Page 1 :
Class 10 Math’s Formula, CBSE Class 10 Math’s Summary, This pdf list all the Class 10 CBSE math’s formula in a concise, manner to help the students in revision and examination as, per NCERT syllabus

Page 2 :
1, , Real Numbers, S.no, , Type of Numbers, , Description, , 1, , Natural Numbers, , 2, , Whole number, , 3, , Integers, , N = {1,2,3,4,5……….}, It is the counting numbers, W= {0,1,2,3,4,5……..}, It is the counting numbers + zero, Z={…-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6…}, , 4, , Positive integers, , Z+= {1,2,3,4,5……..}, , 5, , Negative integers, , Z-={…-7,-6,-5,-4,-3,-2,-1}, , 6, , Rational Number, , A number is called rational if it can be expressed, in the form p/q where p and q are integers ( q>, 0)., Example : ½ , 4/3 ,5/7 ,1 etc., , 7, , Irrational Number, , A number is called rational if it cannot be, expressed in the form p/q where p and q are, integers ( q> 0)., Example : √3, √2, √5,  etc, , 8., , Real Numbers:, , All rational and all irrational number makes the, collection of real number. It is denoted by the, letter R, , S.no, , Terms, , Descriptions, , 1, , Euclid’s Division, Lemma, , For a and b any two positive integer, we can always, find unique integer q and r such that, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 3 :
2, , a=bq + r, , ,, , 0≤r≤b, , If r =0, then b is divisor of a., , 2, , HCF (Highest, common factor), , HCF of two positive integers can be find using the, Euclid’s Division Lemma algorithm, We know that for any two integers a, b. we can write, following expression, a=bq + r, , ,, , 0≤r≤b, , If r=0 ,then, HCF( a, b) =b, If r≠0 , then, HCF ( a, b) = HCF ( b,r), Again expressing the integer b,r in Euclid’s Division, Lemma, we get, b=pr + r1, HCF ( b,r)=HCF ( r,r1), Similarly successive Euclid ‘s division can be written, until we get the remainder zero, the divisor at that, point is called the HCF of the a and b, 3, , HCF ( a,b) =1, , Then a and b are co primes., , 4, , Fundamental, Theorem of, Arithmetic, , Composite number = Product of primes, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 4 :
3, , HCF and LCM by, prime factorization, method, , HCF = Product of the smallest power of each, common factor in the numbers, , 6, , Important Formula, , HCF (a,b) X LCM (a,b) =a X b, , 7, , Important concept, for rational Number, , Terminating decimal expression can be written in, the form, , 5, , LCM = Product of the greatest power of each prime, factor involved in the number, , p/2n5m, , Polynomial expressions, A polynomial expression S(x) in one variable x is an algebraic expression in x term, as, , , , , , , , , , , , , , ,  ⋯………, , , , , , Where an,an-1,…,a,a0 are constant and real numbers and an is not equal to zero, , Some Important point to Note, S.no, , Points, , 1, , an ,an-1 ,an-2 ,…..a1,a0, , are called the coefficients for xn,xn-1 ,…..x1,x0, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 5 :
4, , 2, , n is called the degree of the polynomial, , 3, , when an ,an-1 ,an-2 ,…..a1,a0, , 4, , A constant polynomial is the polynomial with zero degree, it is a constant, value polynomial, , 5, , A polynomial of one item is called monomial, two items binomial and three, items as trinomial, , 6, , A polynomial of one degree is called linear polynomial, two degree as, quadratic polynomial and degree three as cubic polynomial, , all are zero, it is called zero polynomial, , Important concepts on Polynomial, Concept, , Description, , Zero’s or roots, of the, polynomial, , It is a solution to the polynomial equation S(x)=0 i.e. a number, "a" is said to be a zero of a polynomial if S(a) = 0., If we draw the graph of S(x) =0, the values where the curve, cuts the X-axis are called Zeroes of the polynomial, , Remainder, Theorem’s, , If p(x) is an polynomial of degree greater than or equal to 1, and p(x) is divided by the expression (x-a),then the remainder, will be p(a), , Factor’s, Theorem’s, , If x-a is a factor of polynomial p(x) then p(a)=0 or if p(a), =0,x-a is the factor the polynomial p(x), , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 6 :
5, , Geometric Meaning of the Zeroes of the polynomial, Let’s us assume, y= p(x) where p(x) is the polynomial of any form., Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained, by putting the values. The plot or graph obtained can be of any shapes, The zeroes of the polynomial are the points where the graph meet x axis in the Cartesian plane. If the, graph does not meet x axis ,then the polynomial does not have any zero’s., Let us take some useful polynomial and shapes obtained on the Cartesian plane, S.no, , y=p(x), , 1, , y=ax+b where a and b can be, any values (a≠0), , Graph obtained, , Name of the, graph, , Name of the, equation, , Straight line., , Linear polynomial, , It intersect the xaxis at ( -b/a ,0), Example y=2x+3, , 2, , Example ( -3/2,0), , y=ax2+bx+c, , Quadratic, polynomial, , where, , Parabola, , b2-4ac > 0 and a≠0 and a> 0, , It intersect the xaxis at two points, , Example, Example, y=x2-7x+12, (3,0) and (4,0), , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 7 :
