ÷, è3 3ø, , (b) (0, 0), , (c) (25, 0), , (d) (0, 20), , 2. The slope of the tangent of the curve x = 3t 2 + 13t + 8 and y = 12t 2 + 2t - 15 at t = 5, is, (a), , 7, 6, , (b), , 122, 43, , (c) 1, , (d), , 5, 6, , 3. The slope of the normal to the curve ay 2 = x 3 at the point (am 2 , am 3 ) is, (b), , 1, 3m, , (c), , -3m, 2, , (d), , -2, 3m, , 4. The least value of a such that the function f given by f (x) = x 2 + ax + 1 is strictly, increasing on (1, 2) is, (a) -1, (b) -2, , (a) 1, , (c) 0, , (d) 1, , ì x - ( k + 2) x + 2k, x ¹ 2 is continuous at x = 2, then k is equal to, x-2, ï, x=2, 3, îï, (b) -1, (c) 0, (d) 4, , 5. If f (x) = ïí, , 2, , SAMPLE PAPER 4, , (a) 3m

Page 104 :
96, , CBSE Sample Paper Mathematics Class XII (Term I), , 6. If A is a skew-symmetric matrix of order 3, then the value of|A| is, (a) 3, , é x + y kù, , (b) 0, , (c) 9, , (d) 27, , é 6 5ù, , 7. If ê, ú , then the value of x and y are respectively, ú =ê, ë x - y 1û ë 4 1û, (a) 5 and 1, , 8. If A = [a ij ] 3 ´ 2, (a) 1, , (b) 1 and 5, , (c) 3 and 5, , (d) 5 and 3, , é1 4 ù, is a matrix given by A = ê 5 0 ú , then a 22 + a 31 is equal to, ú, ê, êë 6 -1úû, (b) 6, (c) 0, (d) -1, , é -1 9 2ù, 9. If A = ê a 3 5ú is upper triangular matrix, then a + b is equal to, ê, ú, êë b 0 4úû, (a) 0, (b) 1, (c) 2, 3p ö, ÷ is, 4 ø, p, (b), 4, , (d) 3, , 10. The value of sin -1 æç sin, è, , (a), , 3p, 4, , (c), , 5p, 4, , (d) None of these, , (c), , 1, 2, , (d), , 11. If tan -1 (1) = sin -1 x, then the value of x is, (a), , 3, 2, , (b) 0, , 1, 2, , 12. The relation S is defined on the set of integers Z as xSy, if integer x divides integer y., Then,, (a) S is an equivalence relation, (c) S is only reflexive and transitive, , (b) S is only reflexive and symmetric, (d) S is only symmetric and transitive, , 13. Let P = {1, 2, 3} and a relation on set P is given by the set, R = {(1, 2), (1, 3), (2, 1), (1, 1), (2, 2), (3, 3), (2, 3)}. Then, R is, (a) reflexive, transitive but not symmetric, (b) symmetric, transitive but not reflexive, (c) symmetric, reflexive but not transitive, (d) None of these, , 14. Let A = {1, 2, 3}. We defined R 1 = { (1, 2), (3, 2), (1, 3)} and R 2 = {(1, 3), (3, 6), (2, 1), (1, 2)}., , SAMPLE PAPER 4, , Then, choose the correct option., (a) R1 is relation and R2 is not, (c) R1 and R2 are both non-relation, , (b) R1 and R2 is relation, (d) None of these, , 15. Let W denotes the set of all non-negative integers and Z denotes the set of all integers., The function f : Z ® W given by f ( x) =| x | is, (a) one-one but not onto, (b) onto but not one-one, (c) both one-one and onto, (d) neither one-one nor onto, , 16. Let R be the relation in the set N given by R = {(a , b) : a = b - 2, b > 6}., Choose the correct answer., (a) (2, 4) Î R, (c) (6, 8) Î R, , (b) (3, 8) Î R, (d) (8, 7) Î R

Page 105 :
97, , CBSE Sample Paper Mathematics Class XII (Term I), , 17. The area of the triangle with vertices (- 1, 2), (4, 0) and (3, 9) is k sq units, then value of, k is, (a) 43, , (b), , 43, 2, , (c) 20, , (d), , 43, 3, , 2 1 6, 18. If D = - 1 4 2 , then the value of a 11 M 11 - a 12 M 12 + a 13 M 13 is, 0, , 1 2, , (a) 10, , (b) 8, , (c) 14, , 19. If y = x + 1 + x - 1, then x 2 - 1, (a) y, , (b), , 20. If y = log(cos x 2 ), then, (a) - 2 x cot x 2, , dy, dx, , (d) 4, , is equal to, , y, , (c), , 2, , y, 3, , (d) - y, , dy, , is equal to, dx, (b) - 2 x tan x 2, , (c) 2 x tan x 2, , (d) None of these, , Section B, In this section, attempt any 16 questions out of Questions 21-40. Each question is of 1 mark weightage., , ìx+l , x<3, 21. If f (x) = ïí 4, , x = 3 is continuous at x = 3, then the value of l is, ï 3x - 5 , x > 3, îï, (a) 4, (b) 3, (c) 2, (d) 1, ìx 2 - 9, 22. If the function f (x) = ïí x - 3, ï 2x + k, îï, (a) 3, (b) 0, éa-b, , 2c + wù, , é5, , , x ¹ 3 is continuous at x = 3, then the value of k is, , x=3, (c) - 6, , (d) 1, , 3ù, , 23. If ê, ú , then the value of a , b , c and w are respectively, ú=ê, ë 2a - b 2a + wû ë12 15û, (a) 1, 1, 2 and 7, , (b) 7, 1, 2 and 1, , (c) 1, 2, 7and 7, , (d) 7, 2, 1 and 1, , 24. The relation R in the set Z of integers given by R = {(a , b) : a - b is divisible by 5} is, (a) reflexive, (c) symmetric and transitive, , 3 4ù, 1 0ú , then AB is equal to, ú, 2 3úû, 16 - 24ù, 11 11ù, é 5, (c) ê, ú, ú, - 11 11 û, ë - 11 16 24û, , (d) None of these, , é 2 0 0ù, , 26. If A = ê 0 2 0ú and A 4 = kA, then k is equal to, ê, ú, êë 0 0 2úû, , (a) 10, , (b) 16, , (c) 32, , (d) 8, , SAMPLE PAPER 4, , é 1, é 3 - 1 4ù, ê, 25. If A = ê, ú and B = ê 2, 2, 3, 1, û, ë, êë - 3, é - 11 16 24ù, é - 11, (a) ê, (b) ê, ú, ë 5 11 11û, ë 5, , (b) reflexive but not symmetric, (d) an equivalence relation

Page 106 :
98, , CBSE Sample Paper Mathematics Class XII (Term I), , éa 2ù, 3, ú and| A | = 125, then a is equal to, 2, a, ë, û, (a) ± 3, (b) ± 5, (c) 0, , 27. If A = ê, , é 4 0 0ù, 28. If A = ê 0 4 0ú , then| adj A| is equal to, ê, ú, êë 0 0 4ûú, (c) 4 2, (b) 4 6, (a) 4 3, 1, 1, 1, , then the maximum value of D is, 29. If D = 1 1 + sin q, 1, 1, 1, 1 + cos q, 1, 1, (c), (a) 1, (b), 2, 3, , 30. In the interval, , (d) ± 2, , (d) 4 5, , (d) 0, , 3 ù, é 2 sin x, p, is, < x < p, then value of x for which the matrix ê, 2 sin xúû, 2, ë 1, , singular is, p, (a), 2, , (b), , p, 6, , (c), , ìx 2 - x - 6, 31. If the function f (x) = ïí x - 3, ï, k, îï, (a) 5, (b) 4, , (a) 1, , (a) - y, , 2p, 3, , , x=3, , , x¹0, , ïî 4 + x, (b) 2, , , x=0, , 33. If y = x + x 2 - 1, then (y - x), , (d), , , x ¹ 3 is continuous at x = 3, then the value of k is, , ì sin kx, , 32. If the function f (x) = ïí x, , p, 3, , (c) 3, , (d) 2, , is continuous at x = 0. Then, the value of k is, (c) 3, , (d) 4, , (c) y 2, , (d), , dy, is equal to, dx, , (b) y, , 1, y, , 34. If f (x) = x tan - 1 x, then f ¢ (1) is equal to, (a) 1 +, , SAMPLE PAPER 4, , 35. If, , x2, a, , 2, , p, 4, , +, , (b), , y2, b, , 2, , (a) 1, , = 10, then, , 1 p, 2 4, , (c), , 1 p, +, 2 4, , (d) 2, , dy, , at ( a 2 , b 2 ) is equal to, dx, (b) - 1, (c) 0, , (d) 2, , 36. The function f (x) = x 2 e - x is increasing in the interval, (a) ( - ¥ , ¥ ), , (b) ( - 2 , 0), , (c) (0, 2), x 3, 37. In the interval (- 3, 3), the function f (x) = + , x ¹ 0 is, 3 x, (a) decreasing, (b) increasing, (c) neither increasing nor decreasing, (d) None of these, , (d) (2 , ¥ )

Page 107 :
99, , CBSE Sample Paper Mathematics Class XII (Term I), , 38. The slope of the normal to the curve x = a cos 3 q and y = a sin 3 q at q =, , p, is, 4, , (b) - 1, (d) 2, , (a) 1, (c) 0, , 39. The equation of normal to the curve y = 2x 2 + 3 sin x at (0, 0) is, (a) x + 3y = 0, (c) y - 3x = 0, , (b) x - 3y = 0, (d) y + 3x = 0, , 40. If A 2 - 5A + 7 I = O, then A - 1 is equal to, 1, ( A - 5 I), 7, -1, (c), ( 5 I - A), 7, , 1, (b) ( 5 I - A), 7, , (a), , (d) None of these, , Section C, In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , 41. The function f (x) = x 3 - 3x 2 + 6x + 3 has, (a) maximum value at x = 1, (b) minimum value at x = 0, (c) neither a maximum nor a minimum value, (d) None of the above, , é - 1 - 2 - 2ù, 42. If A = ê 2 1 - 2ú , then adj A is equal to, ê, ú, êë 2 - 2 1 úû, 6 ù, é- 3 6, é - 3 - 6 - 6ù, ê, ú, ê, ú, (a) ê - 6 3 - 6ú, (b) ê 6, 3 - 6ú, êë - 6 - 6 3 úû, êë 6 - 6 3 úû, 6 ù, é- 3 6, ú, ê, (d) None of these, (c) ê - 6 - 6 3 ú, êë - 6 3 - 6úû, , 43. Let f (x) = (1 + b 2 )x 2 + 2bx + 1 and m(b) be the minimum value of f (x). As b varies, the, range of m( b) is, (a) [0, 2], , é 1ù, (b) ê0, ú, ë 2û, , é1 ù, (c) ê , 1ú, ë2 û, , (d) (0, 1], , (a) increasing, (c) neither increasing nor decreasing, , (b) decreasing, (d) None of these, , 45. If f (x) = xe x( 1 - x ) , then f (x) is, é 1 ù, (a) increasing on ê - , 1ú, ë 2 û, , (b) decreasing on R, , (c) increasing on R, , é 1 ù, (d) decreasing on ê - , 1ú, ë 2 û, , SAMPLE PAPER 4, , 44. In the interval [1, ¥), the function f (x) = (x + 1) 3 (x - 3) 3 is

Page 108 :
100, , CBSE Sample Paper Mathematics Class XII (Term I), , CASE STUDY, The feasible solution for a LPP is shown below., Y, 6, , 2y =, , G(0, 5), , 3x +, , 5, , 2, , 13, , 4, , y, x–, , =, , 4, , D, , 3, , E(0, 2), , 2, C(4, 1), , 1, X′, , B(6, 1/2), F(8, 0), , O, , 1, , Y′, , 2, , 3, , 4, , 5, A(5, 0), , 6, , and the objective function is Z = 15x - 4y., Based on the above information, answer the following questions., , 46. The coordinate of point D is, (a) (1, 2), (c) (3, 2), , (b) (2, 2), (d) (4, 2), , 47. The value of (n - 1) 2 , where n is number of corner points, is, (a) 36, (c) 16, , (b) 25, (d) 9, , 48. The maximum of Z is, (a) 88, (c) 56, , (b) 75, (d) 89, , 49. The minimum of Z is, (a) 0, (c) 1, , 50. Z æ, , SAMPLE PAPER 4, , 1ö, ç6, ÷, è 2ø, , (a) 88, (c) 82, , (b) 37, (d) - 8, , + Z ( 0 , 2 ) is equal to, (b) 80, (d) 86, , 7, , X

Page 109 :
OMR SHEET, , SP 4, , Roll No., Sub Code., , Student Name, , Instructions, Use black or blue ball point pens and avoid Gel & Fountain pens for filling the OMR sheet., Darken the bubbles completely. Don’t put a tick mark or a cross mark, half-filled or over-filled bubbles will not be read, by the software., Correct, , ✔, , ✗, , Incorrect, , Incorrect, , Incorrect, , Do not write anything on the OMR Sheet., Multiple markings are invalid., , 1, , 18, , 35, , 2, , 19, , 36, , 3, , 20, , 37, , 4, , 21, , 38, , 5, , 22, , 39, , 6, , 23, , 40, , 7, , 24, , 41, , 8, , 25, , 42, , 9, , 26, , 43, , 10, , 27, , 44, , 11, , 28, , 45, , 12, , 29, , 46, , 13, , 30, , 47, , 14, , 31, , 48, , 15, , 32, , 49, , 16, , 33, , 50, , 17, , 34, , Check Your Performance, Total Questions:, Total Correct Questions:, , If Your Score is, , Score Percentage =, , Total Correct Questions, Total Questions, , × 100, , Less than 60%, > Average (Revise the concepts again), Greater than 60% but less than 75% > Good (Do more practice), Above 75%, > Excellent (Keep it on)

Page 110 :
102, , CBSE Sample Paper Mathematics Class XII (Term I), , Answers, 1. (a), 11. (d), 21. (d), , 2. (b), 12. (c), 22. (b), , 3. (d), 13. (a), 23. (d), , 4. (a), 14. (a), 24. (d), , 5. (b), 15. (b), 25. (a), , 6. (b), 16. (c), 26. (d), , 7. (a), 17. (b), 27. (a), , 8. (b), 18. (b), 28. (b), , 9. (a), 19. (b), 29. (b), , 10. (b), 20. (b), 30. (d), , 31. (a), 41. (c), , 32. (d), 42. (a), , 33. (b), 43. (d), , 34. (c), 44. (a), , 35. (b), 45. (a), , 36. (c), 46. (c), , 37. (a), 47. (b), , 38. (a), 48. (a), , 39. (a), 49. (d), , 40. (b), 50. (b), , SOLUTIONS, 1. Given objective function, Z = x + y, Corner points, , Value of, Z =x + y, , æ 50 40 ö, ç , ÷, è 3 3ø, , 30 (Maximum), , (0,0), , 0, , (25, 0), , 25, , (0, 20), , 20, , Thus, maximum Z = 30 occurs at point, æ 50 40 ö, ç , ÷., è3 3 ø, 2. Given that, x = 3 t 2 + 13 t + 8, y = 12 t 2 + 2 t - 15, dx, Now,, = 6 t + 13, dt, dy, and, = 24 t + 2, dt, \ Slope of the tangent to the curve,, dy, dy dt 24 t + 2, =, =, dx dx 6 t + 13, dt, 24( 5 ) + 2 122, æ dy ö, =, =, ç ÷, è dx ø t = 5 6( 5 ) + 13, 43, and, , SAMPLE PAPER 4, , 3. Given, equation of curve, ay 2 = x 3, On differentiating w.r.t. y, we get, dx, 2 ay = 3 x 2, dy, dx 2 ay, =, Þ, dy 3 x 2, - dx - 2 ay, Slope of normal =, =, dy, 3x2, æ - dx ö, - 2 ´ a2m3 - 2, =, =, ç, ÷, 3m, è dy ø( am 2 , am 3 ), 3 ´ a2m4, , 4. Given, f ( x ) = x 2 + ax + 1 Þ f ¢ ( x ) = 2 x + a, In interval (1, 2) ,, 1 < x < 2 Þ 2 < 2x < 4, Þ, (2 + a) < (2 x + a) < ( 4 + a), Since, f ( x ) is strictly increasing function, then, f ¢( x ) > 0 Þ (2 + a) > 0, Þ, a > -2, Hence, the least value of a = -1., ì x 2 - ( k + 2 )x + 2 k, ï, , x¹2, 5. Given, f ( x ) = í, x -2, ï, x=2, 3,, îï, Now, since f ( x ) is continuous at x = 2., …(i), \, lim f ( x ) = f (2 ), x ®2, , x2 - (k + 2 )x + 2 k, x ®2, x ®2, x -2, 2, ( h + 2 ) - ( k + 2 )( h + 2 ) + 2 k, = lim, h ®0, (h + 2 ) - 2, , Now, lim f ( x ) = lim, , h 2 + 4 + 4 h - ( hk + 2 k + 2 h + 4 ) + 2 k, h ®0, h, 2, h + 2 h - hk, = lim, h ®0, h, = lim ( h + 2 - k ), , = lim, , h ®0, , = 0 +2 - k = 2 - k, and f (2 ) = 3, \ From Eq. (i),, 2 - k = 3 Þ k = -1, 6. It is given that A is a skew-symmetric matrix of, order 3., a bù, é 0, So, let A = ê -a 0 g ú be a skew-symmetric, ê, ú, êë -b -g 0 úû, matrix., 0, a b, \ |A| = -a, -b, , 0, , g, , -g, , 0

