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FY-327, , Reg. No. : ......................................, , Name : ..........................................., , FIRST YEAR HIGHER SECONDARY MODEL EXAMINATION – 2021, Part – III, , Time : 2 Hours, , MATHEMATICS (SCIENCE) Cool-off time : 20 Minutes, Maximum : 60 Scores, , General Instructions to Candidates :, , , , , , , , , , , There is a ‘Cool-off time’ of 20 minutes in addition to the writing time., , Use the ‘Cool-off time’ to get familiar with questions and to plan your answers., Read questions carefully before answering., Read the instructions carefully., , Calculations, figures and graphs should be shown in the answer sheet itself., Malayalam version of the questions is also provided., Give equations wherever necessary., , Electronic devices except non-programmable calculators are not allowed in the, Examination Hall., ,  , ,     20  ‘  ’ ., ,  ‘  ’    ,  ., ,       .,     ., ,   , , ,   , ., ,    ., ,     .,  , ,    ,     ., , FY-327, , 1, , P.T.O.

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(6  3 = 18), , Answer any 6 questions from 1 to 12. Each carries 3 scores., 1., , Let A = {x : x is a prime number less than 10}, (i), , (ii), , Write A in the roster form., , Number of subsets of A = ________., , (a), 2., , 3., , (2), , 16, , (b), , 8, , (c), , 32, , (d), , 4, , (1), , If A and B are two sets such that n(A) = 100, n(B) = 150 and n(A  B) = 50. Find the, following values :, , (i), , n(A  B), , (2), , (ii), , n(A – B), , (1), , Consider the statement p(n) : 7n – 3n is divisible by 4, n  N, (i), , (ii), , Check whether P(1) is true., , (1), , If P(k) is true. Prove that P(k + 1) is also true., , (2), , 4., , , Using Binomial Theorem expand  x 2 , , , 3,  , x  0., x, , 5., , x, , Finds the middle term in the expansion of   9y  ., 3, , , 6., , In an A.P. 8th term is 16 and 16th term is 48., , 5, , 10, , (3), , (3), , (i), , Find the common-difference of the A.P., , (1), , (ii), , Find the 25th term of the A.P., , (2), , FY-327, , 2

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1  12   3  .  6 , 1., , A = {x : x   }   ., , (i), , (ii), , 2., , (6  3 = 18), ,  ., , A    ., , A    ________ ., , (a), , 16, , (b), , 8, , (c), , 32, , (d), , 4, , (1), , A, B   . n(A) = 100, n(B) = 150, n(A  B) = 50 , ,    ., , (i), , (ii), , 3., , n(A  B), , (2), , n(A – B), , (1), ,    . p(n) : 7n – 3n  4 ,   , n  N., , (i), , (ii), , 4., , (2), , P(1)   ., , (1), , P(k)  P(k + 1)    ., , , , , 3, x, , (2), , 5, ,     x 2   , x  0  , ., , (3), , 5., , x, ,   9y     ., 3, , , (3), , 6., ,   8-  16  16-  48  ., , 10, , (i), , (ii), FY-327, ,    .,  25-  ., 3, , (1), (2), P.T.O.

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7., , (i), , The slope of the line joining the points (–2, 6) and (4, 8) is ________., (a), , (ii), , 8., , 9., , 3, , (b), , 1, 3, , (c), , –3, , (d), , –1, 3, , If the above line is perpendicular to the line-joining the points (8, 12) and (x, 24), , then find the value of ‘x’., , (i), , A point (x, y, z) is on the XY-plane, then which of the following is always true :, , (ii), , Find the distance between the points A(–2, 3, 5) and B(1, 2, 3)., , (a), , x=0, , (b), , y=0, , (c), , z=0, , (d), , z0, , Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, –8) is, divided by the YZ plane., , (1), (2), , (1), (2), , (3), , 10. Consider the parabola y2 = 16x., (i), , Length of the latus rectum of the parabola is _______., , (ii), , Write the co-ordinates of the focus and equation of the directrix of the above, , (a), , 4, , (b), , 16, , (c), , 8, , (d), , 32, , parabola., , (1), (2), , 11. Evaluate :, , 5, lim  x – 32 , x  2  x3 – 8 , , , , (3), , 12. Consider the statement ‘p : If a number is divisible by 10, then it is divisible by 5.’, Write the, , (i), , Negation of the statement, , (1), , (iii) Contrapositive of the statement, , (1), , (ii), , FY-327, , Converse of the statement, , (1), 4