6, , 3, , y=ax2+bx+c, , Quadratic, polynomial, , where, , Parabola, , b2-4ac > 0 and a≠0 and a < 0, , It intersect the xaxis at two points, , Example, , 4, , y=-x2+2x+8, , Example (-2,0), and (4,0), , y=ax2+bx+c, , Parabola, , where, , It intersect the xaxis at one points, , Quadratic, polynomial, , b2-4ac = 0 and a≠0 a > 0, , Example, y=(x-2)2, 5, , y=ax2+bx+c, , Parabola, , where, , It does not, intersect the x-axis, , Quadratic, polynomial, , b2-4ac < 0 and a≠0 a > 0, It has no zero’s, Example, y=x2-2x+6, 6, , y=ax2+bx+c, , Parabola, , where, , It does not, intersect the x-axis, , Quadratic, polynomial, , b2-4ac < 0 and a≠0 a < 0, It has no zero’s, Example, y=-x2-2x-6, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 8 :
7, , 7, , y=ax3 +bx2+cx+d, , It can be of any, shape, , It will cut the x-axis, at the most 3 times, , Cubic Polynomial, , It can be of any, shape, , It will cut the x-axis, at the most n times, , Polynomial of n, degree, , where a≠0, , , , 8, , , ,   ,  ⋯………,  , , , , Where an ≠0, , Relation between coefficient and zeroes of the Polynomial:, , S.no, , Type of Polynomial, , General form, , Zero’s, , 1, , Linear polynomial, , ax+b , a≠0, , 1, , 2, , Quadratic, , ax2+bx+c, a≠0, , 2, , Relationship between Zero’s, and coefficients, , ,   ,  , , 3, , Cubic, , ax3+bx2+cx+d, a≠0, , 3, ,   , !!" !, , , !!" !, !!" !, ,   , !!" !, ,     #, !!" !, , !!" !, , , , , , , #, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 9 :
8, ,   #, , , ,   , !!" ! #, ,     #   #, !!" !, !!" ! , , Formation of polynomial when the zeroes are given, Type of, polynomial, , Zero’s, , Polynomial Formed, , Linear, , k=a, , (x-a), , Quadratic, , k1=a and, k2=b, , (x-a)(x-b), Or, x2-( a+b)x +ab, Or, x2-( Sum of the zero’s)x +product of the zero’s, , Cubic, , k1=a ,k2=b, and k3=c, , (x-a)(x-b)(x-c), , Division algorithm for Polynomial, Let‘s p(x) and q(x) are any two polynomial with q(x) ≠0 ,then we can find polynomial s(x) and r(x) such, that, P(x)=s(x) q(x) + r(x), Where r(x) can be zero or degree of r(x) < degree of g(x), , Dividend =Quotient X Divisor + Remainder, This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 10 :
9, , COORDINATE GEOMETRY, S.no, , Points, , 1, , We require two perpendicular axes to locate a point in the plane. One of, them is horizontal and other is Vertical, , 2, , The plane is called Cartesian plane and axis are called the coordinates axis, , 3, , The horizontal axis is called x-axis and Vertical axis is called Y-axis, , 4, , The point of intersection of axis is called origin., , 5, , The distance of a point from y axis is called x –coordinate or abscissa and, the distance of the point from x –axis is called y – coordinate or Ordinate, , 6, , The distance of a point from y axis is called x –coordinate or abscissa and, the distance of the point from x –axis is called y – coordinate or Ordinate, , 7, , The Origin has zero distance from both x-axis and y-axis so that its, abscissa and ordinate both are zero. So the coordinate of the origin is (0,, 0), , 8, , A point on the x –axis has zero distance from x-axis so coordinate of any, point on the x-axis will be (x, 0), , 9, , A point on the y –axis has zero distance from y-axis so coordinate of any, point on the y-axis will be (0, y), , 10, , The axes divide the Cartesian plane in to four parts. These Four parts are, called the quadrants, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 11 :
10, , The coordinates of the points in the four quadrants will have sign according to the, below table, , Quadrant, , x-coordinate, , y-coordinate, , Ist Quadrant, , +, , +, , IInd quadrant, , -, , +, , IIIrd quadrant, , -, , -, , IVth quadrant, , +, , -, , S.no, , Terms, , Descriptions, , 1, , Distance formula, , Distance between the points AB is given by, $, , %, , , , , , , , , ,  &  &, , , , Distance of Point A from Origin, $, 2, , Section Formula, , %, , , ,  &, , A point P(x,y) which divide the line segment AB in, the ratio m1 and m2 is given by, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 12 :
11, , &, ,    ,   , , , ,  &   &,   , , The midpoint P is given by, ', 3, , Area of Triangle, , () *(+, , , ,,, , -) *-+, , , Area of triangle ABC of coordinates A(x1,y1) ,, B(x2,y2) and C(x3,y3), ., , 1, 0 &  &# , 2  , ,  &#, ,  & , , # &, ,  & 1, , For point A,B and C to be collinear, The value of A, should be zero, , LINEAR EQUATIONS IN TWO, VARIABLES, An equation of the form ax + by + c = 0, where a, b and c are real numbers, such, that a and, b are not both zero, is called a linear equation in two variables, Important points to Note, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 13 :