Page 111 :
103, , CBSE Sample Paper Mathematics Class XII (Term I), , Now, expanding along R1 we get, |A| = 0(0 + g 2 ) - a (0 + bg ) + b(ag - 0 ), = - abg + abg = 0, é x + y k ù é6 5ù, 7. Given, ê, ú=ê, ú, ë x - y 1û ë 4 1û, On comparing both the matrices, we get, …(i), x+ y=6, …(ii), x-y=4, Adding Eq. (i) and Eq. (ii), we get, 2 x = 10 Þ x = 5, Now, from Eq. (i),, 5+ y=6 Þ y=6- 5=1, é1 4 ù, 8. Given, A = ê 5 0 ú, ê, ú, êë 6 -1úû, \, a22 = 0 and a31 = 6, Now, a22 + a31 = 0 + 6 = 6, é -1 9 2 ù, 9. Given, A = ê a 3 5 ú, ú, ê, êë b 0 4 úû, Also, given that A is upper triangular matrix, Þ, a=0= b, \ a+b=0+0=0, p öö, æ æ, 10. sin -1 ç sin ç p - ÷ ÷, è è, 4 øø, pö p, æ, [Q sin -1 (sin q ) = q ], = sin -1 ç sin ÷ =, è, 4ø 4, 11. Given, tan -1(1) = sin -1 x, p, p, 1, Þ, = sin -1 x Þ x = sin Þ x =, 4, 2, 4, 12. The relation S is defined on the set of integers, Z as xSy, if integer x divides integer y., Reflexive Since, every integer divides itself., \ Integer x divides integer x Þ xSx, Hence, S is reflexive., Symmetric Let x , y Î Z such that xSy, , Þ Integer x divides integer y and integer y, divides integer z., Þ Integer x divides integer z., Þ xSz, Hence, S is transitive., , 3 R3, where 1, 2 , 3 Î P., So, R is reflexive., Symmetric In given relation R , 1 R 3 Þ, / 3 R1, and 2 R 3 Þ, / 3 R2 ., So, R is not symmetric., Transitive In given relation R, 1 R2 and 2 R3, Þ 1 R 3., So, R is transitive., Thus, R is reflexive, transitive but not, symmetric., 14. Here, R1 Ì A ´ A., So, R1 is a relation on A., But (3, 6) Ï A ´ A, so R2 Ë A ´ A and hence R2, is not relation onA., 15. Let f ( x1 ) = f ( x2 ), Þ, |x1| = |x2| Þ x1 = ± x2, Therefore, the function is not one-one., Since, f ( x ) = |x| is always non-negative., \ Range of f = Set of non-negative integers, = Codomain of f, Þ f ( x ) is onto, Thus, f ( x ) = |x|is onto but not one-one., 16. Given, R = {( a, b) : a = b - 2 , b > 6 }., Now, since b > 6, so (2, 4) Ï R., Also, as 3 ¹ 8 - 2 , so (3, 8) Ï R and as 8 ¹ 7 - 2, Therefore, (8 , 7 ) Ï R., Now, consider (6, 8) we have 8 > 6 and also, 6 = 8 - 2., Therefore, (6, 8) Î R., 17. The area of the triangle with vertices, ( -1, 2 ), ( 4 , 0 ) and (3, 9) is given by, -1 2 1, 1, 4 0 1, k=, 2, 3 9 1, 1, |[ - 1(0 - 9 ) - 2 ( 4 - 3 ) + 1(36 - 0 )]|, 2, 1, = |9 - 2 + 36 |, 2, 43, sq units, =, 2, 43, Þk =, 2, =, , SAMPLE PAPER 4, , i.e. integer x divides integer y., But this does not implies that integer y divides, integer x., Thus, S is not symmetric., Transitive Let x , y , z Î Z such that xSy and ySz, , 13. Given, relation is, R = {(1, 2 ), (1, 3 ), (2 , 1), (1, 1), (2 , 2 ), (3 , 3 ), (2 , 3 )}, and P = {1, 2 , 3 }, Reflexive In given relation R, 1 R 1, 2 R 2 and

Page 112 :
104, , CBSE Sample Paper Mathematics Class XII (Term I), , 2, , ì x2 - 9, ï, 22. Given f ( x ) = í x - 3, ï2 x + k, îï, , 1 6, , 18. We have, D = - 1 4 2, 0, , 1 2, , Now,, , -1 2, , M12 =, , 0, , M13 =, , and, , 2, , x®3, , Now,, = -2 - 0 = - 2, , -1 4, = -1 - 0 = - 1, 0 1, , On differentiating w.r.t. x, we get, dy, 1, 1, =, +, dx 2 x + 1 2 x - 1, ö, æ, y, 1 ç x - 1 + x + 1÷, =, ÷ 2 x2 - 1, 2 çè, x2 - 1, ø, dy y, 2, x -1, =, dx 2, =, , 20. Given, y = log(cos x 2 ), On differentiating w.r.t. x, we get, dy, 1, d, =, (cos x 2 ), dx cos x 2 dx, d, 1, ( - sin x 2 ) ( x 2 ), =, dx, cos x 2, , x®3, , -, , x®3, , +, , Reflexive ( a - a) i.e. 0 is divisible by 5 for all, …(i), , SAMPLE PAPER 4, , x ® 3-, , = lim (3 - h + l ), h®0, , =3+ l, lim f ( x ) = lim (3 x - 5 ), , x®3, , +, , x®3, , +, , = lim [3(3 + h ) - 5 ] = 9 - 5 = 4, h®0, , Þ, f (3 ) = 4, From Eq. (i),, 3+ l = 4 Þ l =1, , Q Both the matrices are equal., \ Their corresponding elements will also be, equal., …(i), Þ, a-b=5, …(ii), Þ, 2 a - b = 12, Subtracting Eq. (i) from Eq. (ii), we get, a=7, \ From Eq. (i), 7 - b = 5 Þ b = 2, Now, again on comparing both sides, we get, 2 a + w = 15, Þ, 2 (7 ) + w = 15 Þ w = 15 - 14 = 1, Again, comparing both sides, we get, 2c + w = 3, Þ, 2c + 1 = 3, Þ, 2c = 2 Þ c = 1, 24. Given, R = {( a, b): a - b is divisible by 5}, , Now, lim f ( x ) = lim ( x + l ), x ® 3-, , x®3, , and, f (3 ) = 2 (3 ) + k = 6 + k, From Eq. (i),, 6 =6 + k Þk =0, é a - b 2c + wù é 5 3 ù, 23. Given, ê, ú=ê, ú, ë2 a - b 2 a + w û ë12 15 û, , y= x+1+ x-1, , = - tan x 2(2 x ) = - 2 x tan x 2, ìx+ l , x<3, ï, 21. Given, f ( x ) = í 4, , x=3, ï3 x - 5 , x > 3, îï, Q f ( x ) is continuous at x = 3., Þ, lim f ( x ) = lim f ( x ) = f (3 ), , x2 - 9, x®3 x -3, (x - 3) (x + 3), = lim, x®3, (x - 3), , lim f ( x ) = lim, , x®3, , = 2 (6 ) - 1( - 2 ) + 6( - 1), = 12 + 2 - 6 = 8, , Þ, , …(i), , = lim ( x + 3 ) = 3 + 3 = 6, , \ a11 M11 - a12 M12 + a13 M13, , 19. Given,, , , x=3, , Q f ( x ) is continuous at x = 3, \, lim f ( x ) = f (3 ), , a11 = 2 , a12 = 1 and a13 = 6, 4 2, M11 =, = 8 - 2 = 6,, 1 2, , Here,, , , x¹3, , a Î Z . So, R is reflexive., Symmetric Let ( a, b) Î R Þ ( a - b) is divisible, by 5., Þ -( b - a) is divisible by 5 Þ ( b, a) Î R, So, R is symmetric., Transitive Let ( a, b) Î R and (b, c) ÎR, Þ ( a - b) and ( b - c) are both divisible by 5., Þ a - b + b - c is divisible by 5., Þ ( a - c) is divisible by 5., Þ ( a, c) Î R, So, R is transitive., Thus, R is reflexive, symmetric and transitive., Hence, R is an equivalence relation.

Page 114 :
106, , CBSE Sample Paper Mathematics Class XII (Term I), , 34. Given, f ( x ) = x tan - 1 x, , 39. Given curve is y = 2 x 2 + 3 sin x, , On differentiating w.r.t. x, we get, æ 1 ö, d, ÷ + tan -1 x(1), f ¢ (x) =, ( x tan -1 x ) = x ç, 2, dx, è1 + x ø, Þ f ¢ (x) =, \, , f ¢ (1) =, x2, , x, 1 + x2, 1, , 1 + (1)2, , + tan - 1(1) =, , 1 p, +, 2 4, , \Slope of normal =, , y2, , = 10, a 2 b2, Now, differentiating both sides w.r.t.x, we get, 2 x 2 y æ dy ö, + 2 ç ÷ =0, a2, b è dx ø, , 35. Given,, , +, , + tan - 1 x, , 2, , 2 y æ dy ö - 2 x, dy - b x, = 2, ÷= 2 Þ, 2 ç, ø, è, dx, dx, b, a, a y, , Þ, \, , - b2 a 2, æ dy ö, = 2 2 = -1, ç ÷, è dx ø( a 2 , b 2 ), a b, , f ( x ) = x 2 e- x, d 2 -x, \, f ¢ (x) =, ( x e ) = x 2( - e- x ) + e- x (2 x ), dx, x(2 - x ), = xe- x ( - x + 2 ) =, ex, Since, ex > 0 " x Î R, and x(2 - x ) is > 0, when 0 < x < 2, \ f ¢ ( x ) > 0, when 0 < x < 2, \ f ( x ) is increasing in the interval (0, 2)., x 3, 37. Given, f ( x ) = +, 3 x, x2 - 9, d æx 3ö 1 3, f ¢ (x) =, \, ç + ÷= - 2 =, dx è 3 x ø 3 x, 3x2, 36. Given,, , Q x 2 - 9 < 0, when - 3 < x < 3, \ f ¢ (x) < 0 " x Î (- 3, 3), \ f ( x ) is decreasing in ( - 3 , 3 )., , SAMPLE PAPER 4, , 38. Given curve is x = a cos 3 q and y = a sin 3 q, On differentiating w.r.t. q, we get, dx, = - 3 a cos 2 q sin q, dq, dy, and, = 3 a sin 2 q cos q, dq, dy dy / dq, 3 a sin 2 q cos q, \, = - tanq, =, =, dx dx / dq - 3 a cos 2 q sin q, æ dy ö, æpö, = - tan ç ÷ = - 1, ç ÷, è4ø, è dx ø q = p, 4, , -1, -1, =1, \ Slope of normal =, =, æ dy ö, ( - 1), ç ÷, è dx ø q = p, 4, , On differentiating w.r.t. x, we get, dy, = 4 x + 3 cos x, dx, æ dy ö, Now, ç ÷, = 4(0 ) + 3 cos(0 ) = 3, è dx ø( 0, 0), -1, -1, =, æ dy ö, 3, ç ÷, è dx ø( 0, 0), , \Equation of normal is given by, -1, -x, y-0=, (x - 0) Þ y =, 3, 3, Þ, 3y = - x Þ x + 3 y = 0, 40. Given, A2 - 5 A + 7 I = O, Post-multiplying by A- 1 on both the sides, we, get, A2 A- 1 - 5 AA- 1 + 7 IA- 1 = OA- 1, Þ, A( AA- 1 ) - 5 I + 7 A- 1 = O [Q AA-1 = I ], Þ, AI - 5 I + 7 A- 1 = O, Þ, A - 5 I + 7 A- 1 = O, Þ, 7 A- 1 = 5 I - A, 1, \ A- 1 = ( 5 I - A), 7, 41. Given, f ( x ) = x 3 - 3 x 2 + 6 x + 3, d 3, (x - 3x2 + 6x + 3), f ¢ (x) =, \, dx, = 3x2 - 6x + 6, = 3( x 2 - 2 x + 2 ), For minimum or maximum values, f ¢ ( x ) = 0, \ x2 - 2 x + 2 = 0, \, , x=, , - ( -2 ) ± ( -2 )2 - 4 (1)(2 ), 2 (1), , 2 ± -4 2 ± 2 i, =, = 1± i, 2, 2, \ f ¢ ( x ) ¹ 0 for any real value of x., \ f ( x ) has neither maximum nor minimum, value., é- 1 - 2 - 2 ù, 42. Given, A = ê 2, 1 - 2ú, ú, ê, êë 2 - 2 1 úû, =, , Now, cofactors of A are, 1 -2, = 1 - 4 = - 3,, C11 =, -2 1, C12 = C13 =, , 2, 2, , -2, = -(2 + 4 ) = - 6, 1, , 2, , 1, , 2, , -2, , = - 4 - 2 = - 6,

Page 115 :
107, , CBSE Sample Paper Mathematics Class XII (Term I), , C21 = C22 =, , -2, , -2, , 1, , -1 -2, 2, , 1, , -2, 1, , C31 =, , 45. Given, f ( x ) = xex ( 1- x ), , = - 1 + 4 = 3,, , On differentiating w.r.t. x, we get, = -(2 + 4 ) = - 6, , -2, , 2, , \ f ¢ ( x ) ³ 0 " x Î [1, ¥ ), , = -( - 2 - 4 ) = 6, , \ f ( x ) is increasing in the interval [1, ¥ )., , -1 -2, , C23 = -, , -2, = (4 + 2 ) = 6, -2, , -1 -2, = -(2 + 4 ) = - 6, 2 -2, , C32 = C33 =, , -2, , -1 -2, 2, , 1, , = ( -1 + 4 ) = 3, , C13 ù é - 3 - 6 - 6 ù, C23 ú = ê 6, 3 - 6ú, ú, ú ê, C33 úû êë 6 - 6 3 úû, 6 ù, é- 3 6, ê, Þ adj A = - 6 3 - 6 ú, ú, ê, êë - 6 - 6 3 úû, , é C11, \ adj A = ê C21, ê, êë C31, , C12, C22, C32, , 2, , T, , 2, , 43. Given, f ( x ) = (1 + b ) x + 2 bx + 1, On differentiating w.r.t. x, we get, f ¢ ( x ) = 2 x(1 + b2 ) + 2 b, For minimum, put f ¢( x ) = 0, Þ, 2 x (1 + b2 ) + 2 b = 0, b, \ x=(1 + b2 ), Now, f ¢ ¢ ( x ) = 2 (1 + b2 ) is always positive, so, that f ( x ) is minimum when, b, x=1 + b2, b2, 2 b2, \Minimum f ( x ) = (1 + b2 ), +1, (1 + b2 )2 (1 + b2 ), =, , \, , (1 + b2 ) - b2, , f ¢ ( x ) = ex( 1- x ) × 1 + x × ex( 1- x )(1 - 2 x ), Þ f ¢ ( x ) = ex ( 1 - x )(1 + x - 2 x 2 ), Þ f ¢( x ) = - ex ( 1 - x )( x - 1)(2 x + 1), 1ö, æ, Þ f ¢ ( x ) = - 2 ex ( 1 - x ) ç x + ÷( x - 1), è, 2ø, x (1 - x ), Þ f ¢ (x) = - 2 e, A, Since, exponential function is always positive, and the sign of f ¢( x ) will be opposite to the, é 1 ù, sign of A which is negative in - , 1 ., êë 2 úû, é 1 ù, Hence, f ¢( x ) is positive in - , 1 , so that f ( x ), êë 2 úû, is an increasing function in this interval., 46. Given, equation of lines 3 x + 2 y = 13, , and, …(ii), 2x - y = 4, The point D is the intersection point of the, above two lines., Multiply by 2 from Eq. (ii), 4 x - 2 y = 8 …(iii), Adding Eqs. (i) and (ii), we get, 7 x = 21 Þ x = 3, From Eq. (ii),, 2x - 4 = y Þ y = 2 ´3 - 4 = 2, Hence, the coordinates of point D is (3 , 2 )., 47. Since, shaded region is OABCDEO., The number of corner points, n = 6, \ ( n - 1)2 = (6 - 1)2 = 5 2 = 25, 48. Value of objective function at all corner points., Corner points, , 1, , =, 1 + b2, (1 + b2 ), 1, m ( b) =, = + ve, 1 + b2, , 15 (0 ) - 4 (0 ) = 0, , A( 5, 0 ), , 15 ´ 5 - 4 ´ 0 = 75, , C (4 , 1), , 15 ´ 4 - 4 ´ 1 = 56, , D( 3, 2 ), , 15 ´ 3 - 4 ´ 2 = 37, , E (0, 2 ), , 15 ´ 0 - 4 ´ 2 = -8, (Minimum), , \ f ¢ ( x ) = 3 ( x + 1)2( x - 3 )3 + ( x + 1)33( x - 3 )2, = 3( x + 1)2( x - 3 )2[( x - 3 ) + ( x + 1)], 2, , 2, , = 6( x + 1) ( x - 3 ) ( x - 1), Q ( x + 1)2( x - 3 )2( x - 1) ³ 0 "x Î [1, ¥ ), , 1, = 88, 2, (Maximum), , 15 ´ 6 - 4 ´, , \ Maximum of Z is 88., 49. Minimum of Z is -8., 50. Z æ, , 1ö, ç 6, ÷, è 2ø, , + Z ( 0, 2) = 88 + ( -8 ) = 80, , SAMPLE PAPER 4, , 44. Given, f ( x ) = ( x + 1)3( x - 3 )3, , Value of Z = 15 x - 4 y, , O(0, 0 ), , æ 1ö, B ç6, ÷, è 2ø, , Clearly, m( b) is always greater than zero and, less than or equal to 1., So, the range of m ( b) is (0, 1]., , = 3( x + 1)2( x - 3 )2(2 x - 2 ), , …(i)

Page 116 :
108, , CBSE Sample Paper Mathematics Class XII (Term I), , SAMPLE PAPER 5, MATHEMATICS, A Highly Simulated Practice Questions Paper, for CBSE Class XII (Term I) Examination, , Instructions, 1. This question paper contains three sections - A, B and C. Each section is compulsory., 2. Section - A has 20 MCQs, attempt any 16 out of 20., 3. Section - B has 20 MCQs, attempt any 16 out of 20., 4. Section - C has 10 MCQs, attempt any 8 out of 10., 5. There is no negative marking., 6. All questions carry equal marks., Maximum Marks : 40, Time allowed : 90 min, , Roll No., , Section A, In this section, attempt any 16 questions out of Questions 1-20. Each question is of 1 mark weightage., , 1. The principal value of cot -1 (-1) is, (a), , p, 4, , (b) -, , p, 4, , é2 3 ù, - 1 3ù, and B = ê 4 - 2ú , then, ú, ê, 5 1úû, êë 1 5 úû, (a) only ABis defined, (c) ABand BA both are defined, , (c), , 3p, 4, , (d) None of these, , SAMPLE PAPER 5, , é 2, 2. If A = ê, ë- 4, , 3. If, , 1 + sin x + 1 - sin x, 1 + sin x - 1 - sin x, , (a) cot -1 q, , (b) only BA is defined, (d) ABand BA both are not defined, , = q, then the value of x is, , (b) 2 cot -1 q, , (c) sin -1 q, , (d) cos -1 q, , 4. If product of rows and colums of matrix is 8, then number of possible different ordered, matrices are, (a) 4, , (b) 3, , (c) 1, , (d) 2, , 2p ö, ÷ is, è 7 ø, , 5. The interval of increase of the function f (x) = x - e x + tan æç, (a) ( -¥ , 0), , (b) (0, ¥ ), , (c) (1, ¥ ), , (d) ( -¥ , 1)

