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NAVODAYA VIDYALAYA SAMITI, 1st PRE-BOARD EXAMINATION SESSION 2021-22, MATEMATICS (041), TERM -I, TIME ALLOWED: -90 Minutes, , CLASS-XII, , MAXIMUM MARKS-40, , GENERAL INSTRUCIONS, 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., , SECTION A, In this section, attempt any 16 questions out of Questions 1-20, Each question is of 1-mark weightage., −1, °, Q.1. If 𝜃 = sin (sin 600 ) then the value of 𝜃 is, 𝜋, , 𝜋, , (b) − 3, , (a) 3, , (c) 0, , (d), , 2𝜋, 3, , Q.2. The function given by 𝑓 (𝑥 ) = tan 𝑥 is discontinuous on the set, (a) {𝑥: 𝑥 = 2𝑛𝜋, 𝑛 ∈ 𝑍}, (b) {𝑥: 𝑥 = (𝑛 − 1)𝜋, 𝑛 ∈ 𝑍}, (c) {𝑥: 𝑥 = 𝑛𝜋, 𝑛 ∈ 𝑍}, , 𝜋, , (d) {𝑥: 𝑥 = (2𝑛 + 1) 2 , 𝑛 ∈ 𝑍}, , Q.3. If P and Q of symmetric matrix of same order then PQ ⎯ QP is a, (a) Zero matrix, (b) Identity matrix, (c) Skew Symmetric matrix, (d)Symmetric matrix, Q.4.The number of all possible matrices of order 3x3 with each entry 1 or 2 is, (a) 27, (b) 18, (c) 81, (d) 512, Q.5. The slope of the normal to the curve 𝑦 = 2𝑥 2 + 3 sin 𝑥 at 𝑥 = 0 is, (a) 3, , 1, , (b) 3, , (c) ⎯3, , 1, , (d) ⎯3, 1

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Q.6. Let A be a non-singular square matrix of order 3× 3. Then |𝑎𝑑𝑗𝐴| is equal to, (a) |A| (b) |A|2, , (c) |A|3, , (d) 3|A|, , Q.7. Let 𝑅 be the relation in the set {1, 2, 3, 4} given by, 𝑅 = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)} then𝑅 is, (a) reflexive and symmetric but not transitive., (b) reflexive and transitive but not symmetric., (c) symmetric and transitive but not reflexive., (d) equivalence relation, Q.8. Which of the given values of 𝑥 and y make the following pair of matrix equal., [, , 3𝑥 + 7, 5, 0 𝑦−2, ],[, ]., 𝑦 + 1 2 − 3𝑥 8, 4, 1, (a) 𝑥 = − 3 , 𝑦 = 7 (b) Not possible to find., 2, , (c) 𝑥 = − 3, , 1, , , 𝑦 = 7 (d) 𝑥 = − 3, , , 𝑦=−, , 2, 3, , Q.9. The tangent to the curve 𝑦 = 𝑒 2𝑥 at the point (0, 1) meets 𝑥-axis at, (a) (0, 1), , 1, , (c) (− 2 , 0), , (b) (0, 2), , (d) (2, 0), , Q.10. tan−1 √3 − sec −1 (−2) is equal to, a), , π, , 𝜋, , (b), , −3, , 𝜋, , (c), , Q.11. If R is a relation from A to B, then, (a) R ⊂ A, (b) R ⊂ B, (c) R ⊂ A × B, Q.12. If 𝑥 = 𝑒 𝑦+𝑒, , 𝑦+⋯……..𝑡𝑜 ∞, , 1, , 2𝜋, 3, , (d) none of these, , 𝑑𝑦, , , 𝑥 > 0, then 𝑑𝑥 is, , 𝑥, , (a) 𝑥, , (d), , 3, , (b) 1+𝑥, , (c), , 1−𝑥, , (d) none of these, , 𝑥, , Q.13.If Y, W, and P are matrices of order 3× k, n × 3, p × k respectively. The restriction on n, k,, and p so that PY+WY will be defined are, (a) k = 3 , p = n, (b) k is arbitrary, p = 2, (c) p is arbitrary, k = 3, (d) k = 2, p = 3, Q.14. The derivative of sin(log 𝑥 ) w.r.t. 𝑥 is, (a), , sin(log 𝑥), 𝑥, , (b) −, , cos(log 𝑥), 𝑥, , (c), , cos(log 𝑥), 𝑥, , (d) none of these, , 2

