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CBSE Class 12 Board Exam 2020 : Mathematics (Series 4)

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SET 1 Series : HMJ/4 . Code No. . 65/4/1 - - Roll No. Candidates must write the Code on the title page of the answer-book. (I) - (I) 15 (II) - (II) - - NOTE Please check that this question paper contains 15 printed pages. Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate. (III) Please check that this question paper contains 36 questions. (IV) Please write down the Serial Number of the question in the answer-book before attempting it. (V) 15 minute time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. - 36 (IV) , (III) (V) - 15 - 10.15 10.15 10.30 - - MATHEMATICS {ZYm [aV g : 3 K Q>o A{YH$V A H$ Time allowed : 3 hours .65/4/1. 333A : 80 Maximum Marks : 80 1 P.T.O. : - , , (i) - 36 (ii) - 1 20 20 (iii) - 21 26 6 (iv) - 27 32 6 (v) - 33 36 4 (vi) - - , - , - - (vii) , , (viii) 1 10 1 1. 3 sin 1 cos 5 (a) .65/4/1. 10 (b) 3 5 (c) 2 10 (d) 3 5 General Instructions : Read the following instructions very carefully and strictly follow them : This question paper comprises four sections A, B, C and D. (i) This question paper carries 36 questions. All questions are compulsory. (ii) Section A Question no. 1 to 20 comprises of 20 questions of one mark each. (iii) Section B Question no. 21 to 26 comprises of 6 questions of two marks each. (iv) Section C Question no. 27 to 32 comprises of 6 questions of four marks each. (v) Section D Question no. 33 to 36 comprises of 4 questions of six marks each. (vi) There is no overall choice in the question paper. However, an internal choice has been provided in 3 questions of one mark, 2 questions of two marks, 2 questions of four marks and 2 questions of six marks. Only one of the choices in such questions have to be attempted. (vii) In addition to this, separate instructions are given with each section and question, wherever necessary. (viii) Use of calculators is not permitted. Section A Question numbers 1 to 10 are multiple choice questions of 1 mark each. Select the correct option : 1. 3 The value of sin 1 cos is 5 (a) .65/4/1. 10 (b) 3 5 (c) 3 10 (d) 3 5 P.T.O. 2. 3 2 2 A = [2 3 4], B = , X = [1 2 3] Y = 3 2 4 (a) [28] 3. (b) [24] (c) 28 (d) 24 (c) 1 (d) 1 2 3 2 x x x + 3 = 0 , x 4 9 1 (a) 3 4. , AB + XY (b) 0 /8 . tan2 (2x) . 0 (a) 5. 4 8 (b) 1 a c + =1 a c (d) 4 2 (b) 30 (c) 60 (d) 90 (b) a c + = 1 a c (c) aa + cc = 1 (d) aa + cc = 1 x 2y + 4z = 10 18x + 17y + kz = 50 , k (a) 4 8. 4 4 x = ay + b, z = cy + d x = a y + b , z = c y + d , (a) 7. (c) a . b = 2 | a | | b| , a b (a) 0 6. 4+ 8 (b) 4 (c) 2 (d) 2 , z = ax + by , , : (a) 0 (b) 2 (c) (d) .65/4/1. 4 2. 3 2 2 If A = [2 3 4], B = , X = [1 2 3] and Y = 3 , then AB + XY equals 2 4 (a) [28] 3. (b) [24] (d) 24 2 3 2 If x x x + 3 = 0, then the value of x is 4 9 1 (a) 3 4. (c) 28 (b) 0 (c) 1 (d) 1 /8 . tan2 (2x) is equal to . 0 (a) 5. 4 8 If a . b = (b) a c + =1 a c (d) 4 2 (b) 30 (c) 60 (d) 90 (b) a c + = 1 a c (c) aa + cc = 1 (d) aa + cc = 1 The two planes x 2y + 4z = 10 and 18x + 17y + kz = 50 are perpendicular, if k is equal to (a) 4 8. 