12, , S.no, , Points, , 1, , A linear equation in two variable has infinite solutions, , 2, , The graph of every linear equation in two variable is a straight line, , 3, , x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis, , 4, , The graph x=a is a line parallel to y -axis., , 5, , The graph y=b is a line parallel to x -axis, , 6, , An equation of the type y = mx represents a line passing through the, origin., , 7, , Every point on the graph of a linear equation in two variables is a solution, of the linear, equation. Moreover, every solution of the linear equation is a point on the, graph, , S.no, , Type of equation, , Mathematical, representation, , Solutions, , 1, , Linear equation in one Variable, , ax+b=0 ,a≠0, , One solution, , a and b are real, number, 2, , Linear equation in two Variable, , ax+by+c=0 , a≠0 and, b≠0, , Infinite solution, possible, , a, b and c are real, number, , 3, , Linear equation in three Variable, , ax+by+cz+d=0 , a≠0, ,b≠0 and c≠0, , Infinite solution, possible, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 14 :
13, , a, b, c, d are real, number, , Simultaneous pair of Linear equation:, A pair of Linear equation in two variables, a1x+b1y+c1=0, a2x +b2y+c2=0, Graphically it is represented by two straight lines on Cartesian plane., Simultaneous pair of, Linear equation, , Condition, , , a1x+b1y+c1=0, , , , a2x +b2y+c2=0, , 2, , 3, 3, , Example, , Graphical, representation, , Algebraic, interpretation, , Intersecting lines. The, intersecting point, coordinate is the only, solution, , One unique solution, only., , Coincident lines. The, any coordinate on the, line is the solution., , Infinite solution., , x-4y+14=0, 3x+2y-14=0, , a1x+b1y+c1=0, a2x +b2y+c2=0, , , , , , 3, 3, , , , , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 15 :
14, , Example, 2x+4y=16, 3x+6y=24, , a1x+b1y+c1=0, a2x +b2y+c2=0, , , , , , 3 , 2, 3 , , Parallel Lines, , No solution, , Example, 2x+4y=6, 4x+8y=18, , The graphical solution can be obtained by drawing the lines on the Cartesian plane., Algebraic Solution of system of Linear equation, , S.no, , Type of method, , Working of method, , 1, , Method of elimination by, substitution, , 1) Suppose the equation are, a1x+b1y+c1=0, a2x +b2y+c2=0, 2) Find the value of variable of either x or y in other, variable term in first equation, 3) Substitute the value of that variable in second, equation, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 16 :
15, , 4) Now this is a linear equation in one variable. Find, the value of the variable, 5) Substitute this value in first equation and get the, second variable, 2, , Method of elimination by, equating the coefficients, , 1) Suppose the equation are, a1x+b1y+c1=0, a2x +b2y+c2=0, 2) Find the LCM of a1 and a2 .Let it k., 3) Multiple the first equation by the value k/a1, 4) Multiple the first equation by the value k/a2, 4) Subtract the equation obtained. This way one, variable will be eliminated and we can solve to get the, value of variable y, 5) Substitute this value in first equation and get the, second variable, , 3, , Cross Multiplication method, , 1) Suppose the equation are, a1x+b1y+c1=0, a2x +b2y+c2=0, , 2) This can be written as, 4), 4+, , (, , 5), 5+, , 6) 5), 6+ 5+, , 6), 6+, , , , 4), 4+, , 3) This can be written as, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 17 :
16, , (, 4) 5+ 4+ 5), , 6) 5+ 6+ 5), , , 6) 4+ 6+ 4), , 4) Value of x and y can be find using the, x => first and last expression, y=> second and last expression, , Quadratic Equations, S.no, , Terms, , Descriptions, , 1, , Quadratic Polynomial P(x) = ax2 +bx+c where a≠0, , 2, , Quadratic equation, , 3, , Solution or root of the A real number α is called the root or solution of the, Quadratic equation, quadratic equation if, , ax2 +bx+c =0, , where a≠0, , aα2 +bα+c=0, , 4, , zeroes of the, polynomial p(x)., , The root of the quadratic equation are called zeroes, , 5, , Maximum roots of, quadratic equations, , We know from chapter two that a polynomial of, degree can have max two zeroes. So a quadratic, equation can have maximum two roots, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 18 :
17, , 6, , A quadratic equation has real roots if b2- 4ac > 0, , Condition for real, roots, , How to Solve Quadratic equation, S.no, , Method, , Working, , 1, , factorization, , This method we factorize the equation by splitting the, middle term b, In ax2+bx+c=0, Example, 6x2-x-2=0, , 1) First we need to multiple the coefficient a and c.In this, case =6X-2=-12, 2) Splitting the middle term so that multiplication is 12, and difference is the coefficient b, 6x2 +3x-4x-2=0, 3x( 2x+1) -2(2x+1)=0, (3x-2) (2x+1)=0, 3) Roots of the equation can be find equating the factors, to zero, 3x-2=0 => x=3/2, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 19 :
18, , 2x+1=0 => x=-1/2, 2, , Square method, , In this method we create square on LHS and RHS and, then find the value., ax2 +bx+c=0, 1) x2 +(b/a) x+(c/a)=0, 2) ( x+b/2a)2 –(b/2a)2 +(c/a)=0, 3) ( x+b/2a)2=(b2-4ac)/4a2, 4), , 47√4+ 865, 6, , Example, x2 +4x-5=0, , 1) (x+2)2 -4-5=0, 2) (x+2)2=9, 3) Roots of the equation can be find using square root on, both the sides, x+2 =-3 => x=-5, x+2=3=> x=1, 3, , Quadratic, method, , For quadratic equation, ax2 +bx+c=0,, roots are given by, 4*√4+ 865, 6, , ,, , 4√4+ 865, 6, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 20 :