Page 117 :
109, , CBSE Sample Paper Mathematics Class XII (Term I), , é 2 3ù, é 1 3 2ù, é1ù, é 4 6 8ù, ,B=ê, , C = ê ú and D = ê, ú, ú, ú , then which of the following is, ë1 2û, ë 4 3 1û, ë 2û, ë 5 7 9û, defined?, (a) A + B, (b) B + C, (c) C + D, (d) B + D, dy, sin x, p, 7. If y =, , then, at x = is equal to, 1 + sin x, 2, dx, , 6. If A = ê, , (a) 1, , 8. If, , (b) 0, , (c) 2, , (d) 3, , (c) - 6, , (d) 0, , x 2, 6 2, , then x is equal to, =, 18 x, 18 6, (b) ± 6, , (a) 6, , ì, 9. If f (x) = ïí 2x + cos 2x , x ¹ 0 is continuous at x = 0, then the value of a is, sin 4 x, , x=0, , a,, , ïî, , (a) 1, , 10. If f (x) =, , (b) 4, , 1 - cos x, x2, , (d) -1, , is continuous at x = 0, then f ( 0) is equal to, , (a) 1, , 11. If y = sin 3 2x, then, (a) 0, , (c) 3, , (b), , 1, 2, , (c), , dy, p, at x = is equal to, dx, 2, (b) 1, , 3, 2, , (c) -1, , (d) 4, , (d) 3, , 12. If the area of a DABC, with vertices A(1, 3), B(0, 0) and C(k , 0) is 3 sq units, then the value, k, is, 2, (a) ±2, , of, , (b) ±1, , é, , æ, , ë, , è, 2p, (b), 3, , (c) 4, , 13. The value of tan - 1 ê 2 sin ç 2 cos - 1, p, (a), 3, , 3 öù, ÷ ú is, 2 øû, (c), , 13p ö ù, ÷ is, è 6 ø úû, p, (b), 6, , -p, 3, , (d) 5, , (d), , p, 6, , (d), , 2p, 3, , 14. The value of cos -1 éê cos æç, 13p, (a), 6, , ë, , (c), , p, 3, , é1 6 1ù, ú, ê, êë 2 2 9úû, , (a) 5, , (b) -5, , (c) 7, , (d) 8, , 16. If the points (2, - 3),(k , - 1) and (0, 4) are collinear, then the value of k is, (a), , 10, 7, , (b), , 7, 140, , (c) 47, , (d), , 40, 7, , SAMPLE PAPER 5, , 15. The minor of a 32 of the matrix ê 5 3 0ú is

Page 118 :
110, , CBSE Sample Paper Mathematics Class XII (Term I), , 17. If y = (1 + x 1/ 6 )(1 + x 1/ 3 )(1 - x 1/ 6 ), then, (a), , 2, 3, , (b) -, , dy, dx, , 2, 3, , at x = 1 is equal to, (c) 3, , (d), , -4, 3, , 18. The conditions x ³ 0, y ³ 0 are called, (a) restrictions only, (c) non-negative restrictions, , (b) negative restrictions, (d) None of these, , 1, 19. The sum of minor of 6 and cofactor of 4 respectively in the determinant D = 4, 7, (a) 0, , (c) -1, , (b) 1, , 2 3, 5 6 is, 8 9, , (d) 4, , 20. Let S be any set and P(S) be its power set. We define a relation R on P(S) by, , A RB, , which, , mean A Í B "A, B Î P(S). Then, R is, (a) equivalence relation, (b) only reflexive and transitive, (c) only reflexive and symmetric, (d) None of the above, , Section B, In this section, attempt any 16 questions out of Questions 21-40. Each question is of 1 mark weightage., , ìx + 1, if x is odd, , then f is, îïx - 1, if x is even, , 21. f : N ® N where f (x) = í, (a) one-one and into, (c) one-one and onto, , (b) many-one and into, (d) many-one and onto, , 22. If y = log x x , then the value of, (a) x x (1 + log x), , 23. If y = sin x + y, then, (a), , cos x, 2y - 1, , dy, dx, , is, , (b) log ( ex), , dy, dx, , (c) log, , e, x, , æ xö, (d) log ç ÷, è eø, , is equal to, , (b), , cos x, 1 - 2y, , (c), , sin x, 1 - 2y, , (d), , sin x, 2y - 1, , SAMPLE PAPER 5, , 24. The set of points, where the function f given by f (x) =|2x - 1|sin x is differentiable, is, ì 1ü, (b) R - í ý, îï 2 þ, , (a) R, , (c) (0 , ¥ ), , (d) None of these, , (c) l ¹ - 2, , (d) None of these, , 2 l -3, 25. If A = 0 2 5 , then A -1 exists, if, 1 1, (a) l = 2, , 3, (b) l ¹ 2, , 26. If x = a secq and y = a cot q, then, (a) - 2, , (b), , 2, , dy, dx, , at q =, , p, is equal to, 4, (c) 1, , (d) -1

Page 119 :
111, , CBSE Sample Paper Mathematics Class XII (Term I), , 27. Total number of possible matrices of order 3 ´ 3 with each entry 2 or 0 is, (a) 9, , (b) 27, , (c) 81, , (d) 512, , 28. If A and B are square matrices of the same order and AB = 3I , then A - 1 is equal to, (a) 3 B, , (b), , 1, B, 3, , (c) 3 B- 1, , (d), , 1 -1, B, 3, , 29. The maximum value of Z = 2x + 4 y, if the feasible region for an LPP is as shown below,, is, , Y, (0, 20), , C(0, 12), , B(8, 8), (15, 0), , O, , (a) 56, , 30. The curve y = x, , (b) 50, 1/ 5, , X, , A, (10, 0), , (c) 36, , (d) 55, , has at (0, 0), , (a) a vertical tangent (parallel to Y-axis), (c) an oblique tangent, , (b) a horizontal tangent (parallel to X-axis), (d) no tangent, , 31. The area of the triangle whose vertices (-2, 6), (3, - 6) and (1, 5) is, (a) 30 sq units, (c) 40 sq units, , (b) 35 sq units, (d) 15.5 sq units, , 32. Which of the given values of x and y make the following pair of matrices equal, 5 ù é 0 y - 2ù, é 3x + 7, ?, ê y + 1 2 - 3xú , ê 8, 4 úû, û ë, ë, -1, (a) x =, and y = 7, 3, -2, (c) y = 7 and x =, 3, , 33. If y = cos -1 x , then the value of, , (b) not possible to find, (d) x =, , d 2y, dx 2, , in terms of y alone is, (b) cosec y cot 2 y, (d) None of these, , 34. The interval in which the function f (x) = 2x 3 + 9x 2 + 12x - 1 is decreasing, is, (a) [ - 1, ¥ ), , (b) [ - 2 , - 1], , (c) ( - ¥ , - 2 ], , (d) [ - 1, 1], , 35. Let R be the relation on the set R of real numbers defined by R = {(a , b)|1 + ab > 0}., Then, R is, (a) reflexive, symmetric but not transitive, (b) reflexive, transitive but not symmetric, (c) transitive but not symmetric and reflexive (d) an equivalence relation, , SAMPLE PAPER 5, , (a) - cot y cosec 2 y, (c) - cot y cosec y, , -1, -2, and y =, 3, 3

Page 120 :
112, , CBSE Sample Paper Mathematics Class XII (Term I), , 36. If f is a function from the set of natural numbers to the set of even natural numbers, given by f ( x) = 2x. Then, f is, (a) one-one but not onto, (b) onto but not one-one, (c) Both one-one and onto, (d) Neither one-one nor onto, , 37. Corner points of the feasible region for an LPP are (0, 5), (6, 0), (6, 8), (0,2), (3, 0). Let, Z = 2x + 3y be the objective function. The minimum value of Z occurs at, (a) only (0, 2), (b) only (3, 0), (c) the mid-point of the line segment joining the points (0, 2) and (3, 0), (d) any point on the line joining the points (0, 2) and (3, 0), dy, 38. If y + sin y = cos x, then is equal to, dx, sin x, sin x, (a) , y ¹ (2 n + 1)p, , y = (2 n + 1)p, (b), 1 + cos y, 1 + cos y, sin x, , y ¹ (2 n + 1)p, (d) None of these, (c) 1 + cos y, dy, p, at x =, is equal to, dx, 2, (b) -p, (c) p, , 39. If y = 1 + cos 2 (x 2 ), then, (a) p, , é cos a, ë sin a, , 40. If A = ê, (a), , p, 6, , (d) - p, , - sin aù, , such that A + A ¢ = I , then the value of a is, cos aúû, p, (b), (c) p, 3, , (d), , 3p, 2, , Section C, In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , 41. The feasible region for an LPP is shown in the following figure. Then, the minimum, value of Z = 11x + 7 y is, , Y, , SAMPLE PAPER 5, , C(0, 5), (0, 3) B, , A(3, 2), , X′, Y′, , (a) 21, , (b) 47, , x+, 3, x+y=5, , X, , y=, 9, , (c) 20, , (d) 31, , 42. The tangent to the curve y = e 2x at the point (0, 1) meets X-axis at, (a) (0, 1), , æ 1 ö, (b) ç - , 0÷, è 2 ø, , (c) (2, 0), , (d) (0, 2 )

Page 121 :
113, , CBSE Sample Paper Mathematics Class XII (Term I), , 43. The function f (x) = 4 sin 3 x - 6 sin 2 x + 12 sin x + 100 is strictly, æ 3p ö, (a) increasing in ç p , ÷, è 2 ø, é -p p ù, (c) decreasing in ê, ,, ë 2 2 úû, , æp ö, (b) decreasing in ç , p ÷, è2 ø, é pù, (d) decreasing in ê0, ú, ë 2û, , 44. The function f : R ® R defined by f (x) = 4 x + 4|x| is, (a) one-one and into, (c) one-one and onto, , (b) many-one and into, (d) many-one and onto, , 45. An optimisation problem may involve finding, (a) maximum profit, (c) minimum use of resources, , (b) minimum cost, (d) All of these, , CASE STUDY, P( x) = - 6x 2 + 120x + 25000 (in `) is the total profit function of a company where x denotes the, production of the company., , Based on the above information, answer the following questions., , 46. When the profit is maximum, production will be, (a) 8, , (b) -8, , (c) 10, , (d) -10, , 47. The interval in which the profit is strictly increasing in, (a) (0, 10), , (c) (10, ¥), , (d) (12, ¥), , (b) ` 25500, , (c) ` 25550, , (d) ` 25600, , (b) 60, , (c) 80, , (d) 100, , (b) (0, 12), , 48. The maximum profit is, (a) ` 25450, , 49. Value of P ¢(5) is, 50. When the production is 3 units, the profit of the company will be, (a) ` 25106, , (b) ` 25206, , (c) ` 25306, , (d) ` 24306, , SAMPLE PAPER 5, , (a) 40

Page 122 :
OMR SHEET, , SP 5, , Roll No., Sub Code., , Student Name, , Instructions, Use black or blue ball point pens and avoid Gel & Fountain pens for filling the OMR sheet., Darken the bubbles completely. Don’t put a tick mark or a cross mark, half-filled or over-filled bubbles will not be read, by the software., Correct, , ✔, , ✗, , Incorrect, , Incorrect, , Incorrect, , Do not write anything on the OMR Sheet., Multiple markings are invalid., , 1, , 18, , 35, , 2, , 19, , 36, , 3, , 20, , 37, , 4, , 21, , 38, , 5, , 22, , 39, , 6, , 23, , 40, , 7, , 24, , 41, , 8, , 25, , 42, , 9, , 26, , 43, , 10, , 27, , 44, , 11, , 28, , 45, , 12, , 29, , 46, , 13, , 30, , 47, , 14, , 31, , 48, , 15, , 32, , 49, , 16, , 33, , 50, , 17, , 34, , Check Your Performance, Total Questions:, Total Correct Questions:, , If Your Score is, , Score Percentage =, , Total Correct Questions, Total Questions, , × 100, , Less than 60%, > Average (Revise the concepts again), Greater than 60% but less than 75% > Good (Do more practice), Above 75%, > Excellent (Keep it on)

Page 123 :
115, , CBSE Sample Paper Mathematics Class XII (Term I), , Answers, 1. (c), 11. (a), 21. (c), , 2. (c), 12. (b), 22. (b), , 3. (b), 13. (a), 23. (a), , 4. (a), 14. (b), 24. (b), , 5. (a), 15. (b), 25. (d), , 6. (d), 16. (a), 26. (a), , 7. (b), 17. (b), 27. (d), , 8. (b), 18. (c), 28. (b), , 9. (c), 19. (a), 29. (a), , 10. (b), 20. (b), 30. (a), , 31. (d), 41. (a), , 32. (b), 42. (b), , 33. (a), 43. (b), , 34. (b), 44. (a), , 35. (a), 45. (d), , 36. (c), 46. (c), , 37. (d), 47. (a), , 38. (c), 48. (d), , 39. (d), 49. (b), , 40. (b), 50. (c), , SOLUTIONS, 1. Let cot -1( -1) = q Þ cotq = - 1 and 0 < q < p, 3p, q=, \, 4, 3p, -1, \ cot ( -1) =, 4, , æ2 p ö, 5. We have, f ( x ) = x - ex + tan ç ÷, è 7 ø, On differentiating w.r.t. x, we get, f ¢ ( x ) = 1 - ex, For f ( x ) to be increasing, we must have, , 2. Let A = [ aij ]2 ´ 3 and B = [ bij ]3 ´ 2., Since, number of columns of A = number of, rows of B, \ AB is defined, Also, as number of columns of B = number of, rows of A, \ BA is defined., Hence, both AB and BA are defined., 1 + sin x + 1 - sin x, 3. Given,, =q, 1 + sin x - 1 - sin x, x, x, x, x, + sin 2 + 2 sin cos, 2, 2, 2, 2, x, x, x, x, + cos 2 + sin 2 - 2 sin cos, 2, 2, 2, 2 =q, x, x, 2x, 2x, cos + sin, + 2 sin cos, 2, 2, 2, 2, x, x, x, x, - cos 2 + sin 2 - 2 sin cos, 2, 2, 2, 2, x, x, x, x, cos + sin + cos - sin, 2, 2, 2, 2 =q, x, x, x, x, cos + sin - cos + sin, 2, 2, 2, 2, é, x, xù, 2x, 2x, êQ cos + sin + 2 sin cos ú, 2, 2, 2, 2, ú, ê, cos 2, , Þ, , Þ, , 2, x, xö ú, æ, = ç cos + sin ÷ ú, è, 2, 2ø û, , x, 2 = q Þ cot x = q Þ x = 2 cot -1 q, Þ, x, 2, 2 sin, 2, 2 cos, , 4. Possible different orders are given by, 1 ´ 8, 2 ´ 4, 4 ´ 2 , 8 ´ 1, Therefore, number of possible different, ordered matrices are 4., , Þ, , 1 - ex > 0, , Þ, , ex < 1, , Þ, , x<0, , Þ, , x Î ( -¥ , 0 ), , 6. Only B + D is defined because matrices of the, same order can only be added., sin x, 7. Given, y =, 1 + sin x, On differentiating w.r.t. x, we get, dy (1 + sin x )cos x - sin x(cos x ), =, dx, (1 + sin x )2, dy cos x + sin x cos x - sin x cos x, =, dx, (1 + sin x )2, cos x, =, (1 + sin x )2, p, cos, 0, æ dy ö, 2, Now, ç ÷, =, =, =0, 2, è dx ø x = p æ, (, 1, 1)2, +, pö, ç1 + sin ÷, 2, è, 2ø, 8. Given,, Þ, , x 2, 6 2, =, 18 x 18 6, x 2 - 36 = 36 - 36, , x 2 - 36 = 0, x 2 = 36, x= ±6, ìï sin 4 x, + cos 2 x , x ¹ 0, 9. Given, f ( x ) = í 2 x, x=0, a,, ïî, ï, Þ, Þ, Þ, , Also, f ( x ) is continuous at x = 0., , \ lim f ( x ) = f (0 ), x ®0, , SAMPLE PAPER 5, , ê, ê, ë, , f ¢ (x) > 0

Page 126 :
118, , CBSE Sample Paper Mathematics Class XII (Term I), , Now,, , 5 ù é0 y - 2 ù, é3 x + 7, 32. Consider, ê, ú=ê, 4 úû, ë y + 1 2 - 3 x û ë8, , dy ( dy / dq ) - acosec2q, =, =, dx ( dx / dq ) a secq tan q, =-, , æ dy ö, \ç ÷, è dx ø q = p, , 4, , 1, 2, , =, , sin q tan q, -1, =- 2, =, æ 1 ö, ç, ÷, è 2ø, , - cot 2 q, sin q, , On equating the corresponding elements,, we get, , 27. Number of entries in 3 ´ 3 matrix is 9., Since, each entry has 2 choices, namely 2 or 0., Therefore, number of possible matrices, 9, , = 2 ´ 2 ´ 2 ... ´ 2 = 2 = 512, 1442443, 9 times, , AB = 3 I, 1, ( AB) = I, 3, 1, æ1 ö, Aç B÷ = I Þ A- 1 = B, è3 ø, 3, , 28. Given,, Þ, Þ, , 29. Since, the feasible region is bounded., Therefore, maximum of Z must occurs at the, corners points of the feasible region., Corner points Value of Z = 2 x + 4 y, O(0, 0 ), , 4 (0 ) + 3(0 ) = 0, , A(10, 0 ), , 4 (10 ) + 3(0 ) = 40, , B(8, 8 ), , 4 (8 ) + 3(8 ) = 56 (Maximum), , C (0, 12 ), , 4 (0 ) + 3(12 ) = 36, , Hence, the maximum value of Z is 56., 30. We have,, , y = x 1/ 5, , On differentiating w.r.t. x, we get, 1, 1 5-1, x, , \, , dy, 1, = x -4 / 5, =, 5, dx 5, 1, æ dy ö, = ´ (0 )-4/ 5 = ¥, ç ÷, è dx ø( 0, 0) 5, , SAMPLE PAPER 5, , So, the curve y = x 1/ 5 has a vertical tangent at, (0, 0), which is parallel to Y-axis., 31. Let D be the area of the triangle., -2 6 1, 1, \ D=, 3 -6 1, 2, 1, 5 1, Now, expanding along R1, 1, D = |-2 ( -6 - 5 ) - 6(3 - 1) + 1(15 + 6 )|, 2, 1, 1, = |22 - 12 + 21|= |31| = 15.5 sq units, 2, 2, , 3 x + 7 = 0; 5 = y -2 ; y + 1 = 8 and 2 - 3 x = 4, -7, -2, ; y = 7; y = 7 and x =, Þ, x=, 3, 3, Hence, we have two different values of x,, which is not possible., 33. Given, y = cos -1 x, Þ, x = cos y, On differentiating w.r.t. y, we get, dx, = - sin y, dy, dy, …(i), = - cosec y, Þ, dx, Again, differentiating w.r.t. x, we get, d 2y, dy, d, =, ( - cosec y ) = - ( - cosec y cot y ), 2, dx, dx, dx, = cosec y cot y ( - cosec y ), [from Eq. (i)], = - cot y × cosec2 y, 34. We have, f ( x ) = 2 x 3 + 9 x 2 + 12 x - 1, On differentiating w.r.t. x, we get, f ¢ ( x ) = 6 x 2 + 18 x + 12, = 6 ( x 2 + 3 x + 2 ) = 6 ( x + 2 )( x + 1), So, f ¢ ( x ) £ 0 , for decreasing., On drawing number lines as below, –, , +, –2, , +, –1, , We see that f ¢( x ) is decreasing in [- 2 , - 1 ]., 35. Let a Î R be any element., Then, 1 + a × a = 1 + a2 > 0, [Q a2 ³ 0 , " a Î R ], Þ, ( a, a) Î R, Hence, R is reflexive., Let ( a, b) Î R., Then,, 1 + ab > 0, Þ, 1 + ba > 0, Þ, ( b, a ) Î R, Q, ( a , b) Î R Þ ( b, a ) Î R, Hence, R is symmetric., æ 1ö, æ1, ö, Now,, ç1, ÷ Î R and ç , - 1÷ Î R, è 2ø, è2, ø, 1 3, 1 1, As,, 1 + = > 0 and 1 - = > 0, 2 2, 2 2, But (1, - 1) Ï R, as 1 + (1)( -1) = 1 - 1 = 0 >| 0, Hence, R is not transitive.

