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Subject Code - 041, , Sample Question Paper, CLASS: XII, Session: 2021-22, Mathematics (Code-041), Term - 1, Time Allowed: 90 minutes, , Maximum Marks: 40, , General 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. All questions carry equal marks., 6. There is no negative marking., , SECTION – A, In this section, attempt any 16 questions out of Questions 1 – 20., Each Question is of 1 mark weightage., , 1., , 𝜋, , 1, , 1, , sin [ 3 −sin-1 (− 2)] is equal to:, 1, , 2., , 1, , a) 2, b) 3, c) -1, d) 1, The value of k (k < 0) for which the function 𝑓 defined as, 1−𝑐𝑜𝑠𝑘𝑥, , 𝑓 (𝑥 ) = { 1𝑥𝑠𝑖𝑛𝑥, 2, , 1, , ,𝑥 ≠ 0, ,𝑥 = 0, , is continuous at 𝑥 = 0 is:, a) ±1, 1, c) ± 2, , 3., , 4., , b) −1, 1, d) 2, , 1, 𝑤ℎ𝑒𝑛 𝑖 ≠ 𝑗, If A = [aij] is a square matrix of order 2 such that aij = {, , then, 0, 𝑤ℎ𝑒𝑛 𝑖 = 𝑗, A2 is:, 1, a) [, 1, , 0, ], 0, , b) |, , 1, 0, , 1, |, 0, , 1, c) |, 1, , 1, |, 0, , 1, d) [, 0, , 0, ], 1, , Value of 𝑘, for which A = [, a) 4, c) ±4, , 𝑘, 4, , 8], is a singular matrix is:, 2𝑘, b) -4, d) 0, , 1, , 1

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5., , Find the intervals in which the function f given by f (x) = x 2 – 4x + 6 is strictly, increasing:, b) (2, ∞), d) (– ∞, 2]∪ (2, ∞), , a) (– ∞, 2) ∪ (2, ∞), c) (−∞, 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, , 7., , 8., , 1, , b) 4, d) 16, , 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), 2𝑎 + 𝑏 𝑎 − 2𝑏, 4 −3, ]=[, ], then value of a + b – c + 2d is:, If [, 5𝑐 − 𝑑 4𝑐 + 3𝑑, 11 24, a) 8, c) 4, , 9., , 1, , 1, , 1, , b) 10, d) –8, 1, , The point at which the normal to the curve y = 𝑥 + 𝑥, x > 0 is perpendicular to, , 1, , the line 3x – 4y – 7 = 0 is:, , 10., , a) (2, 5/2), b) (±2, 5/2), c) (- 1/2, 5/2), d) (1/2, 5/2), -1, sin (tan x), where |x| < 1, is equal to:, a), c), , 11., , 𝑥, , b), , √1−𝑥 2, 1, , d), , √1+𝑥 2, , 12., , 1, √1−𝑥 2, 𝑥, √1+𝑥 2, , Let the relation R in the set A = {x ∈ 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}, c) 𝜙, , 1, , b) {0, 1, 2, 5}, d) A, 𝑑𝑦, , 1, , If ex + ey = ex+y , then 𝑑𝑥 is:, a) e y - x, c) – e y - x, , 1, , b) e x + y, d) 2 e x - y

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13., , Given that matrices A and B are of order 3×n and m×5 respectively, then the, order of matrix C = 5A +3B is:, a) 3×5, c) 3×3, , 14., , b) 5×3, d) 5×5, , 𝑑2 𝑦, , b) y, d) 9y, , 2, 5],, (𝑎𝑑𝑗𝐴)′ is equal to:, −11 7, 7 5], −2 −5], b) [, a) [, 11 2, 11 −7, , For matrix A =[, , 7 11, ], c) [, −5 2, 16., , 17., , 18., , 19., , 1, , If y = 5 cos x – 3 sin x, then 𝑑𝑥 2 is equal to:, a) - y, c) 25y, , 15., , The points on the curve, axis are:, , 7, d) [, 11, 𝑥2, 9, , 𝑦2, , + 16 = 1 at which the tangents are parallel to y-, , a) 7, c) 0, 𝑑𝑦, If y = log(cos 𝑒 𝑥 ), then 𝑑𝑥 is:, , 1, , −5], 2, , b) (±4,0), a) (0,±4), c) (±3,0), d) (0, ±3), Given that A = [𝑎𝑖𝑗 ] is a square matrix of order 3×3 and |A| = −7, then the, value of ∑3𝑖=1 𝑎𝑖2 𝐴𝑖2 , where 𝐴𝑖𝑗 denotes the cofactor of element 𝑎𝑖𝑗 is:, , 1, , 1, , b) -7, d) 49, 1, , b) 𝑒 −𝑥 cos 𝑒 𝑥, a) cos 𝑒 𝑥−1, c) 𝑒 𝑥 sin 𝑒 𝑥, d) − 𝑒 𝑥 tan 𝑒 𝑥, 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?, , a) Point B, c) Point D, , 1, , b) Point C, d) every point on the line, segment CD, , 1

