Class 12 TERM-I SET-3 Series NBSE/XII/2021 Code No. 041/12/3 Roll No. Candidates must write the Code No. on the title page of the OMR sheet. l Please check that this question paper contains 8 pages. l Code number given on the right hand side of the question paper should be written on the title page of the OMR sheet. l Please check that this question paper contains 50 questions. l 15 minutes time has been allotted to read this question paper. MATHEMATICS Time Allowed : 90 Minutes Maximum Marks : 40 General 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. 7. Use of calculator is not permitted. NBSE 2021 1 [P.T.O.

SECTION-A In this section, attempt any 16 questions out of Questions 1–20. Each Question is of 1 mark weightage. 1. If R is a relation on the set of all straight lines drawn in a plane defined by l1 R l2. If l1 is perpendicular to l then R is 1 2 (a) Reflexive (b) Symmetric (c) Transitive (d) An equivalence relation 2. The number of all possible matrices of order 2 × 3 with each entry 1or 2 is 1 (a) 12 (b) 64 (c) 36 (d) 8 x + λ , x<3 3. If f (x) = 4 x = 3 is continuous at x = 3, then λ = , 1 3x − 5 , x>3 (a) 4 (b) 3 (c) 2 (d) 1 1 0 0 (b) Symmetric matrix 1 4. The matrix A = 0 2 0 is a 1 0 0 3 1 (a) Scalar matrix 1 1 (c) Skew-symmetric matrix (d) None of these 1 5. T he function f (x) = x2 e–x strictly increases on (a) (0, 2) (b) [0, ∞) (c) (– ∞, 0 ] ∪ (2, ∞ ] (d) None of these 6. If A is a square matrix of order 3 such that | A | = 5, then the value of | –2A–1 | is (a) 4 (b) − 4 (c) − 8 (d) 8 5 5 5 5 7. Let A = {1, 2, 3} then number of equivalence relations containing (2, 3) is (a) 1 (b) 2 (c) 3 (d) 4 8. If 2x −7 = 6 −7 , then the value of x + 5y is 5 y 3x + y −5 8 (a) 2 (b) –2 (c) 3 (d) –3 9. If x = t2 and y = t3, then d 2 y is equal to dx2 (a) 3 (b) 3 t (c) 3 (d) 3 2 2 2t 4t NBSE 2021 2

10. If A and B are invertible matrices of same order, then which of the following statements is not true? 1 (a) | A–1 | = | A |–1 (b) adj A = | A | A–1 (c) (A + B)–1 = B–1 + A–1 (d) (AB)–1 = B–1 A–1 11. If the tangent to the curve x = t2 – 1, y = t2 – t is parallel to x-axis, then 1 1 (a) t = 0 (b) t = 2 (c) t = 1 (d) t = − 1 2 2 12. T he point on the curve y = 6x – x2 where the tangent is parallel to the line 4x – 2y – 1 = 0 is (a) (2, 8) (b) (8, 2) (c) (6, 1) (d) (4, 2) 13. If cos sin −1 2 + cos−1 x = 0, then x is equal to 1 5 (a) 0 (b) 1 (c) 2 5 5 (d) 1 14. If A = {0, 1, 2, 3, 4} and B = {a, b}, then the number of onto functions from A to B is 1 (a) 5P2 (b) 25 – 1 (c) 25 – 2 (d) None of these 15. If the area of the triangle with vertices (1, –1), (–4, k) and (–3, –5) is 24 sq. units, then the values of k are 1 (a) 2, –6 (b) –2, 6 (c) –6, 18 (d) 6, –18 16. If y = f (x2) and f ' (x) = e x ,then dy is equal to 1 dx (a) 2xe2 x (b) 2xex (c) 4xe x (d) 4xex 17. The objective function of an LPP is 1 (a) A constant (b) A linear function to be optimized 1 (d) A quadratic expression 1 (c) An inequality 18. The maximum value of f (x) = – |x + 2 | + 5 is (a) –5 (b) 5 (c) 7 (d) 3 19. The derivate of sec (tan–1 x) w.r.t. x is (a) 1 x (b) 1 + x2 1 + x2 x (d) x 1 + x2 (c) 1 + x2 20. If A is any m × n matrix and B is a matrix such that AB and BA are both defined, then B is a matrix of order 1 (a) n × n (b) m × m (c) m × n (d) n × m NBSE 2021 3 [P.T.O.

SECTION-B In this section, attempt any 16 questions out of the Questions 21–40. Each Question is of 1 mark weightage. n − 1, when n is odd 2 is 1 21. A function f from the set of natural numbers to integers f (n) = −n, when n is even 2 (a) one-one but not onto (b) onto but not one-one (c) one-one and onto both (d) neither one-one nor onto 1 −1 0 2 2 −4 22. If A = 2 4 0 3 and B = −4 2 −4 , then 1 1 1 2 2 −1 5 (d) B–1 = 1 A (a) A–1 = B (b) A–1 = 6B (c) B–1 = B 6 23. The interval in which the function f (x) = 2x3 + 3x2 – 12x + 1 is strictly increasing is (a) [–2, 1] (b) (–∞, –2] ∪ [1, ∞) (c) (–∞, 1] (d) [–∞, –1] ∪ [2, ∞) 24. If sin–1 x + sin–1 y = 2π , then cos–1 x + cos–1 y is equal to 1 3 (a) p (b) 2π π (d) π 3 (c) 3 6 25. If y = loge x2 ,then d2y is equal to 1 e2 dx2 (a) − 1 (b) − 1 (c) 2 (d) − 2 x x2 x2 x2 26. The feasible region for an LPP is shown below. Let Z = 5x + 7y be the objective function. Maximum of Z occurs at 1 B (3, 4) C (0, 2) (a) (0, 0) O (0, 0) A (6, 0) (d) (0, 2) NBSE 2021 (b) (6, 0) (c) (3, 4) 4

