NBSE Question Papers Mathematics Term-1 (Set-1) for Class 12

Class 12  TERM-I SET-1 Series NBSE/XII/2021 Code No. 041/12/1 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 the Questions 1–20. Each Question is of 1 mark weightage. 1. Let R be relation on set T of all triangles drawn in a plane defined by aRb. If a is similar to b for all a, b ∈ T, then R is  1 (a) Reflexive but not transitive (b) Reflexive but not symmetric (c) Symmetric but not transitive (d) An equivalence relation x + λ , x<3 2. 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 3. 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  0 −1 3x  y −5 is skew symmetric, then the value of (5x – y) is 4. If the matrix  1 1 −6 5 0 (a) 10 (b) –10 (c) 11 (d) 12 5. The function f (x) = x4 – 4x is strictly 1 (a) Decreasing in [1, ∞) (b) Increasing in [1, ∞) (c) Increasing in (–∞, 1] (d) Increasing in [–1, 1] 6. If A and B are square matrices of order 3 such that | A | = –2 and | B | = 5, then the value of | 2AB | is 1 (a) 40 (b) 60 (c) –80 (d) 80 7. Let A = {1, 2, 3}, then number of equivalence relations containing (2, 3) is 1 (a) 1 (b) 2 (c) 3 (d) 4 8. If 2x −7  = 6 −7 , then the value of x + 5y is 1 5 y 3x + y −5 8  (a) 2 (b) –2 (c) 3 (d) –3 9. T he point on the curve y = 6x – x2 where the tangent is parallel to the line 4x – 2y – 1 = 0 is 1 (a) (2, 8) (b) (8, 2) (c) (6, 1) (d) (4, 2) 10. I f 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 NBSE 2021 2

11. If A = {0, 1, 2, 3, 4} and B = {a, b}, then the number of onto functions from A to B is 1 1 (a) 5P2 (b) 25 – 1 (c) 25 – 2 (d) None of these 12. T he derivate of sec (tan–1 x) w.r.t. x is (a) 1 x (b) 1 (c) x x 1 + x2 + x2 1 + x2 1 + x2 (d) 13. 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 14. If x = t2 and y = t3, then d 2 y is equal to 1 dx2 (a) 3 (b) 3 t (c) 3 (d) 3 2 2 2t 4t 15. If A and B are invertible matrices of same order, then which of the following statement 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 16. If the tangent to the curve x = t2 – 1, y = t2 – t is parallel to x-axis, then 1 (a) t = 0 (b) t = 2 (c) t = 1 (d) t = − 1 2 2 17. 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 18. 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 19. The objective function of an LPP is 1 1 (a) A constant (b) A linear function to be optimized (d) A quadratic expression (c) An inequality 20. The maximum value of f (x) = – | x + 2 | + 5 is (a) –5 (b) 5 (c) 7 (d) 3 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 21. A function f from the set of natural numbers to integers f (n) =  1  −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 NBSE 2021 3 [P.T.O.

22. If y = loge  x2  ,then d2y is equal to 1  e2  dx2 (a) − 1 (b) − 1 (c) 2 (d) − 2 x x2 x2 x2 23. 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) O (0, 0) A (6, 0) (a) (0, 0) (b) (6, 0) (c) (3, 4) (d) (0, 2) 24. The derivative of sin–1  1 2 x  w.r.t. tan–1  1 2 x  is 1  + x2   − x2  1 (a) 1 (b) –1 (c) 1− x2 (d) 1 1 2 1+ x2 1 1  1 −1 0  2 2 −4 1 25. If A = 2 4 3 and B =  −4 2 −4 , then  0 1 2  2 −1 5 (d) B–1 = 1 A (a) A–1 = B (b) A–1 = 6B (c) B–1 = B 6 26. 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, ∞) 27. If sin–1 x + sin–1 y = 2π , then cos–1 x + cos–1 y is equal to 3 (a) p (b) 2π π (d) π 3 (c) 3 6 28. If A is a square matrix of order 3 such that | A | = –4, then | adj A | is (a) 16 (b) –16 (c) 64 (d) –64 29. If function f : R → R is defined by f (x) = 2x + cos x, then (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 NBSE 2021 4

30. 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} 31. 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 32. 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 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. 34. 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 35. The values of x, if tan–1 x is defined as 1 (a) x ∈ (–∞, ∞) (b) x ∈  − π , π  (c) x ∈ − π , π x ∈ − π, π   2 2  2 2  (d) 4 4 36. The function f (x) = [x], where [x] denotes the greatest integer function, is continuous at 1 1 (a) –2 (b) 1 (c) 4 (d) 1.5 37. For real x, let f (x) = x3 + 5x + 1. Then (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 NBSE 2021 5 [P.T.O.

40. If A is square matrix such that A2 = A, then (I – A)3 + A is equal to  1 (a) I (b) O (c) I – A (d) I + A 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. 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 42. 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) 43. The minimum value of x2 + 250 is  1 x (a) 75 (b) 55 (c) 50 (d) 25 44. 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 45. L et matrix A = 1 y 2 , if xyz = 1, 6x + 12y + 3z = 21, then A(adj A) is equal to 1 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 NBSE 2021 6

Case Study Mr. Dhar is an architect. He designed a building and provided an entry door in the shape of a rectangle surmounted by a semicircular opening. The perimeter of the door is 10 m. Based on the above information, answer the following: 46. If 2x metres and y metres be the breadth and the height of the rectangular part of the door respectively, then the relation between x and y is 1 (a) y = 5 – 1 (p + 2)x (b) y = 10 – 1 (p + 2)x 2 2 (c) y = 5 – x – p + 2x (d) y = 10 – (p + 2)x 47. If A(sq. m) is the area enclosed by the door then 1 (a) A = 10x – (p + 2) x2 (b) A = 10x – 1 (p + 2)x2 2 (c) A = 10x – 1 (p + 4)x2 (d) A = 10x – 1 (p – 4)x2 22 48. To allow maximum airflow inside the building, the width of the door is 1 10 20 20 40 (a) 4 + π m (b) 4 + π m (c) 2 + π m (d) 2 + π m 49. To allow maximum airflow inside the building, the height of the door is 1 5 10 20 30 (a) 4 + π m (b) 4 + π m (c) 4 + π m (d) 4 + π m 50. The area of the door which permits the maximum airflow inside the building is 1 100 m2 (b) 200 m2 (c) 80 m2 (d) 50 m2 (a) 4 + π 4+π 4+π 4+π NBSE 2021 7 [P.T.O.

ROUGH WORK NBSE 2021 8