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Q.15. Let A be a square matrix of order 3× 3, then |kA| is equal to, (a) k |A|, (b) 𝑘 2 |A|, (c) 𝑘 3 |A|, (d) 3k |A|, Q.16. The two curve 𝑥 3 − 3𝑥𝑦 2 + 2 = 0 and 3𝑥 2 𝑦 − 𝑦 3 = 2, 𝜋, (a) touch each other, (b) cut at an angle 3, 𝜋, , (c) cut at right angle, , (d) cut at an angle 6, , Q.17. If matrix A = [𝑎𝑖𝑗]2x2 ,where 𝑎ij = 1, 𝑖𝑓 i ≠ jand 𝑎ij = 0, if i = j, then A2 is equal to, (a) I, , (b) A, , (c) 0, , (d) None of these., , 𝑑2 𝑦, , Q.18. If 𝑒 𝑦 (𝑥 + 1) = 1, then 𝑑𝑥2 is equal to, 1, , 𝑥, , 𝑑𝑦 2, , 1, , (a) 𝑥+1 (b) 1+𝑥, , (d) (𝑑𝑥), , (c) (1−𝑥)2, , Q.19. Based on the given shaded region as the feasible region in the graph, at which point(s) is, the objective function Z = 22𝑥 + 18𝑦 maximum?, , 3𝑥 + 2𝑦 = 48, , (0, 20)B, C(8, 12), 𝑥 + 𝑦 = 20, , (0, 0)O, , (a)Point A, , (b) point B, , A, (16, 0), , (c) Point C, , (d) Point O, , Q.20. The function 𝑓 (𝑥 ) = 𝑥 2 − 𝑥 + 1 in (−1, 1) is, (a) increasing, (b) decreasing, (c) neither increasing nor decreasing, (d) none of these, , 3

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SECTION B, In this section, attempt any 16 questions out of Questions 21-40, Each question is of 1-mark weightage., Q.21. Let 𝑓: 𝑅 → 𝑅 be defined as 𝑓 (𝑥 ) = 𝑥 4 is, (a), 𝑓 is one-one onto, (b), 𝑓 is many one onto., (c), 𝑓 is one-one but not onto, (d), 𝑓 is neither one-one nor onto., Q.22. If 𝑥 = 𝑎 cos 3 𝜃 and 𝑦 = 𝑎 sin3 𝜃, then find the value of, (a) 1, , (c) ⎯1, , (b) 0, , 𝑑𝑦, 𝑑𝑥, , 𝜋, , at 𝜃 = 4 ., , (d) ⎯∞, , Q.23.In the given graph, the feasible reason for a LPP is shaded. The objective function, Z = 12𝑥 + 16𝑦, will be maximum at, (a)Point A, (b) point B, (c) Point C, (d) Point O, , 𝑥 = 2𝑦, , 𝑥 − 3𝑦 = 600, , (0, 1200)B, , C (800, 400), B (1050, 150), A, (600, 0), , (0, 0)O, , Q.24.The derivative oftan−1 (, (a) 1, , X, , 𝑥 + 𝑦 = 1200, , √1+𝑥 2 −1, 𝑥, , ) with respect to tan−1 𝑥, when 𝑥 ≠ 0 is, (c) ⎯1, , (b) 0, , 1, , (d) 2, , 1 3 2 1, Q.25. If [1 𝑥 1] [ 2 5 1] [2] = 0, then value 𝑥 is/are, 15 3 2 𝑥, 1, (a) −2, −14 (b) 14, (c) 2, (d) 2, Q.26.The function 𝑓 (𝑥 ) = 2𝑥 2 − 3𝑥 is, (a) Strictly increasing on (0, ∞), 3, , (b) Strictly increasing on (4 , 6), Strictly decreasing on (−∞, 1), 3, , 3, , 4, , 4, , (c) Strictly increasing on ( , ∞), Strictly decreasing on (−∞, ), (d) none of these, 4