19, , For b2 -4ac > 0, Quadratic equation has two real roots of, different value, For b2-4ac =0, quadratic equation has one real root, For b2-4ac < 0, no real roots for quadratic equation, , Nature of roots of Quadratic equation, S.no, , Condition, , Nature of roots, , 1, , b2 -4ac > 0, , Two distinct real roots, , 2, , b2-4ac =0, , One real root, , 3, , b2-4ac < 0, , No real roots, , Triangles, S.no, , Terms, , Descriptions, , 1, , Congruence, , Two Geometric figure are said to be congruence if, they are exactly same size and shape, Symbol used is ≅, Two angles are congruent if they are equal, Two circle are congruent if they have equal radii, Two squares are congruent if the sides are equal, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 21 :
20, , 2, , Triangle Congruence, , •, , Two triangles are congruent if three sides, and three angles of one triangle is, congruent to the corresponding sides and, angles of the other, A, , B, •, •, •, •, , •, , 3, , Inequalities in, Triangles, , D, , C, , E, , F, , Corresponding sides are equal, AB=DE , BC=EF ,AC=DF, Corresponding angles are equal, ∠. ∠$, ∠; ∠<, ∠, ∠=, We write this as, .; ≅ $<=, The above six equalities are between the, corresponding parts of the two congruent, triangles. In short form this is called, C.P.C.T, We should keep the letters in correct order, on both sides, , 1) In a triangle angle opposite to longer side is, larger, 2) In a triangle side opposite to larger angle is, larger, 3) The sum of any two sides of the triangle is, greater than the third side, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 22 :
21, , In triangle ABC, AB +BC > AC, , Different Criterion for Congruence of the triangles, , N Criterion, , 1 Side angle, Side (SAS), congruence, , Description, , •, , •, , Two triangles are congruent if the, two sides and included angles of, one triangle is equal to the two, sides and included angle, It is an axiom as it cannot be, proved so it is an accepted truth, , Figures and, expression, , A, , D, E, , •, , ASS and SSA type two triangles, may not be congruent always, , C, , B, , F, , If following, condition, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 23 :
22, , AB=DE, BC=EF, ∠>, , ∠?, , Then, , 2 Angle side, angle (ASA), congruence, , •, , •, , Two triangles are congruent if the, two angles and included side of, one triangle is equal to the, corresponding angles and side, It is a theorem and can be proved, , .; ≅ $<=, A, C, , B, D, , F, , E, , If following, condition, BC=EF, ∠>, , ∠?, ∠@, , Then, , 3 Angle angle, side( AAS), congruence, , •, , •, , Two triangles are congruent if the, any two pair of angles and any, side of one triangle is equal to the, corresponding angles and side, It is a theorem and can be proved, , ∠A, , .; ≅ $<=, , C, , B, D, E, , F, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 24 :
23, , If following, condition, BC=EF, ∠B, , ∠C, ∠@, , ∠A, , Then, , 4 Side-Side-Side, (SSS), congruence, , •, , Two triangles are congruent if the, three sides of one triangle is equal, to the three sides of the another, , .; ≅ $<=, A, C, , B, , D, , F, , E, If following, condition, , BC=EF,AB=DE,DF, =AC, Then, , .; ≅ $<=, 5 Right angle –, hypotenuseside(RHS), , •, , Two right triangles are congruent if, the hypotenuse and a side of the, one triangle are equal to, , A, , D, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 25 :
24, , congruence, , corresponding hypotenuse and side, of the another, , B, , C E, , F, , If following, condition, AC=DF,BC=EF, Then, , .; ≅ $<=, , Some Important points on Triangles, Terms, Orthocenter, Equilateral, Median, , Altitude, , Isosceles, Centroid, , In center, Circumcenter, , Description, Point of intersection of the three altitude, of the triangle, triangle whose all sides are equal and all, angles are equal to 600, A line Segment joining the corner of the, triangle to the midpoint of the opposite, side of the triangle, A line Segment from the corner of the, triangle and perpendicular to the, opposite side of the triangle, A triangle whose two sides are equal, Point of intersection of the three median, of the triangle is called the centroid of, the triangle, All the angle bisector of the triangle, passes through same point, The perpendicular bisector of the sides, of the triangles passes through same, point, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 26 :
25, , Scalene triangle, , Triangle having no equal angles and no, equal sides, Right triangle has one angle equal to 900, One angle is obtuse angle while other, two are acute angles, All the angles are acute, , Right Triangle, Obtuse Triangle, Acute Triangle, , Similarity of Triangles, S.no, , Points, , 1, , Two figures having the same shape but not necessarily the same size are called similar, figures., , 2, , All the congruent figures are similar but the converse is not true., , 3, , If a line is drawn parallel to one side of a triangle to intersect the other two sides in, distinct points, then the other two sides are divided in the same ratio, , 4, , If a line divides any two sides of a triangle in the same ratio, then the line is parallel to, the third side., , Different Criterion for Similarity of the triangles, N Criterion, 1 Angle Angle, angle(AAA), similarity, , Description, •, , Two triangles are similar if, corresponding angle are equal, , Expression, If following, condition, ∠B, , ∠C, , ∠@, , ∠A, , ∠>, , ∠?, , Then, , B>, C?, , >@, ?A, , B@, CA, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 27 :