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7., , (i), , (–2, 6), (4, 8)       , ,  ? ________., , (a), , 8., , (b), , 1, 3, , (c), , –3, , (d), , –1, 3, , (1), , (ii), ,    (8, 12), (x, 24)   , , (i), , (x,, , y,, , (a), , x=0, , (ii), 9., , 3, ,    ‘x’   ., z), ,    XY-,   ., (b), , y=0, , (c), , z=0, , , (d), , (2), , , , z0, , (1), , A(–2, 3, 5), B(1, 2, 3)     ., , (2), , (4, 8, 10), (6, 10, –8)       YZ, ,    ., , (3), , 10. y2 = 16x   ., (i), ,      ., , (ii), ,      ,, , (a), , 4, , (b), , 16, , (c), , 8, , (d), , 32, , (1), ,   ., 11., , 5, lim  x – 32 , x  2  x3 – 8 , , , , (2), (3), , ., 12., ,    . ‘p :   10 ,      5   .’,  , , (i), , (ii), , (1), ,  ,  , , (iii)    ., FY-327, , 5, , (1), (1), P.T.O.

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13., , (6  4 = 24), , Answer any 6 questions from 13 to 24. Each carries 4 scores., , U = {1, 2, 3, 4, 5, 6} be the Universal set and A = {1, 2, 3, 4}, B = {3, 4, 5} are two, subsets of U., , (i), , (ii), , Write the sets A' and B'., , (2), , Verify that A – B = A  B'., , (2), , 14. Let A = {–1, 0, 1, 2, 3} B = {1, 2, 3, 4, 5} be two sets. R be a relation defined from, A to B by, , (i), , (ii), , 15. (i), , R = {(x, y) : x + y = 3, x  A, y  B}, , Write R in the Roster form., , Write the domain and range of R., , (2), , Choose one of the possible value of f : R  R, f(x) = sin x from the following, values :, (a), , (ii), , (2), , –1, 2, , If sin x =, , (b), , 3, , (c), , –3, , (d), , 2, , 3, , x in the second quadrant. Find the values of cos x and tan x., 5, , (1), (3), , 16. Using principle of Mathematical Induction prove that for all n  N, 1 + 2 + 3 + .... + n =, , 17. (i), , (ii), FY-327, , If i =, (a), , 1, , n(n  1), 2, , (4), , – 1 , then i4 = _______., , Write z =, , (b), , –1, , (c), , 2  3i, in the form a + ib., 1  2i, , i, , (d), , –i, , (1), (3), , 6

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13  24   4  .  6 , 13., , (6  4 = 24), ,  ., , U = {1, 2, 3, 4, 5, 6}    A = {1, 2, 3, 4}, B = {3, 4, 5}, ,   ., , (i), , (ii), , A', B'   ., , (2), , A – B = A  B'  ., , (2), , 14. A = {–1, 0, 1, 2, 3} B = {1, 2, 3, 4, 5}   . A   B, ,  , , R = {(x, y) : x + y = 3, x  A, y  B}, ,   ., , (i), , (ii), , 15. (i), , R    ., , (2), , R  ,   ., , (2), ,     f : R  R, f(x) = sin x ,     ., (a), , (ii), , –1, 2, , sin x =, , (b), , 3, , (c), , –3, , (d), , 2, , 3,  x   cos x, tan x, 5, ,   ., 16., , (1), , (3), ,      n  N , 1 + 2 + 3 + .... + n =, , 17. (i), , i=, , (ii), , z=, , FY-327, , (a), , n(n  1),  ., 2, , – 1  i4 = _______., 1, , (b), , –1, , (c), , (4), , i, , 2  3i,  a + ib  ., 1  2i, , 7, , (d), , –i, , (1), (3), P.T.O.