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20., , 𝜋, , The least value of the function 𝑓(𝑥 ) = 2𝑐𝑜𝑠𝑥 + 𝑥 in the closed interval [0,2 ], is:, 𝜋, , a) 2, c), , 1, , b) 6 + √3, d) The least value does not, exist., , 𝜋, 2, , SECTION – B, In this section, attempt any 16 questions out of the Questions 21 - 40., Each Question is of 1 mark weightage., , 21., , The function 𝑓: R⟶R defined as 𝑓(𝑥 ) = 𝑥 3 is:, a) One-on but not onto, c) Neither one-one nor onto, , 22., , 𝑑2 𝑦, , c), , −3√3𝑏, 𝑎2, −3√3𝑏, , 𝜋, 6, , 1, , is:, , b), d), , 𝑎, , 23., , 24., , b) Not one-one but onto, d) One-one and onto, , If x = a sec 𝜃, y = b tan 𝜃, then 𝑑𝑥 2 at 𝜃 =, a), , 1, , −2√3𝑏, 𝑎, −𝑏, 3√3𝑎 2, , In the given graph, the feasible region for a LPP is, shaded., The objective function Z = 2x – 3y, will be minimum, at:, , a) (4, 10), b) (6, 8), c) (0, 8), d) (6, 5), -1, 2, The derivative of sin (2𝑥√1 − 𝑥 ) w.r.t sin-1x,, , 1, √2, , <𝑥<, , 1, √2, , ,, , is:, , 1, , 𝜋, , a) 2, c), , −, , 1, , b) 2 − 2, d) −2, , 𝜋, 2, , 25., , 1, 1, If A = [2, 0, , −1, 3, 1, , a) A-1 = B, c) B-1 = B, , 0, 2, 4] and B = [−4, 2, 2, , 2, 2, −1, , −4, −4], then:, 5, b) A-1 = 6B, 1, d) B-1 = 6A

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26., , The real function f(x) = 2x3 – 3x2 – 36x + 7 is:, , 1, , 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) ∪ (3, ∞), , 27., , Simplest form of tan-1 (, a), , 𝜋, 4, , √1+𝑐𝑜𝑠𝑥+√1−𝑐𝑜𝑠𝑥, ),𝜋, √1+𝑐𝑜𝑠𝑥−√1−𝑐𝑜𝑠𝑥, , 𝑥, , −2, 𝑥, , c) − 2, , 28., , <𝑥<, b), , 3𝜋, 2, , 3𝜋, , is:, , 2, , 𝑥, , −2, 𝑥, , d) 𝜋 − 2, , Given that A is a non-singular matrix of order 3 such that A2 = 2A, then value, of |2A| is:, a) 4, c) 64, , 1, , 1, , b) 8, d) 16, , 29., , The value of 𝑏 for which the function 𝑓(𝑥 ) = 𝑥 + 𝑐𝑜𝑠𝑥 + 𝑏 is strictly, decreasing over R is:, b) No value of b exists, a) 𝑏 < 1, c) 𝑏 ≤ 1, d) 𝑏 ≥ 1, , 1, , 30., , Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}, then:, a) (2,4) ∈ R, b) (3,8) ∈ R, c) (6,8) ∈ R, d) (8,7) ∈ R, , 1, , 31., , 𝑥, , ,𝑥 < 0, The point(s), at which the function f given by 𝑓(𝑥) ={ |𝑥|, −1, 𝑥 ≥ 0, is continuous, is/are:, a) 𝑥𝜖R, c) 𝑥𝜖 R –{0}, , 32., , 0, If A = [, 3, are:, , 2, 0, ] and 𝑘A = [, −4, 2𝑏, , 1, , b) 𝑥 = 0, d) 𝑥 = −1and 1, 3𝑎, ], then the values of 𝑘, 𝑎 and 𝑏 respectively, 24, , 1