26, , Then, , 2 Angle angle, (AA) similarity, , •, , Two triangles are similar if the two, corresponding angles are equal as, by angle property third angle will, be also equal, , .; ~ $<=, , If following, condition, ∠B, , ∠>, , ∠C, ∠?, , Then, ∠@, , ∠A, , Then, , B>, C?, , Then, , 3 Side side, side(SSS), Similarity, , Two triangles are similar if the, sides of one triangle is, proportional to the sides of other, triangle, , >@, ?A, , B@, CA, , .; ~ $<=, , If following, condition, B> >@ B@, C? ?A CA, Then, ∠B, , ∠C, , ∠@, , ∠A, , ∠>, , ∠?, , Then, , .; ≅ $<=, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 28 :
27, , 4 Side-AngleSide (SAS), similarity, , •, , Two triangles are similar if the one, angle of a triangle is equal to one, angle of other triangles and sides, including that angle is proportional, , If following, condition, B> B@, C? CA, , And ∠B, Then, , ∠C, , .; ≅ $<=, , Area of Similar triangles, If the two triangle ABC and DEF are similar, , .; ≅ $<=, , Then, , . ! E" FG .;, . ! " FG $<=, , .;, , $<, , , , ;, , <=, , , , ., , $=, , , , Pythagoras Theorem, , S.no, , Points, , 1, , If a perpendicular is drawn from the vertex of the right angle of a right triangle to the, hypotenuse, then the triangles on both sides of the perpendicular are similar to the, whole triangle and also to each other., , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 29 :
28, , 2, , In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the, other two sides (Pythagoras Theorem)., (hyp)2 =(base)2 + (perp)2, , 3, , If in a triangle, square of one side is equal to the sum of the squares of the other two, sides, then the angle opposite the first side is a right angle, , Arithmetic Progression, S.no, , Terms, , Descriptions, , 1, , Arithmetic, Progression, , An arithmetic progression is a sequence of, numbers such that the difference of any two, successive members is a constant, Examples, 1), 1,5,9,13,17…., 2), 1,2,3,4,5,…, , 2, , common difference of, the AP, , the difference between any successive members is a, constant and it is called the common difference of AP, , 1) If a1, a2,a3,a4,a5 are the terms in AP then, D=a2 -a1 =a3 - a2 =a4 – a3=a5 –a4, 2) We can represent the general form of AP in, the form, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 30 :
29, , a,a+d,a+2d,a+3d,a+4dJJJ.., Where a is first term and d is the common, difference, , nth term = a + (n - 1)d, , 3, , nth term of Arithmetic, Progression, , 4, , Sum of nth item in Sn =(n/2)[a + (n-1)d], Arithmetic, Or, Progression, Sn =(n/2)[t1+ tn], , Trigonometry, S.no, , Terms, , Descriptions, , 1, , What is, Trigonometry, , Trigonometry from Greek trigõnon, "triangle" and, metron, "measure") is a branch of mathematics, that studies relationships involving lengths and, angles of triangles. The field emerged during the, 3rd century BC from applications of geometry to, astronomical studies., Trigonometry is most simply associated with, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 31 :
30, , planar right angle triangles (each of which is a, two-dimensional triangle with one angle equal to, 90 degrees). The applicability to non-right-angle, triangles exists, but, since any non-right-angle, triangle (on a flat plane) can be bisected to, create two right-angle triangles, most problems, can be reduced to calculations on right-angle, triangles. Thus the majority of applications relate, to right-angle triangles, 2, , Trigonometric, Ratio’s, , In a right angle triangle ABC where B=90°, , We can define following term for angle A, Base : Side adjacent to angle, Perpendicular: Side Opposite of angle, Hypotenuse: Side opposite to right angle, We can define the trigonometric ratios for, angle A as, sin A= Perpendicular/Hypotenuse =BC/AC, cosec A= Hypotenuse/Perpendicular, =AC/BC, cos A= Base/Hypotenuse =AB/AC, sec A= Hypotenuse/Base=AC/AB, tan A= Perpendicular/Base =BC/AB, cot A= Base/Perpendicular=AB/BC, Notice that each ratio in the right-hand column is, the inverse, or the reciprocal, of the ratio in the, left-hand column., , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 32 :
31, , 3, , Reciprocal of, functions, , The reciprocal of sin A is cosec A ; and viceversa., The reciprocal of cos A is sec A, And the reciprocal of tan A is cot A, These are valid for acute angles., We can define tan A = sin A/cos A, And Cot A =cos A/ Sin A, , 4, , Value of of sin and, cos, , Is always less 1, , 5, , Trigonometric ration, from another angle, , We can define the trigonometric ratios for angle C, as, , 6, , Trigonometric ratios, of complimentary, , sin C= Perpendicular/Hypotenuse =AB/AC, cosec C= Hypotenuse/Perpendicular =AC/ABB, cos C= Base/Hypotenuse =BC/AC, sec C= Hypotenuse/Base=AC/BC, tan A= Perpendicular/Base =AB/BC, cot A= Base/Perpendicular=BC/AB, Sin (90-A) =cos(A), Cos(90-A) = sin A, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 33 :