Page 127 :
119, , CBSE Sample Paper Mathematics Class XII (Term I), , 36. Let A = {Set of even natural number}, Given, f : N ® A and f ( x ) = 2 x, For one-one Let x 1 , x 2 Î R, Considering f ( x 1 ) = f ( x 2 ), Þ, 2 x 1 = 2 x2 Þ x 1 = x 2, So, f is one-one., For onto Let y = f ( x ), y, Then,, y = 2x Þ x =, 2, \ x Î N for every y ÎA., [every element of codomain has, pre-image in domain], So, f is onto., Hence, f is both one-one and onto., , 40. Given, A + A¢ = I, é cos a - sin a ù é cos a, Þ ê, ú+ê, ë sin a cos a û ë - sin a, é2 cos a, Þ ê, ë 0, Þ, , Value of Z = 2 x + 3 y, , (0, 5), , 2 ´ 0 + 3 ´ 5 = 15, , (6, 0), , 2 ´ 6 + 3 ´ 0 = 12, , (6, 8), , 2 ´ 6 + 3 ´ 8 = 36, , (0, 2), , 2´0+ 3´2 = 6, (Minimum), , (3, 0), , 2´3+ 3´0 = 6, (Minimum), , Hence, minimum of Z occurs at any point on, the line joining the points (0, 2) and (3, 0)., 38. Given, y + sin y = cos x, On differentiating w.r.t. x, we get, dy, d, d, +, (sin y ) =, (cos x ), dx, dx dx, [by chain rule of derivative], dy, dy, Þ, + cos y ×, = - sin x, dx, dx, dy, sin x, =Þ, dx, 1 + cos y, where,, , y ¹ (2 n + 1)p, 2, , 2, , 39. Given, y = 1 + cos ( x ), , 41. The values of Z at the corner points are given by, Corner points, , Value of Z = 11 x + 7 y, , (3, 2), , 11( 3) + 7( 2) = 47, , (0, 3), , 11( 0) + 7( 3) = 21, (Minimum), , (0, 5), , 11( 0) + 7( 5) = 35, , 2, , æ 1 ö æ -1 ö, = 2ç, ÷ç, ÷( p ) = - p, è 2 øè 2 ø, , From the above table, we see that the, minimum value of Z is 21., 42. The equation of curve is y = e2x, On differentiating w.r.t. x, we get, dy, = e2 x × 2 = 2 × e2 x, dx, Since, it passes through the point (0, 1)., æ dy ö, = 2 × e2× 0 = 2 = Slope of tangent to, \ç ÷, è dx ø( 0, 1), the curve., \ Equation of tangent is y - 1 = 2 ( x - 0 ), Þ, y = 2x + 1, Since, tangent to curve y = e2x at the point, (0, 1) meets X-axis., i.e., y=0, 1, 2, æ -1 ö, So, the required point is ç , 0 ÷., è2 ø, , \, , 0 = 2x + 1Þ x = -, , 43. We have, f ( x ) = 4 sin 3 x - 6 sin 2 x + 12 sin x + 100, On differentiating w.r.t. x, we get, f ¢ ( x ) = 12 sin 2 x × cos x - 12 sin x × cos x + 12 cos x, = 12 [sin 2 x × cos x - sin x × cos x + cos x ], = 12 cos x [sin 2 x - sin x + 1], ...(i), Þ f ¢ ( x ) = 12 cos x [sin 2 x + (1 - sin x )], Q, 1 - sin x ³ 0 and sin 2 x ³ 0, \, sin 2 x + 1 - sin x ³ 0, Hence, f ¢ ( x ) > 0, when cos x > 0, æ p pö, i.e., x Î ç - , ÷., è 2 2ø, , SAMPLE PAPER 5, , On differentiating w.r.t. x, we get, dy, d, = 2 cos( x 2 ) ((cos( x 2 )), dx, dx, dy, Þ, = 2 cos( x 2 )( - sin( x 2 ))(2 x ), dx, p öæ p ö, æ dy ö, æ p öæ, Now, ç ÷, = 2 cos ç ÷ ç - sin ÷ ç2, ÷, p, è, ø, è, è dx ø x =, 4, 4 øè 2 ø, , 0 ù é1 0ù, =, 2 cos a úû êë0 1úû, 2 cosa = 1, 1, p, cosa = Þ a =, 2, 3, , Þ, , 37., Corner points, , sin a ù, cos a úû

Page 128 :
120, , CBSE Sample Paper Mathematics Class XII (Term I), , æ p pö, So, f ( x ) is increasing when x Î ç - , ÷ and, è 2 2ø, æ p 3p ö, f ¢ ( x ) < 0, when cos x < 0 i.e. x Î ç , ÷., è2 2 ø, æ p 3p ö, Hence, f ( x ) is decreasing when x Î ç , ÷, è2 2 ø, æ p ö æ p 3p ö, Since, ç , p ÷ Î ç , ÷, è2 ø è2 2 ø, æp ö, Hence, f ( x ) is decreasing in ç , p ÷., è2 ø, 44. Given, f : R ® R, , If x ³ 0,, Þ, , ì 4x + 4x ,, f (x) = í x, -x, îï 4 + 4 ,, ì 2 ( 4 x ),, ï, =í x 1, ïî 4 + 4 x ,, ï, , x³0, x<0, x³0, x<0, , f (x) = 2 ´ (4x ), f ( x ) Î [2 , ¥ ), , If x < 0, f ( x ) =, , 4 2x + 1, 4x, , >0, , Since, for different values of x, 2 ( 4 x ) and, æ x 1 ö, ç 4 + x ÷ are different positive numbers, è, 4 ø, \ f is one one., Also, f is not onto as its range is (0 , ¥ ) and it is, subset of its codomain R., , SAMPLE PAPER 5, , 45. An optimisation problem may involve finding, maximum profit, minimum cost, or minimum, use of resources etc., , 46. Total profit function,, P( x ) = - 6 x 2 + 120 x + 25000, On differentiating w.r.t. x, we get, P¢ ( x ) = -12 x + 120, For maximum profit, put P¢ ( x ) = 0, Þ, -12 x + 120 = 0, Þ, 12 x = 120, Þ, x = 10, Now, P¢ ¢ ( x ) = -12 < 0, At x = 10, profit function is maximum., 47. For strictly increasing, we must put P¢ ( x ) > 0, Þ -12 x + 120 > 0, Þ, 120 > 12 x, Þ, 12 x < 120, Þ, x < 10, \ x Î(0 ,10 ), when the profit is strictly, increasing., 48. Maximum profit is at critical point., \ P(10 ) = - 6 (10 )2 + 120 (10 ) + 25000, = -600 + 1200 + 25000, = ` 25600, or P(10 ) = ` 25600, 49. P¢( 5 ) = -12 ´ 5 + 120, = -60 + 120, = 60, 50. At x = 3,, P(3 ) = - 6(3 )2 + 120 ´ 3 + 25000, = - 54 + 360 + 25000 = ` 25306

Page 129 :
121, , CBSE Sample Paper Mathematics Class XII (Term I), , SAMPLE PAPER 6, MATHEMATICS, A Highly Simulated Practice Questions Paper, for CBSE Class XII (Term I) Examination, , Instructions, 1. This question paper contains three sections - A, B and C. Each section is compulsory., 2. Section - A has 20 MCQs, attempt any 16 out of 20., 3. Section - B has 20 MCQs, attempt any 16 out of 20., 4. Section - C has 10 MCQs, attempt any 8 out of 10., 5. There is no negative marking., 6. All questions carry equal marks., Maximum Marks : 40, Time allowed : 90 min, , Roll No., , Section A, In this section, attempt any 16 questions out of Questions 1-20. Each question is of 1 mark weightage., , 1. If A is a 3 ´ 2 matrix, B is a 3 ´ 3 matrix and C is a 2 ´ 3 matrix, then the elements in A, B, and C are respectively, (a) 6, 9, 8, (b) 6, 9, 6, , (c) 9, 6, 6, , 2. If x = a sin q and y = a cos 2 q, then, (a) -2 cos q, , dy, dx, , (d) 6, 6, 9, , is equal to, , (b) 2 cos q, , (d) -2 sinq, , (c) - 1, , (d) None of these, , 3. The value of cosæç sin -1, è, , (a) 0, , 1, 1 ö, + cos -1, ÷ is, 2, 2ø, (b) 1, , (c) 2 sinq, , æ, è, , 2p, (a) 3, , 3ö, ÷ is, 2 ø, , p, (b) 3, , é1 -1ù, 2, ú , then A is, 0, 4, û, ë, 5ù, é 1 -5 ù, (b) ê, ú, 16úû, ë0 16 û, , (c), , 4p, 3, , (d), , 5p, 3, , 5. If A = ê, é1, (a) ê, ë0, , é0 16 ù, (c) ê, ú, ë 1 -5 û, , é -5 1ù, (d) ê, ú, ë 16 0û, , SAMPLE PAPER 6, , 4. The principal value of sin -1 ç -

Page 130 :
122, , CBSE Sample Paper Mathematics Class XII (Term I), , é a bù é a - bù, is equal to, úê, aúû, ë - b aû ë b, 0 ù, ú, 2, a + b 2 úû, 0ù, ú, 0úû, , 6. The product ê, éa2, (a) ê, êë, éa2, (c) ê 2, êë a, , + b2, 0, + b2, + b2, , é( a + b) 2, (b) ê, 2, êë( a + b), é a 0ù, (d) ê, ú, ë0 bû, , 0ù, ú, 0úû, , 3ù, é 2, ú , then which of the following is true?, ë - 4 - 6û, (a) A( adj A) ¹ | A| I, (b) A( adj A) ¹ ( adj A) A, (c) A( adj A) = ( adj A) A = | A| I, (d) None of these, dy, 8. If x + y = 1, then at (5, 5) is equal to, dx, (a) 1, (b) 0, (c) 2, (d) - 1, , 7. If A = ê, , 9. If M 11 = - 40, M 12 = - 10 and M 13, , 1, 3 -2, = 35 of the determinant D = 4 - 5, 6 , then the, 3, , value of D is, (a) - 80, (b) 60, 1 a bc, 10. If D = 1 b ca , then the minor M 31 is, 1, , 5, , 2, , (c) 70, , (d) 100, , (c) c ( a 2 + b 2 ), , (d) c ( a 2 - b 2 ), , c ab, 2, , (a) - c ( a - b 2 ), , (b) c ( b 2 - a 2 ), , é 0 0 4ù, 11. The matrix P = ê 0 4 0ú is a, ê, ú, êë 4 0 0úû, (a) square matrix, (c) unit matrix, dy, 1, 12. If y = (x 2 + 1) 2 , then at x = is equal to, dx, 2, 5, (a), (b) 5, 2, , (b) diagonal matrix, (d) None of these, , (c), , 5, 4, , (d), , 2, 5, , 13. The equation of tangent to the curve y = x 2 + x - 2 at (1, 0) is given by, , SAMPLE PAPER 6, , (a) 3x - y = 3, , (b) 3x - y = -3, , (c) x - 3y = 1, , (d) None of these, , 14. Which one of the following statements is correct?, (a) e x is an increasing function, (b) e x is a decreasing function, (c) e x is neither an increasing nor a decreasing function, (d) e x is a constant function, , 15. If sin - 1 x = y, then, (a) 0 £ y £ x, , (b), , -p, 2, , £y£, , p, 2, , (c) 0 < y < p, , (d), , -p, 2, , <y<, , p, 2

Page 132 :
124, , CBSE Sample Paper Mathematics Class XII (Term I), , 26. The maximum value of, (a) 1, , (log x), , (b), , x, , is, , 2, e, , (c) e, , (d), , 1, e, , 27. If y = x (x - 3) 2 decreases for the values of x given by, (a) 1 < x < 3, , (b) x < 0, , (c) x > 0, , (d) 0 < x <, , 3, 2, , 28. The relation R in the set of natural numbers N defined as R = {(x, y) : y = x + 5 and x < 4}, is, (a) reflexive, , (b) symmetric, , (c) transitive, , (d) None of these, , 29. The maximum value of the function f (x) = x 3 + 2x 2 - 4 x + 6 exists at, (a) x = -2, , (b) x = 1, , (c) x = 2, , (d) x = -1, , 30. Area of the triangle whose vertices are (a , b + c), (b , c + a) and (c, a + b), is, (a) 2 sq units, , (b) 3 sq units, , (c) 0, , (d) None of these, , é1 1ù, ú with respect to, ë1 0û, , 31. The set of all 2 ´ 2 matrices which is commutative with the matrix ê, matrix multiplication is, é p qù, é p qù, (a) ê, (b) ê, ú, ú, r, r, û, ë, ë q rû, , é p - q pù, (c) ê, r úû, ë q, , q ù, ép, (d) ê, q, p, qúû, ë, , - 1ö, æ p öö, -1æ 1 ö, -1æ, ÷ + cot ç, ÷ + tan ç sin ç - ÷ø ÷ is, è è 2 ø, è 3ø, è 3ø, p, (b), 12, p, (d), 3, , 32. The value of tan - 1 æç, (a), , p, 6, , (c) -, , p, 12, , é1 1 ù, 2, ú and f ( x) = 1 - x , then f ( A) is, 0, 1, û, ë, 1ù, é0 0ù, é1 2 ù, (b) ê, (c) ê, ú, ú, 0úû, 0, 0, û, ë, ë3 4û, , 33. If A = ê, é1, (a) ê, ë0, , (d) None of these, , 34. For the set A = {1, 2, 3}, define a relation R in the set A as follows, , SAMPLE PAPER 6, , R = {(1, 1), (2, 2), (3, 3), (1, 3)}, Then, the ordered pair to be added to R to make it the smallest equivalence relation is, (a) (1, 3), (b) (3, 1), (c) (2, 1), (d) (1, 2), , a, , h, , 35. If D = h b, g, , f, , g, f , then the cofactor A31 is, c, , (a) - ( hc + fg), , 36. If y = log(sin e x ), then, (a) e x cot( e x ), (c) e x tan( e x ), , (b) hf - bg, , dy, dx, , (c) fg + hc, , is equal to, (b) -e x cot( e x ), (d) None of these, , (d) hc - fg

Page 133 :
125, , CBSE Sample Paper Mathematics Class XII (Term I), , 37. The number of all one-one functions from set A = {1, 2, 3} to itself is, (a) 2, , (b) 6, , (c) 3, , 38. If y = log a x + log x a + log x x + log a a, then, 1, + x log a, x, 1, (c), + x log a, x log a, , dy, dx, (b), , (a), , (d) 1, , is equal to, log a, x, , +, , x, log a, , (d) None of these, , 39. If y = ax 3 + bx 2 + cx + d, then, (a) 1, , d2y, dx, , 2, , at x =, , (b) 2, , -b, is equal to, 3a, (c) 3, , (d) 0, , 40. The differential coefficient of sin (cos(x 2 )) w.r.t. x is., (a) -2 x sin x 2 cos (cos x 2 ), (c) 2 x sin( x 2 ) cos ( x 2 ) cos x, , (b) 2 x sin( x 2 ) cos ( x 2 ), (d) None of these, , Section C, In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , 41. The function f (x) =, , 4 - x2, 4x - x 3, , is, , (a) discontinuous at only one point, (c) discontinuous at exactly three points, , (b) discontinuous at exactly two points, (d) None of these, , 42. The equation of the tangent to the curve x 2 - 2yx + y 2 + 2x + y - 6 = 0 at (2, 2) is, (a) 2 y + x = 6, (c) x + y = 4, , 43. If y x = e y - x , then, (a), (c), , (b) 2 x + y = 6, (d) x = y, , dy, dx, , is equal to, , 1 + log y, , (b), , y log y, 1 + log y, , (d), , (log y) 2, , 44. If y = Ae mx + Be nx and, , d2y, , - ( m + n), , dy, dx, , (a) 1, (c) -1, , (1 + log y) 2, log y, , + mny = k, then k is equal to, (b) 0, (d) None of these, , 45. If cos y = x cos(a + y) with cos a ¹ 1, then, (a), , y log y, , sin 2 ( a + y), sin a, 2, , (c) sin ( a + y) sin a, , dy, is equal to, dx, cos 2 ( a + y), (b), sin a, (d) None of these, , SAMPLE PAPER 6, , dx, , 2, , (1 + log y) 2

Page 134 :
126, , CBSE Sample Paper Mathematics Class XII (Term I), , CASE STUDY, If feasible solution of a LPP is given as follows:, Y, , 60, 2x + y = 80, C, , 50, 40, , 30, B(30, 20), , 20, , 10, X′, , O, , 10, , 20, , 30, , A, , 50, , Y′, , 46. The feasible solution consists, (b) (50, 10), (d) (30, 21), , 47. Objective function is maximum at the point, (a) (0, 0), (c) A, , (b) (30, 20), (d) C, , 48. Objective function has the value 420000 at, (a) point A, (c) point C, , (b) point B, (d) point O, , 49. Z| ( 20 , 20 ) - Z| ( 10 , 10 ) is, (a) 200000, (c) 205000, , (b) 195000, (d) 190000, , SAMPLE PAPER 6, , 50. Sum of values of Z at all corner points is, (a) 1365000, (c) 1355000, , X, , x + y = 50, , And the objective function is Z = 10500x + 9000y., Based on above information, answer the following question., (a) (10, 10), (c) (0, 55), , 60, , (b) 1360000, (d) 1350000

Page 135 :
OMR SHEET, , SP 6, , Roll No., Sub Code., , Student Name, , Instructions, Use black or blue ball point pens and avoid Gel & Fountain pens for filling the OMR sheet., Darken the bubbles completely. Don’t put a tick mark or a cross mark, half-filled or over-filled bubbles will not be read, by the software., Correct, , ✔, , ✗, , Incorrect, , Incorrect, , Incorrect, , Do not write anything on the OMR Sheet., Multiple markings are invalid., , 1, , 18, , 35, , 2, , 19, , 36, , 3, , 20, , 37, , 4, , 21, , 38, , 5, , 22, , 39, , 6, , 23, , 40, , 7, , 24, , 41, , 8, , 25, , 42, , 9, , 26, , 43, , 10, , 27, , 44, , 11, , 28, , 45, , 12, , 29, , 46, , 13, , 30, , 47, , 14, , 31, , 48, , 15, , 32, , 49, , 16, , 33, , 50, , 17, , 34, , Check Your Performance, Total Questions:, Total Correct Questions:, , If Your Score is, , Score Percentage =, , Total Correct Questions, Total Questions, , × 100, , Less than 60%, > Average (Revise the concepts again), Greater than 60% but less than 75% > Good (Do more practice), Above 75%, > Excellent (Keep it on)