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33., , 34., , a) −6, −12, −18, b) −6, −4, −9, c) −6, 4, 9, d) −6, 12, 18, A linear programming problem is as follows:, 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑍 = 30𝑥 + 50𝑦, subject to the constraints,, 3𝑥 + 5𝑦 ≥ 15, 2𝑥 + 3𝑦 ≤ 18, 𝑥 ≥ 0, 𝑦 ≥ 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, 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:, a) 75𝑐𝑚2, c) 75√3𝑐𝑚2, , 1, , 1, , b) 7√3𝑐𝑚2, d) 5𝑐𝑚2, , 35., , If A is square matrix such that A2 = A, then (I + A)³ – 7 A is equal to:, , 1, , 36., , a) A, c) I − A, If tan-1 x = y, then:, , 1, , b) I + A, d) I, , a) −1 < y < 1, c), , 37., , 38., , −𝜋, 2, , <y <, , 2, , ≤y ≤, , 𝜋, 2, , −𝜋 𝜋, , d) y 𝜖{ 2 , 2 }, , 2, , a) Surjective function, c) Bijective function, 3 1, ], then 14A-1 is given by:, For A = [, −1 2, , 2, c) 2 [, 1, , −1, ], 3, , 1, −2, ], 6, , −3, d) 2[, 1, , −1, ], −2, , The point(s) on the curve y = x 3 – 11x + 5 at which the tangent is y = x – 11, is/are:, , Given that A = [, , 1, , b) Injective function, d) function, , 4, b) [, 2, , −1, ], −3, , a) (-2,19), c) (±2, 19), 40., , 𝜋, , −𝜋, , 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:, , 2, a) 14 [, 1, , 39., , b), , 1, , b) (2, - 9), d) (-2, 19) and (2, -9), 𝛼, 𝛾, , 𝛽, ]and A2 = 3I, then:, −𝛼, , 1

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a) 1 + 𝛼 2 + 𝛽𝛾 = 0, c) 3 − 𝛼 2 − 𝛽𝛾 = 0, , b) 1 − 𝛼 2 − 𝛽𝛾 = 0, d) 3 + 𝛼 2 + 𝛽𝛾 = 0, , SECTION – C, In this section, attempt any 8 questions., Each question is of 1-mark weightage., Questions 46-50 are based on a 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, c) 𝑎 + 2𝑏 = 0, , 42., , 1, 2, , c) 2, 43., , 45., , d) 3, 1, , c) 1, , b), , 0≤ 𝑥 ≤ 1 is:, , 1, , 1, 2, 3, , 1, , d) √3, , In a linear programming problem, the constraints on the decision variables x, and y are 𝑥 − 3𝑦 ≥ 0, 𝑦 ≥ 0, 0 ≤ 𝑥 ≤ 3. The feasible region, a) is not in the first, b) is bounded in the first, quadrant, quadrant, c) is unbounded in the, d) does not exist, first quadrant, 1, sinα, 1, Let A = [−sinα, 1, sinα], where 0 ≤ α ≤ 2π, then:, −1, −sinα, 1, a) |A|=0, c) |A| 𝜖(2,4), , 1, , b) 1, , The maximum value of [𝑥(𝑥 − 1) + 1]3 ,, a) 0, , 44., , b) 𝑎 = 3𝑏, d) 2𝑎 − 𝑏 = 0, , For which value of m is the line y = mx + 1 a tangent to the curve y 2 = 4x?, a), , 1, , 1, , 1, , b) |A| 𝜖(2, ∞), d) |A| 𝜖[2,4], CASE STUDY, 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., Assume the speed of the train as 𝑣 km/h.

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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, , a) 3, c) 3, 47., , b), d), , 1, 3, 3, 16, , If the train has travelled a distance of 500km, then the total cost of running, the train is given by function:, a), c), , 15, 16, 5, 16, , 𝑣+, , 600000, , 𝑣2 +, , 𝑣, 150000, 𝑣, , b), d), , 1, , 375, 4, 3, 16, , 𝑣+, , 𝑣+, , 1, , 600000, 𝑣, , 6000, 𝑣, , 48., , The most economical speed to run the train is:, , 1, , 49., , a) 18km/h, b) 5km/h, c) 80km/h, d) 40km/h, The fuel cost for the train to travel 500km at the most economical speed is:, , 1, , 50., , a) ₹ 3750, b) ₹ 750, c) ₹ 7500, d) ₹ 75000, The total cost of the train to travel 500km at the most economical speed is:, , 1, , a) ₹ 3750, c) ₹ 7500, , b) ₹ 75000, d) ₹ 15000, , ---------------------------