32, , angles, , Tan(90-A) =cot A, Sec(90-A)= cosec A, Cosec (90-A) =sec A, , 7, , Trigonometric, identities, , Cot(90- A) =tan A, Sin2 A + cos2 A =0, 1 + tan2 A =sec2 A, 1 + cot2 A =cosec2 A, , Trigonometric Ratios of Common angles, We can find the values of trigonometric ratio’s various angle, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 34 :
33, , Area of Circles, S.no, , Points, , 1, , A circle is a collection of all the points in a plane, which are equidistant, from a fixed point in the plane, , 2, , Equal chords of a circle (or of congruent circles) subtend equal angles at, the center., , 3, , If the angles subtended by two chords of a circle (or of congruent circles), at the center (corresponding center) are equal, the chords are equal., , 4, , The perpendicular from the center of a circle to a chord bisects the chord., , 5, , The line drawn through the center of a circle to bisect a chord is, perpendicular to the chord., , 6, , There is one and only one circle passing through three non-collinear points, , 7, , Equal chords of a circle (or of congruent circles) are equidistant from the, center (or corresponding centers)., , 8, , Chords equidistant from the center (or corresponding centers) of a circle, (or of congruent circles) are equal, , 9, , If two arcs of a circle are congruent, then their corresponding chords are, equal and conversely, if two chords of a circle are equal, then their, corresponding arcs (minor, major) are congruent., , 10, , Congruent arcs of a circle subtend equal angles at the center., , 11, , The angle subtended by an arc at the center is double the angle subtended, by it at any point on the remaining part of the circle, , 12, , Angles in the same segment of a circle are equal, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 35 :
34, , 13, , Angle in a semicircle is a right angle., , 14, , If a line segment joining two points subtends equal angles at two other, points lying on the same side of the line containing the line segment, the, four points lie on a circle., , 15, , The sum of either pair of opposite angles of a cyclic quadrilateral is 180°., , 16, , If the sum of a pair of opposite angles of a quadrilateral is 180°, then the, quadrilateral is cyclic., , S.no, , Terms, , Descriptions, , 1, , Circumference of a circle, , 2 π r., , 2, , Area of circle, , π r2, , 3, , Length of the arc of, the sector of angle, , Length of the arc of the sector of angle θ, H, , 2, , 4, , Area of the sector of, angle, , #I, Area of the sector of angle θ, H,  , #I, , 5, , Area of segment of a, circle, , Area of the corresponding sector – Area of the, corresponding triangle, , Surface Area and Volume, S.no, , Term, , Description, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 36 :
35, , 1, , Mensuration, , It is branch of mathematics which is concerned, about the measurement of length ,area and, Volume of plane and Solid figure, , 2, , Perimeter, , a)The perimeter of plane figure is defined as the, length of the boundary, b)It units is same as that of length i.e. m ,cm,km, , 3, , Area, , a)The area of the plane figure is the surface, enclosed by its boundary, b) It unit is square of length unit. i.e. m2 , km2, , 4, , Volume, , Volume is the measure of the amount of space, inside of a solid figure, like a cube, ball, cylinder, or pyramid. Its units are always "cubic", that is,, the number of little element cubes that fit inside, the figure., , Volume Unit conversion, 1 cm3, 1 Litre, , 1mL, 1000ml, , 1000 mm3, 1000 cm3, , 1 m3, , 106 cm3, , 1000 L, , 1 dm3, , 1000 cm3, , 1L, , Surface Area and Volume of Cube and Cuboid, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 37 :
36, , Cuboid, , Cube, , Type, , Measurement, , Surface Area of Cuboid of Length L,, Breadth B and Height H, , 2(LB + BH + LH)., , Lateral surface area of the cuboids, , 2( L + B ) H, , Diagonal of the cuboids, , %J  ;  K , , Volume of a cuboids, , LBH, , Length of all 12 edges of the cuboids, , 4 (L+B+H)., , Surface Area of Cube of side L, , 6L2, , Lateral surface area of the cube, , 4L2, , Diagonal of the cube, , J√3, , Volume of a cube, , L3, , Surface Area and Volume of Right circular cylinder, , Radius, , The radius (r) of the circular base is called the radius of the, cylinder, , Height, , The length of the axis of the cylinder is called the height (h) of the, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 38 :
37, , cylinder, Lateral, Surface, , The curved surface joining the two base of a right circular cylinder is, called Lateral Surface., , Type, , Measurement, , Curved or lateral Surface Area of, cylinder, , 2πrh, , Total surface area of cylinder, , 2πr (h+r), , Volume of Cylinder, , π r2h, , Surface Area and Volume of Right circular cone, , Radius, , The radius (r) of the circular base is called the radius of the, cone, , Height, , The length of the line segment joining the vertex to the center of, base is called the height (h) of the cone., , Slant, Height, , The length of the segment joining the vertex to any point on the, circular edge of the base is called the slant height (L) of the cone., , Lateral, surface, Area, , The curved surface joining the base and uppermost point of a right, circular cone is called Lateral Surface, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 39 :