Page 140 :
132, , CBSE Sample Paper Mathematics Class XII (Term I), , On differentiating w.r.t. x, we get, æ, æ1ö ö, ç (1 + log y ) - ç ÷ y ÷, è y ø ÷ dy, ç, ç, ÷ dx = 1, (1 + log y )2, çç, ÷÷, è, ø, log y, dy, =1, Þ, (1 + log y )2 dx, , Þ, , 46. From the given graph, we can clearly see that, point (10, 10) is in the feasible region or, feasible solution., Hence, option (a) is correct., , dy (1 + log y )2, =, dx, log y, , Þ, 44. Given, y = A emx + B enx, , …(i), , On differentiating twice w.r.t. x, we get, dy, d, d, = Aemx, ( mx ) + Benx, ( nx ), dx, dx, dx, …(ii), = A emx m + B enx n, 2, d y, …(iii), Þ, = m2 Aemx + n2 Benx, dx 2, Using Eqs. (i), (ii) and (iii), we get, d 2y, dy, - ( m + n), + mny, 2, dx, dx, = m2 Aemx + n2 Benx - ( m + n) { Amemx + nBenx }, + mn { Aemx + Benx }, 2, d y, dy, i.e., - ( m + n), + mny = 0, 2, dx, dx, Þ, , k =0, , 45. Given, cos y = x cos( a + y ), cos y, Þ x=, cos ( a + y ), Þ, , dy, cos 2 ( a + y ), 1, =, =, dx dx, sin a, dy, , dx, d ì cos y ü, =, ý, í, dy dy ïî cos ( a + y )þ, cos( a + y ) ( - sin y ) - cos y( - sin( a + y ) 1), =, cos 2 ( a + y ), sin ( a + y - y ), sin a, =, =, 2, 2, cos ( a + y ), cos ( a + y ), [Qsin( A - B) = sin A cos B- cos A sin B], , 47. To know the maximum value of Z, we need, coordinates of all the corner points., We have, equation of lines x + y = 50, …(i), and, …(ii), 2 x + y = 80, For point A, put y = 0 into Eq. (ii),, 2 x + 0 = 80 Þ x = 40 Þ A( 40 , 0 ), For point C, put x = 0 into Eq. (i),, 0 + y = 50 Þ y = 50 Þ C(0 , 50 ), Now,, Corner, Points, , Z = 10500 x + 9000 y, , O(0, 0), , 10500 ´ 0 + 9000 ´ 0 = 0, , A(40, 0), , 10500 ´ 40 + 9000 ´ 0 = 42000, , B (30, 20), , 10500 ´ 30 + 9000 ´ 20 = 495000, (Maximum), , C (0, 50), , 10500 ´ 0 + 9000 ´ 50 = 450000, , Hence, Z is maximum at B(30 , 20 )., 48. From the above table, we can see that objective, function has value 420000 at point A., 49. Z|( 20, 20)- Z|( 10, 10), = (10500 ´ 20 + 9000 ´ 20 ), -(10500 ´ 10 + 9000 ´ 10 ), = 1000 [(105 ´ 2 + 90 ´ 2 ) - (105 + 90 )], = 1000 [(210 + 180 ) - (195 )], = 1000 [390 - 195 ] = 195000, 50. Required sum = 0 + 420000 + 495000 + 450000, , SAMPLE PAPER 6, , = 1365000

Page 141 :
133, , CBSE Sample Paper Mathematics Class XII (Term I), , SAMPLE PAPER 7, MATHEMATICS, A Highly Simulated Practice Questions Paper, for CBSE Class XII (Term I) Examination, , Instructions, 1. This question paper contains three sections - A, B and C. Each section is compulsory., 2. Section - A has 20 MCQs, attempt any 16 out of 20., 3. Section - B has 20 MCQs, attempt any 16 out of 20., 4. Section - C has 10 MCQs, attempt any 8 out of 10., 5. There is no negative marking., 6. All questions carry equal marks., Maximum Marks : 40, Time allowed : 90 min, , Roll No., , Section A, In this section, attempt any 16 questions out of Questions 1-20. Each question is of 1 mark weightage., , 1. The domain of (cos -1 x) is, (a) [0, p], (c) [0, 1], , (b) [-1, 1], (d) None of these, , 2. On the set N of all natural numbers, define the relation R by aRb if GCD of a and b is 2., Then, R is, (a) reflexive but not symmetric, (b) symmetric only, (c) reflexive and transitive, (d) not reflexive, not symmetric, not transitive, , 3. The solution set of the inequation x + 2y > 4 is, , ì1, if x ¹ 0, is not continuous at, î2, if x = 0, , 4. The function f (x) = í, (a) x = 0, (c) x = - 1, , (b) x = 1, (d) None of these, , SAMPLE PAPER 7, , (a) half plane that contains the origin, (b) open half plane not containing the origin, (c) whole xy-plane except the points lying on the line 2 x + y = 5, (d) None of the above

Page 142 :
134, , CBSE Sample Paper Mathematics Class XII (Term I), , 5. The function f (x) = tan x - x, (a) always increases, (b) always decreases, (c) never increases, (d) sometimes increases and sometimes decreases, , é1, , 2ù, , é1, , 3ù, , 6. If A = ê, ú and B = ê -1 1ú , then the value of |A | + | B | is, ë 3 -1û, ë, û, (a) 28, , (b) 7, , (c) - 3, , (d) 4, , 7. Minor of an element of a determinant of order n(n ³ 2) is a determinant of order, (b) n - 1, , (a) n, , é2 + x, 8. If ê 1, ê, êë x, (a) - 25, , 3, -1, 1, , (c) n - 2, , 4ù, 2 ú is a singular matrix, then 13x is, ú, -5úû, (b) 25, (c) 5, , 9. If f (x) = 2x and g(x) =, , (d) n + 1, , (d) - 5, , 2, , x, + 1, then which of the following can be a discontinuous, 2, , function?, (a) f ( x) + g ( x), , a 11, 10. If D = a 21, a 31, , a 12, a 22, a 32, , (b) f ( x) - g( x), , (c) f ( x) × g ( x), , (d), , g( x), f ( x), , a 13, a 23 and Aij is cofactor of a ij , then value of D is given by, a 33, , (a) a 11 A31 + a 12 A32 + a 13 A33, (c) a 21 A11 + a 22 A12 + a 23 A13, , (b) a 11 A11 + a 12 A21 + a 13 A31, (d) a 11 A11 + a 21 A21 + a 31 A31, , 11. Corner points of the feasible region for an LPP are : (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5)., Let Z = 2x + 3y the objective function. The minimum value of Z occurs at, (a) (0, 2) only, (b) (3, 0) only, (c) the mid-point of the line segment joining the points (0, 2) and (3, 0) only, (d) any point on the line segment joining the points (0, 2) and (3, 0), , SAMPLE PAPER 7, , 12. The objective function of an LPP is, (a), (b), (c), (d), , a constraint, a function to be optimised, a relation between the variables, None of the above, , 13. If y = a + bx 2 , and x, (a) 0, , d2y, , dy, , then k is equal to, dx, dx, (b) 5, (c) 1, 2, , =k, , (d) 2, , -1, , 14. The principal value of cot (- 3) is, (a), , p, 6, , (b), , -p, 6, , (c), , 5p, 6, , (d) None of these

Page 143 :
135, , CBSE Sample Paper Mathematics Class XII (Term I), , 15. If A is any square matrix of order 2 ´ 2 such that |A | = 3, then the value of |adj A | is, (a) 3, , (b), , 1, 3, , (c) 9, , (d) 27, , 16. Matrices A and B will be inverse of each other only, if, (a) AB = BA, (c) AB = O and BA = I, , (b) AB = BA = O, (d) AB = BA = I, , 17. If A is matrix of order m ´ n and B is a matrix such that AB¢ and B ¢ A are both defined,, then order of matrix B is, (a) m ´ m, (b) n ´ n, , (c) n ´ m, , (d) m ´ n, , 18. If A is a square matrix of order 3, with |A| = 9, then the value of | 3 adj A |, (a) 2187, , (b) 81, , (c) 8, , (d) 324, , 19. The element a 23 of a 3 ´ 3 matrix A = [a ij ], whose elements are given by a ij =, (a) 1, , (b) 2, , (c) 3, , | i2 - j2 |, 5, , is, , (d) 0, , 20. If A and B are square matrices of the same order, then (A + B) (A - B) is equal to, (a) A2 - B2, (c) A2 - B2 + BA - AB, , (b) A2 - BA - AB - B2, (d) A2 - BA + B2 + AB, , Section B, In this section, attempt any 16 questions out of Questions 21-40. Each question is of 1 mark weightage., , 21. The interval on which the function f (x) = 2x 3 + 9x 2 + 12x - 1 is decreasing, is, (a) [-1, ¥], , (b) [-2, -1], , (c) [-¥, -2], , (d) [- 1, 1], , é 1 + sin x + 1 - sin x ù, p, ú , 0 < x < is, 2, êë 1 + sin x - 1 - sin x úû, , 22. Derivative of cot -1 ê, (a), , 1, 2, , (b) 1, , (c) 2, , (d) None of these, , 23. The corner points of the feasible region determined by the following system of linear, , inequalities 2x + y £ 10, x + 3y £ 15, x , y ³ 0 are (0 ,0) (5, 0) (3, 4) and (0, 5)., Let Z = px + qy , where p , q > 0. Condition on p and q, so that the maximum of Z occurs, at both (3, 4) and (0, 5), is, (a) p = q, (b) p = 2 q, (c) p = 3 q, (d) q = 3 p, , 24. Let X be the set of all persons living in Delhi. The persons a and b in set X are said to be, , 25. If A and B are invertible matrices, then which of the following is not correct?, (a) adj A = |A|× A-1, (c) ( AB ) -1 = B-1 A-1, , (b) det ( A) -1 = [det ( A )] -1, (d) ( A + B ) -1 = B-1 + A-1, , SAMPLE PAPER 7, , related, if the difference in their ages is atmost 4 yr. The relation is, (a) an equivalence relation, (b) reflexive and transitive but not symmetric, (c) symmetric and transitive but not reflexive, (d) reflexive and symmetric but not transitive

Page 144 :
136, , CBSE Sample Paper Mathematics Class XII (Term I), , 26. If area of a triangle is 35 sq units with vertices (2, - 6), (5, 4) and (k , 4), then k is, (b) - 2, , (a) 12, , (c) - 12 , - 2, , (d) 12 , - 2, , 27. The function f (x) = 3 sin 2x + cos 2x + 10 is one-one in the interval, p p, (a) é - , ù, êë 2 2 úû, , p p, (b) é - , ù, êë 4 4 úû, , p p, (c) é - , ù, êë 3 3 úû, , æ 3x - x 3 ö, dy, 1, 1, ÷,,then, is, <x<, 2 ÷, dx, 3, 3, è 1 - 3x ø, 1, -3, (b), (c), 1 + x2, 1 + x2, , p p, (d) é - , ù, êë 3 6 úû, , 28. If y = tan -1 çç, (a), , 3, 1 + x2, , (d), , 3, 1 - x2, , p, 29. Which of the following functions is decreasing on æç 0, ö÷ ?, è, , (a) sin2 x, , (b) tan x, , 2ø, , (c) cos x, , (d) cos 3x, , 30. Let R be set of real numbers. If f : R ® R is defined by f (x) = e x , then f is, (a) surjective but not injective, (c) bijective, , é 1, , 2ù, , é a 4ù, , (b) injective but not surjective, (d) neither surjective nor injective, , é 5 6ù, , 2, 2, 31. If ê, ú + ê 3 2ú = ê1 0ú , then a + b is equal to, 2, b, ë, û ë, û ë, û, , (a) 20, (c) 12, , 32. If y = 3x + 2 +, , (b) 22, (d) 10, , 1, , ,then, , 2x 2 + 4, 3, 2x, (a), 2, 2 3x + 2 (2 x + 4) 3/ 2, 3, 2, (c), +, 2, 2 3x + 2 (2 x + 4) 3/ 2, , dy, dx, , is equal to, (b), , 3, 2x, +, 2, 2 3x + 2 (2 x + 4) 3/ 2, , (d) None of these, , dy, p, at x = is equal to, dz, 4, (b) - 1, (c) 0, , 33. If y = sin x and z = cos x, then, (a) 1, , (d) 2, , 34. If x + y = K is normal to y 2 = 12x, then K is, (a) 3, , (b) 9, , (c) - 9, , (d) - 3, , SAMPLE PAPER 7, , 35. If A, B are symmetric matrices of same order, then AB - BA is a, (a) skew-symmetric matrix, (c) zero matrix, , (b) symmetric matrix, (d) identity matrix, , 36. Let X be the set of all citizens of India. Elements x, y in X are said to be related, if the, difference of their age is 5 yr. Which one of the following is correct?, (a) The relation is an equivalence relation on X, (b) The relation is symmetric but neither reflexive nor transitive, (c) The relation is reflexive but neither symmetric nor transitive, (d) None of the above

Page 145 :
137, , CBSE Sample Paper Mathematics Class XII (Term I), , 37. If Z = x - 2y be the objective function and min Z = -10. The minimum value occurs at, point, 2 16, (b) æç , ö÷, è3 3 ø, , (a) (14, 2), , (c) (2, 3), , (d) (0, 0), , 38. Solve the linear programming problem, minimise Z = 2x - 3y subject to the constraints, 2x + 3y £ 20, x - y £ 10 and x ³ 0, y ³ 0., (a) Z = 0, (c) Z = -20, , (b) Z = -10, (d) Z = 20, , p, 1, 39. sin æç - sin - 1 æç - ö÷ ö÷ is equal to, è, øø, è, 3, , 2, , (a) 1/2, (c) 1/ 4, , (b) 1/3, (d) 1, , é1 3 ù, -1, ú , then A equals, 2, 2, ë, û, 1 é -2 -3 ù, (a) - ê, 8 ë -2 1 úû, 1 é -1 -3 ù, (c) ê, 8 ë -2 2 úû, , 1 é 3 1ù, (b) - ê, 8 ë -2 2 úû, , 40. If A = ê, , (d) None of these, , Section C, In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , 41. The derivative of cos -1 (2x 2 - 1) w.r.t. cos -1 x is, (a) 2, , (b), , -1, 2 1- x, , 2, , (c), , 2, x, , (d) 1 - x 2, , 42. The points at which the tangents to the curve y = x 3 - 12x + 18 are parallel to X-axis are, (a) (2 , - 2 ) and ( - 2 , - 34), (c) (0, 34) and ( - 2 , 0), , (b) (2 , 34) and ( - 2 , 0), (d) (2 , 2 ) and ( - 2 , 34), , 44. If y = (tan -1 x) 2 , then the value of (x 2 + 1) 2 y2 + 2x(x 2 + 1) y1 is, (a) 2, , 45. If x = e x / y , then, (a), , x-y, x log x, , (b) 3, , dy, dx, , (c) 4, , (d) None of these, , is equal to, (b), , y-x, log x, , (c), , y-x, x log x, , (d), , x-y, log x, , SAMPLE PAPER 7, , ì( x - 1) sin 1, , if x ¹ 1, 43. Let f (x) = ïí, . Then, which of the following is true?, ( x - 1), ïî, , if x = 1, 0, (a) f ( x) is not differentiable at x = 1, (b) f ( x) is differentiable at x = 1, (c) cannot say, (d) None of the above

Page 146 :
138, , CBSE Sample Paper Mathematics Class XII (Term I), , CASE STUDY, In a college, an architecture design a auditorium for its cultural activities purpose. The shape of, the floor of the auditorium is rectangular and it has a fixed perimeter, say P., , Based on the above information, answer the following questions., , 46. If l and b represents the length and breadth of the rectangular region, then relation, between the variable is :, (a) l + b = P, (c) 2( l + b) = P, , (b) l 2 + b 2 = P 2, (d) l + 2 b = P, , 47. The area (A) of the floor, as a function of l can be expressed as, l, 2, Pl - 2 l 2, (c) A =, 2, , (a) A = Pl +, , Pl + l 2, 2, 2, l, (d) A = + Pl 2, 2, , (b) A =, , 48. College manager is interested in maximising the area of floor ‘A’. For this purpose, the, value of l must be, P, (a), 4, P, (c), 2, , (b), , P, 3, , (d) P, , 49. The value of b, for which the area of floor is maximum, is, P, 16, P, (c), 3, (a), , P, 4, P, (d), 2, , (b), , SAMPLE PAPER 7, , 50. Maximum area of floor is, (a), , P2, 64, , (b), , P2, 28, , (c), , P2, 16, , (d), , P2, 4

Page 147 :
OMR SHEET, , SP 7, , Roll No., Sub Code., , Student Name, , Instructions, Use black or blue ball point pens and avoid Gel & Fountain pens for filling the OMR sheet., Darken the bubbles completely. Don’t put a tick mark or a cross mark, half-filled or over-filled bubbles will not be read, by the software., Correct, , ✔, , ✗, , Incorrect, , Incorrect, , Incorrect, , Do not write anything on the OMR Sheet., Multiple markings are invalid., , 1, , 18, , 35, , 2, , 19, , 36, , 3, , 20, , 37, , 4, , 21, , 38, , 5, , 22, , 39, , 6, , 23, , 40, , 7, , 24, , 41, , 8, , 25, , 42, , 9, , 26, , 43, , 10, , 27, , 44, , 11, , 28, , 45, , 12, , 29, , 46, , 13, , 30, , 47, , 14, , 31, , 48, , 15, , 32, , 49, , 16, , 33, , 50, , 17, , 34, , Check Your Performance, Total Questions:, Total Correct Questions:, , If Your Score is, , Score Percentage =, , Total Correct Questions, Total Questions, , × 100, , Less than 60%, > Average (Revise the concepts again), Greater than 60% but less than 75% > Good (Do more practice), Above 75%, > Excellent (Keep it on)