27. The derivative of sin–1 1 2 x w.r.t. tan–1 1 2 x is 1 + x2 − x2 (a) 1 (b) –1 (c) 1− x2 (d) 1 2 1+ x2 28. If A = 3 1 be such that A–1 = kA, then the value of k is 1 2 −3 (a) 11 (b) –11 (c) 1 (d) − 1 11 11 29. If a function f : R → R is defined by f (x) = x2 + 1, then pre-images of 17 and –3 respectively, are 1 (a) f, {4, –4} (b) {3, –3}, f (c) {4, –4}, f (d) {4, –4}, {2, –2} 30. If matrix A = [aij]2 × 2 where aij = 1, i ≠ j , then A3 is equal to 1 0, i= j (d) None of these (a) A (b) I (c) 0 31. The maximum value of f (x) = 4+ x x2 on [–1, 1] is 1 x+ (a) − 1 (b) − 1 (c) 1 1 4 3 6 (d) 5 32. If function f : R → R is defined by f (x) = 2x + cos x, then 1 (a) f has a minimum at x = p (b) f has a maximum at x = 0 (c) f is a decreasing function (d) f is a strictly increasing function 33. The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20, 40), (60, 20), (60, 0). The objective function is Z = 4x + 3y. Compare the quantity in column A and column B. 1 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) Two quantities are equal (d) The relationship cannot be determined on the basis of information given. 1 34. The values of x, if tan–1 x is defined as (a) x ∈ (–∞, ∞) (b) x ∈ − π , π (c) x ∈ − π , π x ∈ − π, π 2 2 2 2 (d) 4 4 35. If A = [aij] is a matrix of order 2 × 2, such that | A |= –15 and Cij represents the cofactor of aij, then a21 C21 + a22C22 equals 1 (a) 0 (b) –15 (c) 15 (d) 225 NBSE 2021 5 [P.T.O.

36. The value of the function f at x = 0, so that the function f (x) = 2x − 2−x , x ≠ 0 , is continuous at x1 x = 0, is (a) 0 (b) log 2 (c) log 4 (d) 22 37. For real x, let f (x) = x3 + 5x + 1. Then 1 (a) f is one-one but not onto in R. (b) f is onto in R but not one-one in R. (c) f is one-one and onto in R. (d) f is neither one-one nor onto in R. 38. If A and B are square matrices of same order such that AB = A and BA = B, then A2 + B2 = 1 (a) AB (b) A + B (c) 2AB (d) 2BA 39. The point(s) on the curve 9y2 = x3 where the normal to the curve makes equal intercepts is/are 1 (a) 4, ± 8 (b) −4, ± 8 3 3 (c) −4, − 8 (d) 8 , 4 3 3 1 −2 3 and B = 2 3 1 40. If A = −4 2 5 4 5 and BA = (bij), then b21 + b32 is equal to 2 1 (a) 12 (b) –12 (c) 18 (d) –18 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. The corner points of the feasible region determined by the system of linear inequalities are (0, 3), (2, 2) and (3, 0). If the minimum value of Z = ax + by, a, b > 0 occurs at both (0, 3) and (2, 2), then 1 (a) a = 2b (b) 2a = b (c) a = b (d) 3a = b x34 1 42. Let matrix A = 1 y 2 , if xyz = 1, 6x + 12y + 3z = 21, then A(adj A) is equal to 33z 500 10 0 0 20 0 0 15 0 0 (a) 0 5 0 (b) 0 10 0 (c) 0 20 0 (d) 0 15 0 005 0 0 10 0 0 20 0 0 15 43. The minimum value of x x is 1 log (a) e (b) 1 (c) e2 (d) 2e e NBSE 2021 6

44. Based on the shaded region as the feasible region in the graph, at which point(s) is the objective function Z = 3x + 9y maximum? 1 Y 40 35 30 x=y 25 C(15, 15) 20 D(0, 20) 15 10 A (0, 10) 5 B(5, 5) (60, 0) X¢ O X 5 10 15 20 25 30 35 40 45 50 55 60 (10, 0) x + 3y = 60 x + y = 10 Y¢ (a) Point B (b) Point C (c) Point D (d) Every point on the line segment CD 45. The coordinates of the point where the tangent to the curve y = 2x2 – x + 1 is parallel to the line y = 3x – 5 are 1 (a) (2, 1) (b) (1, 2) (c) (–1, 2) (d) (1, –2) Case Study A cable network provider in a small town has 500 subscribers and he used to collect ` 300 per month from each subscriber. He proposes to increase the monthly charges and it is believed from past experience that for every increase of ` 1, one subscriber will discontinue the service. Based on the above information, answer the following: 46. I f ` X is the monthly increase in subscription amount, then the number of subscribers are 1 (a) X (b) 500 – X (c) X – 500 (d) None of these 47. Total revenue R is given by 1 (a) R = 300 X + 300 (500 – X) (b) R = (300 + X)(500 + X) (c) R = (300 + X)(500 – X) (d) R = 300X + 500(X + 1) 48. The number of subscribers which gives the maximum revenue is 1 (a) 100 (b) 200 (c) 300 (d) 400 49. What is increase in charges per subscriber that yields maximum revenue? 1 (a) X = ` 100 (b) X = ` 200 (c) X = ` 300 (d) X = ` 400 50. The maximum revenue generated is 1 (a) ` 200000 (b) ` 180000 (c) ` 160000 (d) ` 150000 NBSE 2021 7 [P.T.O.

ROUGH WORK NBSE 2021 8

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