38, , Type, , Measurement, , Curved or lateral Surface Area of, cone, , πrL, , Total surface area of cone, , πr (L+r), , #, , Volume of Cone, ,   L, , Surface Area and Volume of sphere and hemisphere, Sphere, , Hemisphere, , Sphere, , A sphere can also be considered as a solid obtained on, rotating a circle About its diameter, , Hemisphere A plane through the centre of the sphere divides the sphere into two, equal parts, each of which is called a hemisphere, radius, , The radius of the circle by which it is formed, , Spherical, Shell, , The difference of two solid concentric spheres is called a spherical, shell, , Lateral, Surface, Area for, , Total surface area of the sphere, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 40 :
39, , Sphere, Lateral, It is the curved surface area leaving the circular base, Surface, area of, Hemisphere, , Type, , Measurement, , Surface area of Sphere, , 4πr2, , Volume of Sphere, , 4 #, , 3, , Curved Surface area of hemisphere, , 2πr2, , Total Surface area of hemisphere, , 3πr2, , Volume of hemisphere, , 2 #, , 3, , Volume of the spherical shell whose, outer and inner radii and ‘R’ and ‘r’, respectively, , 4, N #   #, 3, , How the Surface area and Volume are determined, , Area of Circle, , The circumference of a circle is 2πr., This is the definition of π (pi). Divide, the circle into many triangular, segments. The area of the triangles, is 1/2 times the sum of their bases,, 2πr (the circumference), times their, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 41 :
40, , height, r., , ., , 1, 2, 2, ,  , , Surface Area of cylinder, , This can be imagined as unwrapping the, surface into a rectangle., , Surface area of cone, , This can be achieved by divide the, surface of the cone into its triangles, or, the surface of the cone into many thin, triangles. The area of the triangles is 1/2, times the sum of their bases, p, times, their height,, ., , 1, 2, 2, , , , Surface Area and Volume of frustum of cone, , h = vertical height of the frustum, l = slant height of the frustum, r1 and r2 are radii of the two bases (ends) of the frustum., , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 42 :
41, , Type, , Measurement, , Volume of a frustum of a cone, , , L, #, , %L    , , Slant height of frustum of a cone, Curved surface area of a frustum of a cone, Total surface area of frustum of a cone, ,     , , G  , , , , G      , , Statistics, S.no, , Term, , 1, , Statistics, , 2, , Data, , Description, Statistics is a broad mathematical discipline, which studies ways to collect, summarize, and, draw conclusions from data, A systematic record of facts or different values of, a quantity is called data., Data is of two types - Primary data and Secondary, data., Primary Data: The data collected by a researcher, with a specific purpose in mind is called primary, data., Secondary Data: The data gathered from a, source where it already exists is called secondary, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 43 :
42, , data, 3, , Features of data, , •, , •, , •, •, , •, , •, , •, , •, , •, •, , •, , Statistics deals with collection,, presentation, analysis and interpretation of, numerical data., Arranging data in an order to study their, salient features is called presentation of, data., Data arranged in ascending or descending, order is called arrayed data or an array, Range of the data is the difference, between the maximum and the minimum, values of the observations, Table that shows the frequency of different, values in the given data is called a, frequency distribution table, A frequency distribution table that shows, the frequency of each individual value in the, given data is called an ungrouped frequency, distribution table., A table that shows the frequency of groups, of values in the given data is called a, grouped frequency distribution table, The groupings used to group the values in, given data are called classes or classintervals. The number of values that each, class contains is called the class size or, class width. The lower value in a class is, called the lower class limit. The higher value, in a class is called the upper class limit., Class mark of a class is the mid value of, the two limits of that class., A frequency distribution in which the upper, limit of one class differs from the lower limit, of the succeeding class is called an, Inclusive or discontinuous Frequency, Distribution., A frequency distribution in which the upper, limit of one class coincides from the lower, limit of the succeeding class is called an, exclusive or continuous Frequency, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 44 :