Page 148 :
140, , CBSE Sample Paper Mathematics Class XII (Term I), , Answers, 1. (b), 11. (d), 21. (b), , 2. (b), 12. (b), 22. (a), , 3. (b), 13. (c), 23. (d), , 4. (a), 14. (c), 24. (d), , 5. (a), 15. (a), 25. (d), , 6. (c), 16. (d), 26. (d), , 7. (b), 17. (d), 27. (d), , 8. (a), 18. (a), 28. (a), , 9. (d), 19. (a), 29. (c), , 10. (d), 20. (c), 30. (b), , 31. (a), 41. (a), , 32. (a), 42. (d), , 33. (b), 43. (a), , 34. (b), 44. (a), , 35. (a), 45. (a), , 36. (b), 46. (c), , 37. (b), 47. (c), , 38. (c), 48. (a), , 39. (d), 49. (b), , 40. (a), 50. (c), , SOLUTIONS, 1. Cosine function is one-one and onto function, in the interval [-p, 0], [0, p], [p, 2p], ...etc., The domain of cos -1 x is [-1, 1]., 2. aRa, then GCD of a and a is a., \ R is not reflexive., Now, aRb Þ bRa, , 7. Minor of an element aij of a determinant is the, , If GCD of a and b is 2, then GCD of b and a is 2., \ R is symmetric., If GCD of a and b is 2 and GCD of b and c is 2,, then it need not be GCD of a and c is 2., \ R is not transitive., e.g. 6 R2 , 2R12 but 6 R 12., 3. Given, inequation, x + 2 y > 4, Let write the inequation as a line to get the, graph., Y, , (0, 2), X, O, , (4, 0), x + 2y = 4, , Y′, , At x = 0, y = 2 Þ (0, 2), , SAMPLE PAPER 7, , At y = 0 , x = 4 Þ (4, 0), At (0, 0) Þ 0 + 2 ´ 0 >/ 4, Hence, solution set is open half plane not, containing the origin., 4. lim, , x ® 0-, , f ( x ) = 1 = lim, , x ® 0+, , f ( x ) ¹ f (0 ) = 2, , 5. We have, f ( x ) = tan x - x, \, , determinant obtained by deleting ith row and, jth column in which element aij lies. It is, denoted by Mij ., The minor of an element of a determinant of, order n( n ³ 2 ) is a determinant of order ( n - 1)., , Now, aRb, bRc Þ, / aRc, , X′, , é 1 3ù, é1 2 ù, and B = ê, 6. Given, A = ê, ú, ú, ë -1 1û, ë3 -1û, Now, | A | = -1 - 6 = - 7 and | B | = 1 + 3 = 4, \| A | + | B| = -7 + 4 = - 3, , f '( x ) = sec2 x - 1, , Þ, f ¢ (x) ³ 0, " x Î R, So, f ( x ) always increases., , 8. Given, matrix is singular., 2+x, \, , 1, x, , 3, , 4, , -1 2 = 0, 1 -5, , Þ (2 + x )( 5 - 2 ) - 3 (-5 - 2 x) + 4(1 + x ) = 0, Þ, 3 (2 + x ) + 15 + 6 x + 4 + 4 x = 0, Þ, 13 x = -25, 25, Þ, x=13, \, 13 x = -25, 9. We know that, if f and g are continuous, functions, then, (a) f + g is continuous, (b) f - g is continuous., (c) fg is continuous, f, (d) is continuous at these points,, g, where g( x ) ¹ 0., x2, + 1 x2 + 2, g( x ), Here,, = 2, =, 2x, 4x, f (x), which is discontinuous at x = 0., 10. D = Sum of product of elements of any row (or, column) with their corresponding cofactors., Hence, a11 A11 + a21 A21 + a31 A31 = 0

Page 149 :
141, , CBSE Sample Paper Mathematics Class XII (Term I), , 11. Given objective function, Z = 2 x + 3 y, Corner Points, , Value of Z = 2 x + 3 y, , (0, 2), , Z =0+6=6, , (3, 0), , Z =6+0=6, , (6, 0), , Z = 12 + 0 = 12, , (6, 8), , Z = 12 + 24 = 36, , (0, 5), , Z = 0 + 15 = 15, , From the obtained value at (0, 2) and (3, 0) we, can say that minimum of Z occurs at any point, on the line segment joining the points (0, 2), and (3, 0)., 12. The objective function of an LPP is a function, which is to be optimised. It has either a, maximum or minimum value or has no, solution., 13. Given,, , y = a + bx 2, , On differentiating w.r.t. x twicely, we get, dy, ...(i), \, = 2 bx, dx, d 2y, = 2b, Þ, dx 2, d 2y, Now, x 2 = 2 bx, ...(ii), dx, From Eqs. (i) and (ii), we get, d 2y dy, x 2 =, Þ k =1, dx, dx, 14. Let cot -1( - 3 ) = q, cotq = - 3, p, cot q = cot æç p - ö÷, Þ, è, 6ø, 5p, Þ, q=, 6, 5p, ., Hence, cot -1( - 3 ) =, 6, 15. We know, | adj A | = | A | n - 1, Here, n = 2, | adj A | = 3 2- 1 = 3 1 = 3, , 17. Let A =[aij]m´ n and B = [bij]p ´ q, \, , B¢= [bji]q ´ p, , Now, AB¢ is defined, so n = q, and B¢ A is also defined, so p = m, \ Order of B is m ´ n., , Here, n = 3, \, | adjA| = [9 ]3 - 1 = 9 2 = 81, Now, | 3 adj A| = 3 3 | adj A |, = 27 ´ 81, = 2187, |i 2 - j 2|, 5, |4 - 9| |- 5| 5, =, =, = =1, 5, 5, 5, =1, , 19. Given, aij =, \, , a23, , Þ, , a23, , 20. ( A + B) ( A - B) = A( A - B) + B( A - B), = A2 - AB + BA - B2, 21. We have, f ( x ) = 2 x 3 + 9 x 2 + 12 x - 1, On differentiating w.r.t. x, we get, \ f ¢ ( x ) = 6 x 2 + 18 x + 12 = 6( x 2 + 3 x + 2 ), For decreasing function, f ¢ (x) £ 0, Þ 6( x + 1) ( x + 2 ) £ 0, +, , –, –2, , +, –1, , \ f ( x ) is decreasing in [- 2, - 1]., x, x, 22. 1 + sin x = cos + sin, 2, 2, x, x, and 1 - sin x = cos - sin, 2, 2, x, 2 cos, 1 + sin x + 1 - sin x, 2 = cot x, =, \, 1 + sin x - 1 - sin x 2 sin x, 2, 2, æ 1 + sin x + 1 - sin x ö, xö, -1 æ, Þ cot -1 ç, ÷ = cot çè cot ÷ø, 2, è 1 + sin x - 1 - sin x ø, x, p, = , if 0 < x <, 2, 2, d æxö 1, Þ, ç ÷=, dx è 2 ø 2, 23. The maximum value of Z is unique., It is given that the maximum value of Z occurs, at two points (3, 4) and (0, 5)., Value of Z at (3, 4) = Value of Z at (0, 5), Þ p(3 ) + q( 4 ) = p(0 ) + q( 5 ), Þ, 3p + 4 q = 5 q, Þ, 3p = q, 24. Given, R = {( a, b) :|a - b| £ 4 }, Reflexive Let ( a, a) Î R, then, ( a, a) = | a - a | = 0, 0 £ 4, ( a, a) Î R, Hence, R is reflexive., , SAMPLE PAPER 7, , 16. Suppose A is a non-zero square matrix of order, n and there exists matrix B of same order n, such that AB = BA = I, then such matrix B is, called an inverse of matrix A., , 18. We know that, | adjA | = | A | n - 1

Page 151 :
143, , CBSE Sample Paper Mathematics Class XII (Term I), , 1, , 32. Let y = 3 x + 2 +, 1, 2 )2, , = (3 x +, , 35. ( AB - BA)¢ = ( AB)¢ - ( BA)¢, , 2 x2 + 4, -, , + (2 x 2 + 4 ), , = B¢ A¢ - A¢ B¢, = BA - AB [Q A¢ = A and B¢ = B], = - ( AB - BA), , 1, 2, , On differentiating w.r.t. x, we get, , 36. Given that, X = {Set of all citizens of India}, , 1, , -1 d, dy 1, = (3 x + 2 ) 2, × (3 x + 2 ), dx, dx 2, 1, , - -1 d, 1, + æç - ö÷(2 x 2 + 4 ) 2 × (2 x 2 + 4 ), è 2ø, dx, 1, 2) 2, , 1, 1, (3 x +, × (3 ) - æç ö÷ (2 x 2 +, è2 ø, 2, 3, 2x, =, 3, 2 3x + 2, (2 x 2 + 4 ) 2, =, , 3, 4) 2, , × 4x, , 33. Given, y = sin x and z = cos x, On differentiating w.r.t. x, we get, dy, dz, = cos x and, = - sin x, dx, dx, dy dy dx, =, ´, \, dz dx dz, dy, 1, \, = cos x ´, = - cot x, dz, - sin x, p, æ dy ö, \, ç ÷ p = - cot = -1, è dz ø x =, 4, 4, , 34. We have, y 2 = 12 x, On differentiating w.r.t. x, we get, dy, 2y, = 12, dx, dy 6, =, Þ, dx y, Let x + y = K be normal to y 2 = 12 x at point, P( x1 , y1 ), then, æ -1 ö, ÷, ç, è dy / dx ø at, , \, xRx Ï R, So, R is not reflexive., Symmetric Again, xR y, x-y = 5, Þ, Þ, |y - x|= 5, Þ, yRx, So, R is symmetric., Transitive Let x , y , z Î X, Then,, , xRy Þ|x - y| = 5, , and, , yRz Þ|y - z| = 5, , But, , |x - z| ¹ 5, , So, R is not transitive., Hence, the relation is symmetric but neither, reflexive nor transitive., 37. Given objective function,, Z = x -2y, Corner Points, , Value of Z = x - 2 y, Z = 14 - 4 = 10, , (14, 2), , 2 32, 3, 3, -30, =, = -10, 3, (Minimum), , Z =, æ 2 16 ö, ç , ÷, è3 3 ø, , 12 x1 = 36, , Þ, x1 = 3, Also P( x1 , y1 ) lies on x + y = K, therefore, x1 + y1 = K, K = 3 +6 = 9, , (2, 3), , Z = 2 - 6 = -4, , (0, 0), , Z =0-0=0, , 2 16, Thus, minimum Z = -10 occurs at point æç , ö÷., è3 3 ø, 38. We have to minimize, Z = 2 x - 3y, Subject to the constraints are, x - y £ 10, 2 x + 3 y £ 20, x ³ 0 and y ³ 0, , SAMPLE PAPER 7, , y12 = 12 x1, , Þ, , Reflexive |x - x| = 0 ¹ 5, , = (Slope of the line x + y = K), p, , y, - 1 = -1, Þ, 6, Þ, y1 = 6, Since, ( x1 , y1 ) lies on y 2 = 12 x, therefore, Þ, , R = {(x , y ) : x , y Î X,|x - y| = 5}, , and

Page 153 :
145, , CBSE Sample Paper Mathematics Class XII (Term I), , Again, differentiating w.r.t. x, we get, dy, 2, d, (1 + x 2 ) 1 + y1, (1 + x 2 ) =, dx, dx, 1 + x2, 2, Þ, (1 + x 2 )y2 + y1 (0 + 2 x ) =, 1 + x2, éQ d y = y ù, êë dx 1 2 úû, Þ, , (1 + x 2 )2 y2 + 2 x (1 + x 2 )y1 = 2, , 45. Given that , x = ex / y, Taking log on both sides, we get, x, x, log x = × log e =, y, y, , [Q log e = 1], , Þ, x = y log x, Now, differentiating w.r.t. x, we get, dy, 1, 1 = y × + log x ×, x, dx, dy, x-y, Þ, =, dx x log x, , 47. Area, A = length ´ breadth, Q, Þ, Þ, Þ, , P = 2( l + b), P, = l+b, 2, P, -l= b, 2, P -2l, =b, 2, , Þ, , P - 2lö, A = l æç, ÷, è 2 ø, A=, , Pl - 2 l 2, 2, , Pl - 2 l 2, 2, On differentiating w.r.t. x, we get, dA 1, = ( P - 4 l), dl 2, dA, For maximum area of floor, put, =0, dl, 1, \ ( P - 4 l) = 0, 2, Þ, P - 4l = 0, P, Þ, l=, 4, , 48. A =, , P d2 A, ,, = -2 < 0, 4 dl 2, P, \Area is maximum at l = ., 4, Clearly at l =, , 46. Perimeter of rectangular floor, = 2 (length + breadth), Þ, P = 2( l + b), A= l´b, , From Eq. (i),, , ...(i), , 49. We have, b =, , P -2l P, = -l, 2, 2, , Since, area of floor is maximum at l =, \, , b=, , p, 4, , P P P, - =, 2 4 4, , 50. We have, A = l ´ b, For maximum area of floor,, P P P2, A= ´ =, 4 4 16, , SAMPLE PAPER 7

Page 155 :
147, , CBSE Sample Paper Mathematics Class XII (Term I), , 5. If x = sin q, y = tan q, then, (a) 1, , dy, dx, , at q =, , (b) 8, , p, is equal to, 3, (c) 3, , (d) 4, , 6. If A is a square matrix such that A 2 = I, then A + A -1 is equal to, (a) A + I, , (b) A, , (c) 0, , (d) 2 A, , 7. The equation of tangent to the curve y = 2x 2 + 3 sin x and (0, 0) is, (a) y = 3x, , (b) y = - 3x, , é1, , (c) x = 3y, , (d) x = - 3y, , (c) 8, , (d) 1, , 0ù, , 2, 8. If [x 1] ê, ú = 0, then x is equal to, 2, 0, ë, û, , (a) 2, , (b) 4, , æ1, è2, , 9. The value of cosç sin -1, (a), , 1, 2, , (b), , 3ö, ÷ is, 2 ø, 3, 2, , æ, , 10. The principal value of sin -1 ç (a), , è, p, (b) 3, , - 2p, 3, , (c), , 1, 2, , (c), , 4p, 3, , (d) None of these, , 3ö, ÷ is, 2 ø, (d), , 5p, 3, , 11. Let X be the set of all persons living in a city. Persons x, y in X are said to be related as, x < y, if y is atleast 5 yr older than x. Which one of the following is correct?, (a) The relation is an equivalence relations on X, (b) The relation is transitive but neither reflexive nor symmetric, (c) The relations is reflexive but neither transitive nor symmetric, (d) The relation is symmetric but neither transitive nor reflexive, , 12. Let S denote set of all integers. Define a relation R on S as ‘aRb’ if ab ³ 0, where a, b Î S., Then, R is, (a) reflexive but neither symmetric nor transitive relation, (b) reflexive, symmetric but not transitive relation, (c) an equivalence relation, (d) symmetric but neither reflexive nor transitive relation, é x + yù, , é 2 1ù é 1 ù, , 13. If ê, ú =ê, ú ê ú , then ( x , y) is, ë x - yû ë 4 3û ë - 2û, (b) (1, - 1), , (a) (1, 1), , 1, (a) 1, , 15. If, , 5, , 7, , M 21, M 32 - 1, , (d) ( - 1, - 1), , (c) 3, , (d) 4, , (c) - 3, , (d) 3, , is equal to, , (b) 2, , x -1, 6 2, , then x is equal to, =, 9 x, 9 6, , (a) 6, , (b) ± 3, , SAMPLE PAPER 8, , 2 -3 5, 0 4 , then, , 14. If D = 6, , (c) ( - 1, 1)

Page 156 :
148, , CBSE Sample Paper Mathematics Class XII (Term I), , 16. The function f (x) = log x is strictly increasing on, (a) [ - 1, 0], , (b) (0, ¥ ), 3, , (c) ( - ¥ , ¥ ), , (d) None of these, , 2, , 17. The function f (x) = x - 3x + 3x - 10 in the interval (- ¥, ¥) is, (a) decreasing, , (b) increasing, , (c) strictly increasing, , (d) strictly decreasing, , 18. The slope of the tangent to the curve x = 3t 2 + 1, y = t 3 - 1 at x = 1 is, (a) - 1, , (b) 1, , é 2x, , 19. If x ê, 3, ë, , (a) 4, , (c) 0, , é x 2 + 8 24ù, 2ù, é 8 5x ù, 2, 2, 2, +, =, ú , then x is equal to, ê, ê 4 4 xú, xúû, 6xû, ë, û, ë 10, (b) 16, (c) 2, , (d) 2, , (d) 1, , 20. The point on the curve y = x 2 - 4 x + 5, where tangent to the curve is parallel to the, X-axis is, (a) (0, 5), , (b) ( - 1, 0), , (c) (1, 2), , (d) (2, 1), , Section B, In this section, attempt any 16 questions out of Questions 21-40. Each question is of 1 mark weightage., , 21. If y = (cot -1 x) 2 and (x 2 + 1) 2, (a) 1, , d2y, dx, , 2, , dy, = K, then K is equal to, dx, (c) 5, (d) 7, , + 2x( x 2 + 1), , (b) 2, , é1 1ù, , then A 2 + 3A - 2I is equal to, ú, ë 0 2û, 6ù, é2 0ù, (b) ê, ú, ú, 8û, ë 4 6û, , 22. If A = ê, é2, (a) ê, ë0, , é0 6 ù, (c) ê, ú, ë2 4û, , é 1 1ù, (d) ê, ú, ë2 3û, , 23. The equation of normal at the point (1, 1) on the curve 2y + x 2 = 3 is, (a) x + y = 0, , (b) x - y = 0, , 24. If x = a(t - sin t), y = a(1 + cos t), then, (a) - 1, , dx, , at t =, , (b) 0, , 25. If y = log(xy), then, (a) 1, , SAMPLE PAPER 8, , dy, , dy, dx, , (c) x + y + 1 = 0, , (d) x - y = 1, , p, is equal to, 2, (c) 3, , (d) 8, , (c) 3, , (d) 4, , at (1, 2) is equal to, (b) 2, , 26. The maximum value of the function f (x) = - (x - 1) 2 + 8 is, (a) 7, , (b) 8, , ép, , æ, , ë2, , è, , 27. The value of sin ê - sin -1 ç (a), , 1, 2, , (b) -, , (d) 1, , (c) 1, , (d) - 1, , 3 öù, ÷ ú is, 2 øû, , 1, 2, , 28. The function f : R ® R defined by f (x) =, (a) one-one, , (c) 0, , (b) not one-one, , x, 2, , x +1, , , " x Î R is, , (c) bijective, , (d) None of these

Page 157 :
149, , CBSE Sample Paper Mathematics Class XII (Term I), , 29. Let f : [0, 1] ® [0, ¥) be defined by f (x) =, , x, , then f is, 1+x, , (a) one-one but not onto, (c) both one-one and onto, , (b) onto but not one-one, (d) neither one-one nor onto, , é1 5ù, é1 0ù, and B = ê, ú, ú , then, ë 3 9û, ë 0 1û, (a) AB = BA, (b) AB ¹ BA, , (c) A2 = B, , (d) None of these, , é1 2ù, ú , then| 3A| is equal to, ë 3 4û, (a) 3| A|, (b) 9| A|, , (c) | A|, , (d) 27 | A|, , 30. If A = ê, , 31. If A = ê, , 32. If at x = 1, then function f (x) = x 4 - 62x 2 + ax + 9 attains its maximum value on the, interval [ 0, 2], then the value of a is, (b) 120, , (a) 124, , 33. If y = 3 cos x + 3 sin x, then, (a) 0, , 34. If y = e - 3 x and, (a), , d y, dx 2, , (c) 2, , (d) 3, , (c) 2, , (d) 1, , (c) 2 cot 2 x, , (d) sec 2 x, , + y is equal to, , (b) 9, , d2y, , is equal to, dx 2, (b) 2 cosec 2 x, , 36. If x = at 2 and y = at 3 , then, , d2y, dx, , 2, , at t =, , 3, is equal to, 4, , 1, (b), a, , (c) 4, , 37. If y = a cos 3 t and x = a sin 3 t, then, (a) 1, , (d) 128, , = Ky, then K is equal to, , 35. If y = 2 log sin x, then, , (a) a, , dx 2, , (b) 1, 2, , 1, 9, , (a) - 2 cosec 2 x, , d2y, , (c) - 120, , dy, dx, , at t =, , (b) -1, , p, is, 4, (c) 0, , (d) -1, , (d) 2, , x, , 38. The function f (x) = (x - 1)e + 2 on [0, ¥) is, (a) increasing, , (b) decreasing, , (c) strictly decreasing, , (d) None of these, , number. The, the relation R is, (a) reflexive, (c) transitive, , (b) symmetric, (d) an equivalence relation, , 40. The function f (x) = x 3 + x 2 + x + 1 has, (a) maximum value at x = - 1, (c) neither maximum nor minimum value, , (b) minimum value at x = - 1, (d) None of these, , SAMPLE PAPER 8, , 39. For real numbers x and y, define a relation R, xRy if only if x - y + 2 is an irrational