4 4 The two lines x = ay + b, z = cy + d; and x = a y + b , z = c y + d are perpendicular to each other, if (a) 7. (c) 1 | a | | b|, then the angle between a and b is 2 (a) 0 6. 4+ 8 (b) 4 (c) 2 (d) 2 In an LPP, if the objective function z = ax + by has the same maximum value on two corner points of the feasible region, then the number of points at which zmax occurs is (a) 0 .65/4/1. (b) 2 (c) finite 5 (d) infinite P.T.O. 9. a { 1,2,3,4,5 } a b (a b) b : (a) 10. 1 3 (b) 1 4 (c) 1 2 (d) 3 5 3 , 4 2 2 , : (a) 1 18 (b) 1 36 (c) 1 12 (d) 1 24 11 15 / : 11. f : R R, f(x) = (3 x3)1/3 , fof (x) = _________ 12. 9 x+y 7 2 = x y 9 7 . 4 , x y = _________ 13. f(x) = |x| |x + 1| f _________ 14. y = x3 x (2, 6) _______. , r , r = 3 , _________. 15. ^ k^ _________. a , ( a .^i) ^i + ( a .^j) ^j + ( a .k) ^i + ^j ^i ^j _________. .65/4/1. 6 9. From the set { 1,2,3,4,5 }, two numbers a and b (a b) are chosen at a random. The probability that is an integer is : b (a) 1 3 (b) 1 4 (c) 1 2 (d) 3 5 10. A bag contains 3 white, 4 black and 2 red balls. If 2 balls are drawn at random (without replacement), then the probability that both the balls are white is (a) 1 18 (b) 1 36 (c) 1 12 (d) 1 24 In Q. Nos. 11 to 15, fill in the blanks with correct word / sentence : 11. If f : R R be given by f(x) = (3 x3)1/3, then fof (x) = ________ x + y 7 2 = 12. If 9 x y 9 7 . 4 , then x y = ________ 13. The number of points of discontinuity of f defined by f(x) = |x| |x + 1| is ____________. 14. The slope of the tangent to the curve y = x3 x at the point (2, 6) is _______. OR The rate of change of the area of a circle with respect to its radius r, when r = 3 cm, is _________. ^ ^ 15. If a is a non-zero vector, then ( a .^i) ^i + ( a .^j) ^j + ( a .k) k equals _________. OR The projection of the vector ^i ^j on the vector ^i + ^j is _________. .65/4/1. 7 P.T.O. 16 20 2 1 3 16. A = 4 17. . , adj A . 2x + 1 5x 1 dx 10x 2 18. . : . |sin x|dx 0 a 19. . dx . 2 = 8 , a 1 + 4x 0 . dx x+x : . 20. y = ax + 2a2, dy dy 2 2 dx + x dx y = 0 21 26 2 21. A = { 1, 2, 3, 4, 5, 6 } R = { (x, y) : y, x } R (i) (ii) : 2 2 9 9 1 9 sin 1 = sin 1 8 4 3 4 3 .65/4/1. 8 Q. 16 to 20 are very short answer questions. 2 16. Find adj A, if A = 4 1 3 . 2x + 1 5x 1 17. Find . dx 10x 2 . 18. Evaluate . |sin x|dx 0 a . dx 19. If . = , then find the value of a. 1 + 4x2 8 0 OR . dx Find . x+x 20. Show that the function y = ax + 2a2 is a solution of the differential equation dy 2 dy 2 + x y = 0. dx dx Section B Q. Nos. 21 to 26 carry 2 marks each. 21. Check if the relation R on the set A = { 1, 2, 3, 4, 5, 6 } defined as R = { (x, y) : y is divisible by x } is (i) symmetric (ii) transitive OR Prove that : 2 2 9 9 1 9 sin 1 = sin 1 8 4 3 3 4 .65/4/1. 9 P.T.O. 22. = dy , x = cos cos 2 , y = sin sin 2 . 3 dx 23. f f(x) = (x 1) ex + 1, x > 0 24. | a| = 2| b| ( a + b) ( a b) = 12 , | a| | b| a = 4^i + 3^j + k^ b = 2^i ^j + 2k^ 25. y- 3 xz 26. P(A) = 10 , P(B) = 5 P(A B) = 5 , [P(B/A) + P(A/B)] 3 2 3 27 32 4 27. Z R = {(x, y) : (x y) 5 } R, 28. y = sin 1 1+x+ 2 1 x dy 1 , = dx 2 1 x2 2 , 2 f(x) = ex cos x .65/4/1. 10 22. Find the value of dy at = , if x = cos cos 2 , y = sin sin 2 . 3 dx 23. Show that the function f defined by f(x) = (x 1) ex + 1 is an increasing function for all x > 0. 