43, , Distribution, • ., A bar graph is a pictorial representation of data in, which rectangular bars of uniform width are drawn, with equal spacing between them on one axis,, usually the x axis. The value of the variable is, shown on the other axis that is the y axis., , 4, , Bar graph, , 5, , Histogram, , A histogram is a set of adjacent rectangles whose, areas are proportional to the frequencies of a, given continuous frequency distribution, , 6, , Mean, , The mean value of a variable is defined as the, sum of all the values of the variable divided by the, number of values., O, , 7, , Median, , , , , , , , , 4, , #, , , , 8, , ∑, , , The median of a set of data values is the middle, value of the data set when it has been arranged in, ascending order. That is, from the smallest value, to the highest value, Median is calculated as, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 45 :
44, , 1,   1, 2, , Where n is the number of values in the data, If the number of values in the data set is even,, then the median is the average of the two middle, values., , 8, , Mode, , Mode of a statistical data is the value of that, variable which has the maximum frequency, , S.no, , Term, , Description, Mean is given by, , 1, , Mean for, Ungroup, Frequency table, , 2, , Mean for group, Frequency table, , Q, , ∑ !R R, ∑ !R, , In these distribution, it is assumed that frequency of each class, interval is centered around its mid-point i.e. class marks, G   , , STT G  G""  JU G  G"", 2, , Mean can be calculated using three method, a) Direct method, , Q, , b) Assumed mean method, , Q, , ∑ !R R, ∑ !R, , , ∑ !R VR, ∑ !R, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 46 :
45, , Where, a=> Assumed mean, di => xi –a, , c) Step deviation Method, , Q, , Where, , , , ∑ !R WR, L, ∑ !R, , a=> Assumed mean, ui => (xi –a)/h, , 3, , Mode for, grouped, frequency table, , Modal class: The class interval having highest frequency is, called the modal class and Mode is obtained using the modal, class, , QX, , GY, , Where, , !  !, ZL, 2!  !  !, , l = lower limit of the modal class,, h = size of the class interval (assuming all class sizes to, be equal),, f1 = frequency of the modal class,, f0 = frequency of the class preceding the modal class,, f2 = frequency of the class succeeding the modal class., 4, , Median of a, grouped data, frequency table, , For the given data, we need to have class interval, frequency, distribution and cumulative frequency distribution, Median is calculated as, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 47 :
46, , QO, , ,  !, 2, G[, \L, !, , Where, l = lower limit of median class,, n = number of observations,, cf = cumulative frequency of class preceding the median class,, f = frequency of median class,, h = class size (assuming class size to be equal), 3 Median=Mode +2 Mean, , 5, , Empirical, Formula between, Mode, Mean and, Median, , Probability, S.n, o, , Term, , Description, , 1, , Empirical, probability, , It is a probability of event which is calculated based on, experiments, , ?]^_`abc d_efbf`c`gh, ie ej g_b`ck lm`am no^nagnp eqgae]n ab]n, regbc iq]fn_ ej g_`bck, , Example:, , A coin is tossed 1000 times; we get 499 times head and, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 48 :
47, , 501 times tail,, So empirical or experimental probability of getting head is, calculated as, T, , 499, 1000, , .499, , Empirical probability depends on experiment and, different will get different values based on the, experiment, , 2, , Important point, about events, , If the event A, B, C covers the entire possible outcome in, the experiment. Then,, P (A) +P (B) +P(C), =1, , 3, , impossible, event, , The probability of an event (U) which is impossible to, occur is 0. Such an event is called an impossible event, P (U)=0, , 4, , Sure or certain, event, , The probability of an event (X) which is sure (or certain), to occur is 1. Such an event is called a sure event or a, certain event, P(X) =1, , 5, , Probability of, any event, , Probability of any event can be as, , Term, , Description, , S.n, o, , 0 v w< v 1, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.

Page 49 :
48, , The theoretical probability or the classical probability of the event is, defined as, , 1, , Theoretical, Probability, , 2, , Elementary, events, , d?, , iq]fn_ ej eqgae]n jbxeq_bfcn ge ?, iq]fn_ ej bcc ^ekk`fcn eqgae]n ej gmn no^n_`]nyg, , An event having only one outcome of the experiment is called an, elementary event., “The sum of the probabilities of all the elementary events of, an experiment is 1.”, I.e. If we three elementary event A,B,C in the experiment ,then, P(A)+P(B) +P(C)=1, , 3, , Complementar, y events, , The event Ᾱ, representing ‘not A’, is called the complement of the event, A. We also say that Ᾱ and A are complementary events. Also, P(A) +P(Ᾱ)=1, , 4, , Sure or certain, event, , The probability of an event (X) which is sure (or certain), to occur is 1. Such an event is called a sure event or a, certain event, P(X) =1, , 5, , Probability of, any event, , Probability of any event can be as, 0 v w< v 1, , This material is created by http://physicscatalyst.com/ and is for your personal and non-commercial use, only.