Page 158 :
150, , CBSE Sample Paper Mathematics Class XII (Term I), , Section C, In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , 41. If x = tan æç log yö÷ and (1 + x 2 ), 1, èa, , (a) a, , ø, , (b), , a, 2, , d2y, dx, , 2, , + 2x, , dy, dx, , =k, , é1 1ù, , then A 50 is equal to, ú, ë1 1û, 49, (a) 2 A, (b) 2 A, , dy, , , then k is equal to, dx, (c) 2 a, (d) 1, , 42. If A = ê, , (c) 49 A, , (d) 2 99 A, , 43. The function f given by f (x) = x 2 - x + 1 on (- 1, 1) is, (a) strictly decreasing, (c) neither strictly increasing nor strictly decreasing, , 44. If y = sin -1 x and (1 - x 2 ), (a) x, , d2y, , dx, (b) x 2, , 2, , =k, , 1, 45. If l = - 2, then the value of l2, 2, 3, , (a) - 11, , dy, dx, , (b) strictly increasing, (d) None of these, , , then k is equal to, (c) 1, , (d) 0, , 2l 1, 1 3l2 is, 2l, , 1, , (b) - 12, , (c) - 13, , (d) 0, , CASE STUDY, Suppose a dealer in rural area wishes to propose a number, of sewing machines. He has some money to invest and has, space for few items for storage., Let x denotes the number of electronic sewing machines and, y denotes the number of manually operated sewing, machines purchased by the dealer. For the same, constraint, related to investment is given by 3x + 2 y £ 48., And objective function is Z = 22 x + 18y., And other constraints consists the following x + y £ 20, x, y ³ 0., Based on above information, answer the following questions., , 46. Number of corner points of the feasible region is, (a) 3, , (b) 4, , (c) 5, , (d) 6, , (c) 1100, , (d) 1108, , 47. Sum of values of Z at all the corner points is, , SAMPLE PAPER 8, , (a) 1008, , (b) 1104, , 48. To get the maximum profit (i.e. maximise Z) how many electronic sewing machines, should be purchased by the dealer., (a) 12, (b) 8, , (c) 10, , (d) 5, , 49. To get the maximum profit (i.e. maximise Z) how many manually operated sewing, machines should be purchased by the dealer., (a) 10, (b) 5, (c) 8, , (d) 12, , 50. Z| max - Z| min is equal to, (a) 360, , (b) 392, , (c) 352, , (d) None of these

Page 159 :
OMR SHEET, , SP 8, , Roll No., Sub Code., , Student Name, , Instructions, Use black or blue ball point pens and avoid Gel & Fountain pens for filling the OMR sheet., Darken the bubbles completely. Don’t put a tick mark or a cross mark, half-filled or over-filled bubbles will not be read, by the software., Correct, , ✔, , ✗, , Incorrect, , Incorrect, , Incorrect, , Do not write anything on the OMR Sheet., Multiple markings are invalid., , 1, , 18, , 35, , 2, , 19, , 36, , 3, , 20, , 37, , 4, , 21, , 38, , 5, , 22, , 39, , 6, , 23, , 40, , 7, , 24, , 41, , 8, , 25, , 42, , 9, , 26, , 43, , 10, , 27, , 44, , 11, , 28, , 45, , 12, , 29, , 46, , 13, , 30, , 47, , 14, , 31, , 48, , 15, , 32, , 49, , 16, , 33, , 50, , 17, , 34, , Check Your Performance, Total Questions:, Total Correct Questions:, , If Your Score is, , Score Percentage =, , Total Correct Questions, Total Questions, , × 100, , Less than 60%, > Average (Revise the concepts again), Greater than 60% but less than 75% > Good (Do more practice), Above 75%, > Excellent (Keep it on)

Page 165 :
157, , CBSE Sample Paper Mathematics Class XII (Term I), , Intervals, , Sign of f ¢ ( x ), , Nature of, f (x ), , (–1, 1/2), , – ve, , Strictly, decreasing, , (1/2, 1), , + ve, , Strictly, increasing, , \ f ¢( x ) does not have same sign throughout, the interval ( -1, 1)., Thus, f ( x ) is neither increasing nor decreasing, strictly in the interval ( -1, 1)., 44. Given, y = sin -1( x ), dy, 1, \, =, dx, 1 - x2, Þ, , æ d 2y ö, x, æ dy ö, 1- x ç 2÷ =, ç ÷, 2 è dx ø, è dx ø, 1-x, , Þ, 1, , æ dy ö, = xç ÷, 2, è dx ø, dx, , k=x, 2l 1, , Now, expanding along R1, we get, 2, , 2, , 3, , 1(1 - 6 l ) - 2 l( l - 6 l ) + 1(2 l - 2 ), = 1 - 6 l3 - 2 l3 + 12 l3 + 2 l3 - 2, = 6 l3 - 1, = 6( - 2 ) - 1, = - 12 - 1, = - 13, , B (8, 12) Feasible region, , X, , A (16,0), , x+y=20, , 47. The value of Z at the corner points are, 0, 352, 392 and 360, respectively., \ Required sum = 0 + 352 + 392 + 360 = 1104, 48. The coordinates of the corner points A, B, C, and O are (16, 0), (8, 12), (0, 20) and (0, 0),, respectively., Corner Points, , d 2y, , 45. We have, l2 1 3 l2, 2 2l 1, 3, , C (0, 20), , \ Number of corner points are 4., , 2, , (1 - x 2 ), , Y, , 3x+2y=48, , æ dy ö, 1 - x2 ç ÷ = 1, è dx ø, , or, , Subject to constraints, x + y £ 20, 3 x + 2 y £ 48, x , y ³ 0, , O, (0, 0), , æ d 2y ö æ dy ö 1( -2 x ), Now, 1 - x 2 ç 2 ÷ + ç ÷, =0, è dx ø è dx ø 2 1 - x 2, Þ, , 46. Objective function, Z = 22 x + 18 y, , Z = 22 x + 18 y, , (0, 0), , 0 (Minimum), , (16, 0), , 352, , (8, 12), , 392 (Maximum), , (0, 20), , 360, , Z is maximum at the point (8, 12)., \To get maximum profit 8 electronic sewing, machines should be purchased by the dealer., 49. Q Z is maximum at the point (8, 12)., \To get maximum profit 12 manually operated, sewing machines should be purchased by the, dealer., 50. Z|max - Z|min = 392 - 0 = 392, , SAMPLE PAPER 8

Page 166 :
158, , CBSE Sample Paper Mathematics Class XII (Term I), , SAMPLE PAPER 9, MATHEMATICS, A Highly Simulated Practice Questions Paper, for CBSE Class XII (Term I) Examination, , Instructions, 1. This question paper contains three sections - A, B and C. Each section is compulsory., 2. Section - A has 20 MCQs, attempt any 16 out of 20., 3. Section - B has 20 MCQs, attempt any 16 out of 20., 4. Section - C has 10 MCQs, attempt any 8 out of 10., 5. There is no negative marking., 6. All questions carry equal marks., Maximum Marks : 40, Time allowed : 90 min, , Roll No., , Section A, In this section, attempt any 16 questions out of Questions 1-20. Each question is of 1 mark weightage., , 3p ö, ÷ is, 4 ø, p, (b), 4, , 1. The value of tan -1 æç tan, è, , 3p, (a), 4, , (c), , 5p, 4, , 2. If A is an invertible matrix of order 3 and|A| = 5, then, (a) 5, , 3. The value of, , SAMPLE PAPER 9, , (a) tan -1 y, , (b) 25, , (d) -, , |adj A|, |A|, , p, 4, , is equal to, , (c) 1, , (d) 0, , 1 + cos x, is y, then the value of x in terms of y is, 1 - cos x, (b) 2 tan -1 y, , (c) 2 cot -1 y, , (d) cot -1 y, , 4. Let N be the set of natural numbers and f : N ® N be a function given by f (x) = x + 1 for, x Î N. Which one of the following is correct?, (a) f is one-one and onto, (b) f is one-one but not onto, (c) f is only onto, (d) f is neither one-one nor onto, , 5. If f be given by f (x) = x + 2, x Î(0, 1), then, (a) the function f has not a local maximum value, (b) the function f has not a local minimum value, (c) Both (a) and (b) are true, (d) Both (a) and (b) are false

Page 167 :
159, , CBSE Sample Paper Mathematics Class XII (Term I), , 6. If A is a non-singular matrix of order 3 and adj A = A a , then the value of, (a) 0, , (b) 1, , (c) 2, , -1 ö, ÷ is, è 3ø, , a, is, 2, , (d) 3, , 7. The principal value of cot -1 æç, (a) -, , p, 4, , (b), , 5p, 6, , 8. If x = a cos 3 q and y = a sin 3 q, then, (a) 1, , (c), , dy, dx, , at q =, , (b) - 1, , 2p, 3, , p, is equal to, 4, (c) 0, , é 6 xù, x, and A = A T , then is equal to, ú, y, ë y 0û, (a) 0, (b) 1, (c) 2, , (d) None of these, , (d) 4, , 9. If A = ê, , é 2 - 1ù, ú is given by, ë4 3 û, é -4 2 ù, é 3 1ù, (b) ê, (c) ê, ú, ú, ë 3 1û, ë -4 2 û, , (d) - 1, , 10. The adjoint of the matrix ê, é3 1 ù, (a) ê, ú, ë2 -4û, , (d) None of these, , 11. How many tangents are parallel to X-axis for the curve y = x 2 - 4 x + 3?, (a) 1, (b) 2, (c) 3, (d) No tangent is parallel to X-axis, , 12. The graph of inequations is given below., Y, , x = 20, , 200 (0, 200), C(20,180), 180, B(40,160), 160, 140, 120, 100, A, 80, 60, 40, 20, (200, 0), X, X′ O, 20 40 60 80 100120 140160180200, (0,0), Y′, y –4x = 0, x + y = 200, , é 5 3 8ù, 13. If D = ê 2 0 1ú , then the difference of minor of the element a 23 and minor of element, ú, ê, êë1 2 3úû, a 32 is, (a) 10, (b) 15, (c) 18, (d) 20, , SAMPLE PAPER 9, , The feasible region consists the corner point, (a) (20, 40), (b) (200, 0), (c) (20, 60), (d) (20, 80)

Page 168 :
160, , CBSE Sample Paper Mathematics Class XII (Term I), , 14. If A, , T, , é2, ê, (a) ê 1, êë 1, , é 3 4ù, é -1, = ê -1 2ú and B = ê, ú, ê, ë1, êë 0 1úû, 5ù, é5, ú, ê, (b) ê 1, 4ú, êë 1, 4úû, , 2 1ù, , then find A T + B T is equal to, 2 3úû, 2ù, ú, 4ú, 4úû, , é2 5 1ù, (c) ê, ú, ë 1 3 4û, , (d) None of these, , 15. Let D be the domain of the real valued function f defined by f (x) = 25 - x 2 . Then, D is, equal to, (a) [ -5, 5], , (b) [ -2.5, 2.5], , (c) [ -25, 25], , (d) [ -0.5, 0.5], , 16. Suppose P and Q are two different matrices of order 4 ´ n and n ´ p, then the order of, the matrix P ´ Q is, (a) 4 ´ p, , (b) p ´ 4, , (c) n ´ n, , (d) 4 ´ 4, , 17. Let A = {a, b, c } and the relation R be defined on A as follows, R = {( a , a), ( b , c), ( a , b)}, Then, minimum number of ordered pairs to be added in R to make R reflexive and, transitive is, (a) 1, (b) 2, (c) 3, (d) 4, é, , 8, , x + 2ù, , 18. If ê, ú is a symmetric matrix, then the value of 3x is, ë 2x - 3 x + 1û, (a) 10, , (b) 15, , (c) 8, , (d) 6, , -1, , 19. The principle value of cosec (- 2) is, (a), , p, 4, , (b), , p, 2, , (c) -, , p, 4, , (d) 0, , 20. The number of all possible matrices of order 3 ´ 3 with each entry 3 or 4 is, (a) 27, , (b) 18, , (c) 81, , (d) 512, , Section B, In this section, attempt any 16 questions out of Questions 21-40. Each question is of 1 mark weightage., , 21. Divide 20 into two parts such that the product of one part and the cube of the other is, , SAMPLE PAPER 9, , maximum, then the two parts are, (a) {10, 10}, (b) {12, 8}, , (c) {15, 5}, , 22. The graphical representation of an LPP is the following, (0, 1800), , (0, 450), , (1080, 180), x+4y=1800, , O (0, 0), , (1200, 0), , (1800, 0), 3x+2y=3600, , (d) {5, 10}

Page 169 :
161, , CBSE Sample Paper Mathematics Class XII (Term I), , æ Maximum Z - Minimum Z ö, If Z = 100x + 170y, then the ç, ÷ is equals to, ø, è, 100, (a) 1200, , (b) 1386, , (c) 756, , (d) 1400, , 23. Let T be the set of all triangles in the Euclidean plane and let a relation R on T be, defined as aRb, if a is congruent to b , " a, b ÎT . Then, R is, (a) reflexive but not transitive, (b) transitive but not symmetric, (c) equivalence relation, (d) None of these, , 24. If the set P contains 3 elements and the set Q contains 4 elements, then the number of, one-one and onto mappings from P to Q is, (a) 160, (b) 80, (c) 0, (d) None of these, , 25. The function f (x) = log e (x 3 + x 6 + 1 ) is, (a) even, (c) decreasing, , (b) odd, (d) None of these, , 26. The graph of inequations is shown below, Y, 8 (0,8), , x=5, , 7, 6, E(0,5), , 5, 4, F(0,4), 3, 2, , D(3,5), , y=5, C(5,3), , y=, , y=, x+, , x+, , 1, , 2, , Y′, , B(5,0), 5 6 7, 3 4, A(4,0), , 8, , 4, , 1, X′, (0,0) O, , (8,0), X, 8, , If Z = x - 7 y + 190, then Z maximum - Z minimum is equal to, (a) 40, (b) 50, (c) 60, , (d) 70, , 27. The graph of inequations is shown below, Y, 12, 10, , (0, 9), , 8, (0, 6)C 6, B, , 2, X′, , –4, , 10, , X, , If Z = 7 x + 4 y, then the value of Z|at C +Z|at B is, (a) 10, (b) 30, (c) 50, , 2, , Y′, , 8, , 1, y=, +2, 3x, y=9, 3x+, , (4, 0), –4 –2 O, 2A 4 6, (0, 0) –2 (3,0), , (d) 70, , SAMPLE PAPER 9, , 4

Page 170 :
162, , CBSE Sample Paper Mathematics Class XII (Term I), , 28. If the function f is given by f (x) = x 3 - 3x 2 + 4 x , x Î R, then, (a), (b), (c), (d), , f is strictly increasing on R, f is decreasing on R, f is neither increasing nor decreasing on R, f is strictly decreasing on R, 3, , 2, é, dy ù 2, 29. If (x - a) 2 + (y - b) 2 = c2 for some c > 0, then the value of ê1 + æç ö÷ ú is, êë è dx ø úû, c3, c, c2, c, (b), (d), (c), (a), ( y - b), ( y - b) 3, ( y - b) 2, ( y - b) 2, , 30. Graph of inequations is given below, Y, , x+, y=, 24, (0, 16) C, , X′, , B, , (0, 0) O, , 1, 2 x+y, =, , 16, X, , (24, 0) A, , Y′, , If Z = 9x - 5y, then the value of Z at point B is, (a) 102, (b) 144, (c) 94, , 31. The coordinates of the point on the curve, inclined to the axes is, (a) (2, 2), (b) (2,4), , x + y = 4 at which tangent is equally, (c) (3, 4), , (d) (4, 4), , 4, , x, - x 3 - 5x 2 + 24 x + 12, then the critical numbers are, 4, (a) -3, 2 and 4, (b) 2, 3 and 4, (c) -3, - 2 and 4, , 32. If f (x) =, , (d) 104, , (d) -3, 3 and 4, , é 3 5ù, é 1 - 1ù, = 3ê, ú , then x , y, z and t are respectively, ú, 2û, ë 4 6û, (a) 3, 6, 9 and 6, (b) 3, 9, 6 and 6, (c) 3, 6, 6 and 9, (d) 6, 9, 3 and 6, é x zù, , 33. If 2 ê, ú + 3 ê0, ë, ë y tû, , 34. If the curve ay + x 2 = 7 and x 3 = y, cut orthogonally at (1, 1), then the value of a is, , SAMPLE PAPER 9, , (a) 1, , (b) 0, , (c) - 6, , (d) 6, , 35. The bookshop of a particular school has 10 dozen Chemistry books, 8 dozen Physics, books, 10 dozen Economics books. Their selling prices are ` 80, ` 60 and ` 40 each, respectively. The total amount, the bookshop will receive from selling all the books, using matrix algebra, is, (a) ` 21160, (b) ` 20610, (c) ` 26100, (d) ` 20160, dy, 36. If y = x + x 2 + a 2 , then is equal to, dx, y, y, x, x, (c), (d), (a), (b), 2, 2, 2, 2, x+a, x+a, x +a, x +a

Page 171 :
163, , CBSE Sample Paper Mathematics Class XII (Term I), , 37. If y = (tan -1 x) 2 , then the value of (x 2 + 1) 2 y2 + 2x(x 2 + 1) y1 is, (a) 2, , (b) 3, , (c) 4, , (d) None of these, , 2y, zù, y - zú satisfies the equation A¢ A = I , then the values of x , y and z are, ú, ê, zúû, ëê x - y, 1, 1, 1, 1, 1, 1, (b) x = ±, (a) x = ±, ,y= ±, and z = ±, ,y= ±, and z = ±, 3, 6, 2, 2, 3, 6, 1, 1, 1, 1, 1, 1, (d) x = ±, (c) x = ±, ,y= ±, and z = ±, ,y= ±, and z = ±, 6, 6, 2, 2, 6, 3, é0, , 38. If A = ê x, , 39. A point on the curve y = (x - 2) 2 at which the tangent is parallel to the chord joining the, points ( 2, 0) and ( 4 , 4) is, (a) (2, 1), (b) (3, 1), , (c) (4, 1), , (d) (1, 2), , 40. The graph of inequations is drawn below, Y, 30, 25, 20, 15, , C(0, 10), , 10, B (12, 6), , A (20, 0), , 5, , x + 3y = 30, , X′, (0, 0)O, , X, 5, , 10, , Y′, , 15 20, , 3x, , 25 30, +, 4y, =6, 0, , If Z = 8000x + 12000y, then Z| at A + Z| at B - Z| at C is, (a) 168000, (b) 198000, (c) 200000, , (d) 208000, , Section C, In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , 41. The angle of intersection of the curves y = 4 - x 2 and y = x 2 is, æ 2ö, (b) tan -1 ç, ÷, è 7 ø, , æ 4ö, (c) tan -1 ç ÷, è7 ø, , (d) tan -1 ( 4 2 ), , dy, t, 42. If x = a æç cos t + log tan ö÷ and y = a sin t, then is equal to, è, , (a) cot t, , 2ø, (b) tan t, , dx, , (c) sec t, , 1, , " x Î R. Then, the range of f is, 2 - cos x, é1 ù, é1 ù, (c) ê , 2 ú, (d) ê , 1ú, ë2 û, ë3 û, , 43. Let f : R ® R be the function defined by f (x) =, é1 ù, (a) ê , 1ú, ë2 û, , é1 ù, (b) ê , 2 ú, ë3 û, , (d) cosec t, , SAMPLE PAPER 9, , æ4 2 ö, (a) tan -1 ç, ÷, è 7 ø