24. Find | a| and | b|, if | a| = 2| b| and ( a + b) . ( a b) = 12. OR Find the unit vector perpendicular to each of the vectors a = 4^i + 3^j + k^ ^ and b = 2^i ^j + 2k. 25. Find the equation of the plane with intercept 3 on the y-axis and parallel to xz plane. 26. Find [P(B/A) + P(A/B)], if P(A) = 3 2 3 , P(B) = and P(A B) = . 10 5 5 Section C Q. Nos. 27 to 32 carry 4 marks each. 27. Prove that the relation R on Z, defined by R {(x, y) : (x y) is divisible by 5} is an equivalence relation. 1+x+ 28. If y = sin 1 2 1 x dy 1 , then show that = dx 2 1 x2 OR Verify the Rolle s Theorem for the function f(x) = ex cos x in , 2 2 .65/4/1. 11 P.T.O. . x sin x 29. : . dx. 1 + cos2 x 0 30. (x + 1) 31. dy = 2e y + 1; y = 0 x = 0 dx I, II III I II 12 III 5 M N , M N ( ) ( ) I II III M 1 2 1 N 2 1 1.25 M ` 600 N ` 400 , , , ? ? 32. 100 5 1000 25 , .65/4/1. 12 . x sin x 29. Evaluate : . dx. 1 + cos2 x 0 30. For the differential equation given below, find a particular solution satisfying the given condition (x + 1) dy = 2e y + 1 ; y = 0 when x = 0. dx 31. A manufacturer has three machines I, II and III installed in his factory. Machine I and II are capable of being operated for atmost 12 hours whereas machine III must be operated for atleast 5 hours a day. He produces only two items M and N each requiring the use of all the three machines. The number of hours required for producing 1 unit of M and N on three machines are given in the following table : Items Number of hours required on machines I II III M 1 2 1 N 2 1 1.25 He makes a profit of ` 600 and ` 400 on one unit of items M and N respectively. How many units of each item should he produce so as to maximize his profit assuming that he can sell all the items that he produced. What will be the maximum profit ? 32. A coin is biased so that the head is three times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails. Hence find the mean of the number of tails. OR Suppose that 5 men out of 100 and 25 women out of 1000 are good orators. Assuming that there are equal number of men and women, find the probability of choosing a good orator. .65/4/1. 13 P.T.O. 33 36 6 33. : a b b c c a b+c c+a a+b a b = a3 + b3 + c3 3 abc. c 1 A = 2 1 34. 3 0 2 2 1 , A3 4A2 3A + 11 I = O A 1 3 f(x) = (x 1)3 (x 2)2 ( ) ( ) 36 , 35. , , , x , y = x x2 + y2 = 32 36. r = a + b r = b + a r . ( a b)= 0 ____________ .65/4/1. 14 Section D Q. Nos. 33 to 36 carry 6 marks each. 33. Using properties of determinates prove that : a b b c c a b+c c+a a+b a b = a3 + b3 + c3 3 abc. c OR 1 If A = 2 1 3 0 2 2 1 , then show that A3 4A2 3A + 11 I = O. Hence find A 1. 3 34. Find the intervals on which the function f(x) = (x 1)3 (x 2)2 is (a) strictly increasing (b) strictly decreasing. OR Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its side. Also, find the maximum volume. 35. Find the area of the region lying in the first quadrant and enclosed by the x axis, the line y = x and the circle x2 + y2 = 32. 36. Show that the lines r = a + b and r = b + a are coplanar and the plane containing them is given by r . ( a b)= 0. ____________ .65/4/1. 15 P.T.O. .65/4/1. 16

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