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18. (i), (ii), , Let z = 1 + i, then | z | = ________., (a), , 1, , 2, , (b), , (c), , 2, , Represent the above complex number z in the polar form., , (d), , 0, , (1), (3), , 19. How many 4 digit numbers can be formed using the digits 1, 2, 3, 4 and 5 assuming, that, , (i), , (ii), , 20. (i), (ii), , Repetition of the digits is allowed., , (2), , Repetition of the digits is not allowed., , (2), , If 20C12 = 20Cr then one of the possible value of r = _________., (a), , 9, , (b), , 8, , (c), , 11, , (d), , 10, , A bag contains 7 red balls and 5 white balls. Determine the number of ways in, , which 3 red balls and 2 white balls can be selected., , 21. Using method of contradiction prove that ‘ 2 is irrational’., 22. A line cuts off equal intercepts on the co-ordinate axes and passes through the point (2, 3)., (i), , (1), (3), (4), , Find the equation of the line., , (3), , 23. (i), , Write the equation of the ellipse with vertices ( 5, 0) and foci ( 4, 0)., , (3), , 24. Find, , dy, x2  1, if y =, ., dx, x 1, , (4), , (ii), , (ii), , FY-327, , Find the slope of the line obtained in (i)., , Find the length of latus rectum of the above ellipse., , 8, , (1), , (1)

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18. (i), (ii), , z = 1 + i  | z | = ________., (a), , 1, , 2, , (b), , (c), , 2, , (d), , 0, ,   z     , ., , (1), (3), , 19. 1, 2, 3, 4, 5        , ,   ., , (i), , (ii), , 20. (i), (ii), , (2), ,   ., , (2), ,   ., 20C, 12, , (a), , = 20Cr  r     _________., , 9, , (b), , 8, , (c), , 11, , (d), , 10, ,   7   5   .  ,  3  , 2    ,  ., , 21., ,    ‘ 2    ’ , ., , 22., , (1), , (3), , (4), ,     X-, Y-,    (2, 3)   ., , (i), , (ii), 23. (i), (ii), 24. y =, FY-327, , (3), ,    ., , (i)     ., , (1), , ( 5, 0) , ( 4, 0)    , ,   .,      ., x2  1, dy, , ., x 1, dx, , (3), (1), (4), , 9, , P.T.O.

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25., , (3  6 = 18), , Answer any 3 questions from 25 to 30. Each carries 6 scores., Consider the function f : R  R, f(x) = | x – 2 |., , (i), , The value of f(0) = ________., , (ii), , Draw the graph of the function f., , (a), , 0, , (b), , 2, , (c), , –2, , (d), , 1, , (iii) Hence write the range of the function f., 26. (i), (ii), , (1), (4), (1), , Find the value of sin 75º., , (3), , Prove that, , (3), , sin 5 x  sin 3 x, = tan 4x, cos 5 x  cos 3 x, , 27. Solve the following system of inequalities graphically :, x + 2y  8, 2x + y  8, , x  0, y  0, 28. (i), (ii), , (6), , If the numbers, , 1, , x, 4 are three consecutive terms of a G.P. Find the value of x., 4, , Find the sum to n terms of the series 5 + 55 + 555 + ..... ., , (2), (4), , 29. Find the variance and standard deviation for the following data :, xi, fi, , 3, 7, , 8, , 10, , 13, 15, , 18, 10, , 23, 6, , (6), , 30. In a class of 60 students 30 opted for NCC, 32 opted for NSS and 24 opted for both, NCC and NSS. One student is selected at random from the class. Find the probability, that student opted for, (i), , (ii), , NCC, , NCC or NSS, , (iii) neither NCC nor NSS, FY-327, , (1), (3), , ___________, 10, , (2)

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25., , 25  30   6  .  3 ,  ., (3  6 = 18), f : R  R, f(x) = | x – 2 |   ., (i) f(0)   = ________., (a) 0, (b) 2, (c) –2, (ii) f    ., (iii) f   ., , 26. (i), , (ii), , 27., , (d), , 1, , (1), (4), (1), , sin 75º   ., sin 5 x  sin 3 x, = tan 4x  ., cos 5 x  cos 3 x, , (3), (3), ,      ,  ., x + 2y  8, 2x + y  8, x  0, y  0, , 28. (i), (ii), 29., , (6), , 1, , x, 4       , 4,  ‘x’   ., (2), 5 + 55 + 555 + .....   n-    . (4), ,      ,   :, xi, fi, , 3, 7, , 8, , 10, , 13, 15, , 18, 10, , 23, 6, , (6), , 30. 60    30  ... , 32 , ..  24   .  , ,     , , ., (i) ..    ., (1), (ii) .. , .. ,  ., (3), (iii) .., .., ,  ., (2), ___________, , FY-327, , 11, , P.T.O.

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FY-327, , 12