Page 172 :
164, , CBSE Sample Paper Mathematics Class XII (Term I), , n, , 44. If y = (1 + x)(1 + x 2 )(1 + x 4 ) . . . . (1 + x 2 ), then the value of, (a) 0, , 45. The derivative of (esec, (a) [1, 1] - {0}, , (b) -1, 2, , x, , dy, dx, , at x = 0 is, , (c) 1, , (d) None of these, , + 3 cos -1 x) is valid in, , (b) ( -1, 1), , (c) ( -1, 1) - {0}, , (d) None of these, , CASE STUDY, Sometimes, x and y are given as functions of one another variable, say x = f( t ), y = y( t ) are two, functions and t is a variable. In such a case, x and y are called parametric functions or parametric, equations and t is called the parameter., To find the derivatives of parametric functions, we use following steps, I. First, write the given parametric functions, Suppose x = f ( t ) and y = g( t ),, where t is a parameter., II. Differentiate both functions separately with respect to parameter t by using suitable formula,, dy, dx, and ., i.e. find, dt, dt, III. Divide the derivative of one function w.r.t. parameter by the derivative of second function, dy, w.r.t parameter, to get required value, i.e. ., dx, dy, dy dt g¢ ( t ), , where f ¢ ( t ) ¹ 0., Thus,, =, =, f ¢ (t), dx dx, dt, Based on above information, answer the following questions., , 46. If x = log t and y = cos t, then, , dy, is equal to, dx, , (a) - t sin t, (c) - t cos t, , (b) t sin t, (d) t cos t, , 47. If x = cos t + sin t and y = sin t - cos t, then, (a) 1, , 1, (a), a, , dy, dx, , at t =, , SAMPLE PAPER 9, , e 2t - 1, e 2t + 1, , (b), , (c) - a, , dy, dx, , (c), , e 2t - 1, , (b) 2 2, , dy, dx, , (d) 2, , (d), , -1, a, , is equal to, , e 2t + 1, , 50. If x = 4 cos t and y = 8 tan t, then, (a) 4 2, , p, is equal to, 2, , 2, is equal to, 3, , (b) a, , 49. If x = e t + e - t and y = e t - e - t , then, (a), , dx, , at t =, (c) - 1, , (b) 0, , 48. If x = at 3 and y = t 2 + 1, then, , dy, , at t =, , et + 1, et - 1, , p, is equal to, 4, (c) - 2, , (d), , et - 1, et + 1, , (d) - 4 2

Page 173 :
OMR SHEET, , SP 9, , Roll No., Sub Code., , Student Name, , Instructions, Use black or blue ball point pens and avoid Gel & Fountain pens for filling the OMR sheet., Darken the bubbles completely. Don’t put a tick mark or a cross mark, half-filled or over-filled bubbles will not be read, by the software., Correct, , ✔, , ✗, , Incorrect, , Incorrect, , Incorrect, , Do not write anything on the OMR Sheet., Multiple markings are invalid., , 1, , 18, , 35, , 2, , 19, , 36, , 3, , 20, , 37, , 4, , 21, , 38, , 5, , 22, , 39, , 6, , 23, , 40, , 7, , 24, , 41, , 8, , 25, , 42, , 9, , 26, , 43, , 10, , 27, , 44, , 11, , 28, , 45, , 12, , 29, , 46, , 13, , 30, , 47, , 14, , 31, , 48, , 15, , 32, , 49, , 16, , 33, , 50, , 17, , 34, , Check Your Performance, Total Questions:, Total Correct Questions:, , If Your Score is, , Score Percentage =, , Total Correct Questions, Total Questions, , × 100, , Less than 60%, > Average (Revise the concepts again), Greater than 60% but less than 75% > Good (Do more practice), Above 75%, > Excellent (Keep it on)

Page 176 :
168, , CBSE Sample Paper Mathematics Class XII (Term I), , 22. Given objective function, Z = 100 x + 170 y, The corner points from the graph are (0, 0),, (1200, 0), (1080, 180) and (0, 450), respectively., Corner points, , Corresponding value of, Z = 100 x + 170 y, , (0, 0), , 0 (Minimum), , (1200, 0), , 1200 ´ 100 = 120000, , (1080, 180), , 100 ´ 1080 + 170 ´ 180 = 138600, (Maximum), , =, , Maximum Z - Minimum Z 138600 - 0, =, 100, 100, = 1386, , SAMPLE PAPER 9, , f ( - x ) = log[ x 6 + 1 - x 3 ], \ f ( - x ) + f ( x ) = log{x 6 + 1 - x 6 }, = log1 = 0, \, , f (- x) = - f (x), , Hence, f ( x ) is odd., , = Positive, " x Î R, , Corner points Values of Z = x - 7 y + 190, A (4 , 0 ), B ( 5, 0 ), C ( 5, 3 ), D ( 3, 5 ), E (0, 5 ), , …(i), , F (0, 4 ), , Z = 4 - 7(0 ) + 190 = 194, Z = 5 - 7(0 ) + 190 = 195, (Maximum), Z = 5 - 7( 3 ) + 190 = 174, Z = 3 - 7( 5 ) + 190 = 158, Z = 0 - 7( 5 ) + 190 = 155, (Minimum), Z = 0 - 7( 4 ) + 190 = 162, , Zmaximum = 195 and Zminimum = 155, …(ii), , \ Zmaximum - Zminimum = 195 - 155 = 40, 27. Given, objective function, Z = 7 x + 4 y, , …(iii), , 24. We know that, if P and Q are two non-empty, finite set containing m and n elements, respectively, then the number of one-one and, onto mapping from P to Q is, n!, if m = n, 0, if m ¹ n, Given that,, m = 3 and n = 4, \, m¹ n, Number of mapping = 0, 25. f ( x ) = log[ x 6 + 1 + x 3 ], , x6 + 1, , The shaded region in the graph represents the, feasible region ABCDEFA and its corner points, are A ( 4 , 0 ), B( 5 , 0 ), C ( 5 , 3 ), D (3 , 5 ), E (0 , 5 ) and, F (0 , 4 )., The values of Z at the corner points are given, below, , 23. Consider that aRb, if a is congruent to b,, " a, b Î T., Then,, aRa Þ a @ a,, which is true for all a Î T, So, R is reflexive., Let, aRb Þ a @ b, Þ, b@a, Þ, bRa, So, R is symmetric., Let aRb and bRc, Þ, a @ b and b @ c, Þ, a @ c Þ aRc, So, R is transitive., Hence, R is equivalence relation., , 3x2, , 26. Given, objective function Z = x - 7 y + 190, , Maximum Z = 138600 and Minimum Z = 0, \, , é, ù, 5, dy, 1, ê3 x 2 + 6 x, ú, =, dx ( x 6 + 1 + x 3 ) ê, 2 x 6 + 1 úû, ë, , Hence, f ( x ) is an increasing function., , 0 + 170 ´ 450 = 76500, , (0, 450), , Again,, , and equation of lines are, …(i), 3 x + 2 y = 12, …(ii), 3x + y = 9, To get point B, we can solve Eqs. (i), and (ii)., subtracting Eq. (ii) from Eq. (i), we get, y=3, Therefore, 3 x + 3 = 9, [from Eq. (ii)], Þ, 3x = 6, Þ, x=2, Point B has coordinate as (2, 3)., Z| at B = 7 ´ 2 + 4 ´ 3 = 14 + 12 = 26, Z| at C = 7 ´ 0 + 4 ´ 6 = 24, \ Z|at B + Z|at C = 26 + 24 = 50, 28. We have, f ( x ) = x 3 - 3 x 2 + 4 x " x Î R, f ¢ (x) = 3x2 - 6x + 4, = 3( x 2 - 2 x + 1) + 1, = 3( x - 1)2 + 1 > 0,, in every interval of R., Therefore, the function f is strictly increasing, on R., , \

Page 181 :
173, , CBSE Sample Paper Mathematics Class XII (Term I), , SAMPLE PAPER 10, MATHEMATICS, A Highly Simulated Practice Questions Paper, for CBSE Class XII (Term I) Examination, , Instructions, 1. This question paper contains three sections - A, B and C. Each section is compulsory., 2. Section - A has 20 MCQs, attempt any 16 out of 20., 3. Section - B has 20 MCQs, attempt any 16 out of 20., 4. Section - C has 10 MCQs, attempt any 8 out of 10., 5. There is no negative marking., 6. All questions carry equal marks., Maximum Marks : 40, Time allowed : 90 min, , Roll No., , Section A, In this section, attempt any 16 questions out of Questions 1-20. Each question is of 1 mark weightage., , 1. If y = log x 2 , then, (a) 1, , dy, at x = 2 is equal to, dx, (b) 2, , (c) 3, , (d) 0, , é 1 0ù, é 2x 0ù, and A -1 = ê, ú , then 2x equals, ú, ë -1 2û, ë x xû, 1, (a) 2, (b) (c) 1, 2, , 2. If A = ê, , 7p ö, ÷ is, 6 ø, p, (b), 6, , (d), , 1, 2, , 3. The value of cos -1 æç cos, è, , 7p, 6, , (c), , 5p, 6, , (d) None of these, , 4. The relation ‘has the same father as’ over the set of children is, (a) only reflexive, (c) only transitive, , (b) only symmetric, (d) an equivalence relation, , 1, 2, , 2, 7, (d) 3, , 5. The elements a ij of a 3 ´ 3 matrix are given by a ij = |-3i + j| , then a 32 is equal to, (a) 0, , (b) 1, , (c) 2, , SAMPLE PAPER 10, , (a)

Page 182 :
174, , CBSE Sample Paper Mathematics Class XII (Term I), , 6. If y = 3e 2 x + e - x and, (a) 1, , d2y, 2, , - y = ke 2 x , then k is equal to, , dx, (b) 2, , (c) 9, , (d) 8, , 7. Consider the linear programming problem., Maximise Z = x + 3y ; Subject to the constraints x + y £ 40, x + y £ 90 and x , y ³ 0,, then maximum value of Z is, (a) 0, (b) 50, (c) 90, (d) does not exists, , 8. The corner point of the feasible region determined by the system of linear constraints, , are (0, 0), (0, 20), (10, 20), (30, 10), (30, 0). The objective function is Z = 2x + 3y. Compare, the quantity in Column A and Column B., Column A, , Column B, , Maximum of Z, , 80, , (a) The quantity in column A is greater, (b) The quantity in column B is greater, (c) The two quantities are equal, (d) The relationship cannot be determined on the basis of information supplied, , dy, -2, at t =, is equal to, dx, 3, (c) 1, (d) -2, , 9. If x = (t - 1)(t 2 + 1 + t) and y = (1 - t)(1 + t) , then, (a) 0, , (b) 5, , dy, at ( a , b) is equal to, dx, (b) 3, (c) 0, , 10. If ay 2 + bx 2 + c = 0, then, (a) 2, , 11. If, , 2x, , 3, , 2, , x, , =, , 6 -2, 7, , 3, , (a) 10, , (d) -1, , 2, , , then the value of, , x +1, , (b) 20, , é 0 -4 1 ù, 12. The matrix ê 4, 0 12ú is a, ú, ê, êë -1 -12 0 úû, (a) diagonal matrix, (c) skew-symmetric matrix, é 3 1 -1ù, ú , then AA¢ is equal to, ë0 1 2 û, 11ù, é0 0ù, (b) ê, ú, ú, -1û, ë0 0û, , 2, , is, (c) 15, , (d) 1, , (b) symmetric matrix, (d) scalar matrix, , SAMPLE PAPER 10, , 13. If A = ê, é1, (a) ê, ë5, , é11 1ù, (c) ê, ú, ë 1 5û, , é 11 -1ù, (d) ê, ú, ë -1 5 û, , 14. Let X = {0, 1, 2, 3, 4, 5} and Y = {-4 , - 1, 0, 1, 4, 9, 16, 25} and f : X ® Y defined by y = x 2 , is, (a) one one onto, , (b) one one into, , (c) many one onto, , (d) many one into, , 15. The function f : R ® R given by f (x) = x 3 + 1 is, (a) one-one but not onto, (c) bijection, , (b) onto but not one-one, (d) neither one-one nor onto

Page 183 :
175, , CBSE Sample Paper Mathematics Class XII (Term I), , é1 - 1ù, ú , then which of the following result is true, ë 2 - 1û, (a) A2 = I, (b) A2 = - I, 2, (c) A = 2 I, (d) None of these, , 16. If A = ê, , 17. If D =, , a + ib c + id, , then D is equal to, -c + id a - ib, , (a) a 2 + b 2 + c 2 + d 2, , (b) a 2 - b 2 - c 2 - d 2, , (c) a 2 - b 2 + c 2 - d 2, , (d) a 2 - b 2 - c 2 + d 2, , 18. Corner points of the feasible region for an LPP are : (0, 2), (3, 0), (6, 0) and (0, 5)., Let Z = 3x + 2y be the objective function. Then, Maximum Z - Minimum Z is equal to, (a) 20, (b) 16, (c) 14, (d) 18, , 19. The minimum value of y = x 4 + 1 is, (a) 1, , (c) - 1, , (b) 0, , (d) None of these, , 2, , | B| + 1, é1 0ù, é 2 4ù, and B = ê, ú then | A| is equal to, ú, 1, 5, 1, 3, û, ë, û, ë, (a) 13, (b) 12, (c) 26, , 20. If A = ê, , (d) 0, , Section B, In this section, attempt any 16 questions out of Questions 21-40. Each question is of 1 mark weightage., , 21. If y = x 2 + 1 +, (a), , 3, 2, , dy, 1, , then, at x = 1 is equal to, dx, x +1, 3, 4, (b), (c), 3, 4 2, , (d) None of these, , 22. The feasible solution for a LPP is shown in following figure. Let Z = 3x + y be the, objective function. Maximum of Z occurs at, Y, , (3, 6), , (7, 4), , (0, 4), (5, 3), , X, (4, 0), , (a) (7, 4), (c) (0, 4), , (b) (5, 3), (d) (3, 6), , é 3 -3ù, l, 2, ú and A = 2 A, then the value of l is, 3, 3, û, ë, (b) 10, (c) 11, , 23. If matrix A = ê, (a) 12, , (d) 14, , SAMPLE PAPER 10, , (0, 0)

Page 185 :
177, , CBSE Sample Paper Mathematics Class XII (Term I), , ì x2, , x ³1, , then at x = 1, îïx + 1 , x < 1, , 36. Let f (x) = í, , (a) LHL = RHL, , (b) LHL ¹ RHL, , (c) LHL = f (1), , (d) None of these, , 37. Let f : R ® R be a function such that f (x) = x 3 + 3x 2 + 5x + sin x, then f (x) is, (a) an increasing function, (b) decreasing function, (c) neither increasing nor decreasing function (d) None of these, , p, 38. The function f (x) = log(cos x) on the interval æç 0, ö÷ is, è, , 2ø, (b) Strictly decreasing, (d) None of these, , (a) Increasing, (c) Strictly increasing, , 39. The slope of the normal to the curve y = 2x 2 + 3 sin x at x = 0 is, (a) 3, , (b), , 1, 3, , (c) - 3, , 40. The slope of the normal to the curve x = 1 - a sin q, y = b cos 2 q at q =, (a), , a, 2b, , (b), , -a, 2b, , (c), , b, 2a, , (d), , -1, 3, , p, is, 2, -b, (d), 2a, , Section C, In this section, attempt any 8 questions. Each question is of 1 mark weightage. Questions 46-50 are based, on Case-Study., , 1, 2, , 41. Let r be the relation on the set R of all real numbers defined by setting arb iff |a - b|£ ., Then, r is, (a) reflexive and symmetric but not transitive, (b) symmertic and transitive but not reflexive, (c) transitive but neither reflexive nor symmetric, (d) None of the above, , 42. If A 2 + A + I = 0, then A - 1 is equal to, (a) A - I, , (b) I - A, , (d) None of these, , ö, ÷, ÷ , x ¹ 0 is continuous at x = 0, then the value of k is, ÷, ø, x=0, (c) 0, , (d) 2, , 44. The function f (x) = 2x 3 - 3x 2 - 36x + 7 is strictly decreasing in the interval, (a) ( - ¥ , - 2 ), , (b) ( - 2 , 3), , (c) (3, ¥ ), , (d) None of these, , 2, , 45. The point on the curve y = (x - 3) , where the tangent is parallel to the chord joining, (3, 0) and (4, 1) is, æ 7 1ö, (a) ç - , ÷, è 2 4ø, , æ 5 1ö, (b) ç , ÷, è2 4 ø, , æ - 5 1ö, (c) ç, , ÷, è 2 4ø, , æ7 1ö, (d) ç , ÷, è2 4 ø, , SAMPLE PAPER 10, , 1, ì æ, ï ç 1 + ex, x, 1, 43. If the function f (x) = ïí çç, ï è 1 - ex, ï, k,, îï, (a) - 1, (b) 1, , (c) - ( A + I )

Page 187 :
OMR SHEET, , SP 10, , Roll No., Sub Code., , Student Name, , Instructions, Use black or blue ball point pens and avoid Gel & Fountain pens for filling the OMR sheet., Darken the bubbles completely. Don’t put a tick mark or a cross mark, half-filled or over-filled bubbles will not be read, by the software., Correct, , ✔, , ✗, , Incorrect, , Incorrect, , Incorrect, , Do not write anything on the OMR Sheet., Multiple markings are invalid., , 1, , 18, , 35, , 2, , 19, , 36, , 3, , 20, , 37, , 4, , 21, , 38, , 5, , 22, , 39, , 6, , 23, , 40, , 7, , 24, , 41, , 8, , 25, , 42, , 9, , 26, , 43, , 10, , 27, , 44, , 11, , 28, , 45, , 12, , 29, , 46, , 13, , 30, , 47, , 14, , 31, , 48, , 15, , 32, , 49, , 16, , 33, , 50, , 17, , 34, , Check Your Performance, Total Questions:, Total Correct Questions:, , If Your Score is, , Score Percentage =, , Total Correct Questions, Total Questions, , × 100, , Less than 60%, > Average (Revise the concepts again), Greater than 60% but less than 75% > Good (Do more practice), Above 75%, > Excellent (Keep it on)

Page 194 :
Downloaded from - Term 1, Books, Join us on Telegram – Click, Here