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𝛽, ] is such that A2 = I(identity matrix) then, 𝛿, (a) 1+𝛼 2 + 𝛽𝛾= 0, (b) 1−𝛼 2 + 𝛽𝛾 = 0, (c) 1−𝛼 2 − 𝛽𝛾 = 0, (d) 1+𝛼 2 − 𝛽𝛾 = 0, , Q.35. If A = [, , 𝛼, 𝛾, , Q.36. The value of sin(2 sin−1 (. 8)) is, (a) sin 1.6, (b) 1.6, (c) .96, , (d) 4.8, , Q.37. Let 𝑓: 𝑅 → 𝑅 be defined by 𝑓 (𝑥 ) = 𝑥 2 + 1. Then, pre-image of 5 is/are, (a) −3, (b) −2, 2, (c) −1, 2, (d) none of these, cos 𝛼, sin 𝛼, 𝜋, (b) 3, , Q.38. If A = [, 𝜋, , (a) 6, , − sin 𝛼, ] then A+AT = I, if the value of 𝛼 is, cos 𝛼, 3𝜋, (c) π, (d) 2, , Q.39. The values of 𝑎 for which 𝑦 = 𝑥 2 + 𝑎𝑥 + 25 touches the x-axis are, (a) 0, (b) ±10, (c) 4, ⎯ 6, (d) ± 5, , Q.40. If matrix A = [𝑎𝑖𝑗]2x2 ,where 𝑎ij = 1, 𝑖𝑓 i ≠ jand 𝑎ij = 0, if i = j, then A2 is equal to, (a) I, , (b) A, , (c) 0, , (d) None of these., , SECTION C, In this section, attempt any 8 questions., Each question is of 1-marks weightage., Questions 46-50 are based on a Case-Study., , Q.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 are (60, 0), (120, 0), (60, 30) and (40, 20). The, condition on 𝑎 and 𝑏 such that the maximum Z occurs at both the points (120, 0) and (60, 30) is, (a) 𝑎 − 2𝑏 = 0, (b) 2𝑎 − 3𝑏 = 0, (c) 2𝑎 − 𝑏 = 0, (d) 𝑎 − 𝑏 = 0, Q.42. The line 𝑦 = 𝑥 + 1 is a tangent to the curve 𝑦 2 = 4𝑥 at the point, (a) (1, 2), (a) (2, 1), (a) (1, −2), (a) (-1, 2), , 6

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Q.43. The function given by 𝑓 (𝑥 ) = 𝑥 3 − 3𝑥 2 + 3𝑥 − 100 is, (a) strictly increasing in R, (b) strictly decreasing in R, (c) neither increasing nor decreasing in R (d) not define in R, Q.44. In a linear programming problem, the constraint on the decision variable x and y are, 𝑥 + 2𝑦 ≤ 8, 3𝑥 + 2𝑦 ≤ 12, 𝑥 ≥ 0, 𝑦 ≥ 0. The feasible region is, (a)is not in the first quadrant, (b) is bounded in the first quadrant, (c) is unbounded in the first quadrant (d) does not exist, 𝑥 2, 6 2, |= |, | then 𝑥 is equal to, Q.45. If |, 18 𝑥 18 6, (a) 6, (b) ± 6, (c) ⎯ 6, (d) 0, , CASE STUDY, Dr. Ritam residing in Delhi went to see an apartment of 3 BHK in Dilshad Garden. The window, of the house was in the form of a rectangle surmounted by a semicircular opening having a, perimeter of the window 10 m. as show in figure., , 𝑦, , 𝑥, , Based on the above information answer the following:, Q.46. If 𝑥 and 𝑦 represent the length and breadth of the rectangular region, then the relation, between the variable is, 𝑥, (a) 𝑥 + 𝑦 + = 10, 2, 𝑥, , (b) 𝑥 + 𝑦 + 2 = 10, (c) 2𝑥 + 4𝑦 + 𝜋𝑥 = 20, 𝑥, (d) 𝑥 + 𝑦 + 2 = 10, Q.47. The area Aof the window is expressed as a function of 𝑥 is, (a) A = 𝑥 −, , 𝑥2, 2, , (b) A = 5𝑥 −, (c) A = 5𝑥 +, (d) A = 5𝑥 −, , −, , 3𝑥 2, 2, 𝑥2, 2, 𝑥2, 2, , 𝜋𝑥 2, 8, , −, , −, −, , 𝜋𝑥 2, , 8, 𝜋𝑥 2, 8, 𝜋𝑥 2, 8, , 7

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Q.48. Dr. Ritam is interested in maximizing the area of the whole window. For this to happen the, value of length𝑥 should be, 20, , (a) 4+𝜋 m, (b), , 20, 𝜋, 20, , m, , (c) 2+𝜋 m, 20, , (d) 4−𝜋 m, Q.49. For maximum value of A, the breadth 𝑦of rectangular part of window is, 10, , (a) 4+𝜋 m, (b), , 10, 𝜋, 20, , m, , (c) 2+𝜋 m, 10, , (d) 4−𝜋 m, , Q.50. The maximum area of window is, 200, , (a) (4+𝜋)2 sq. m, 100, , (b) (4+𝜋)2 sq. m, 200+5𝜋, sq. m, (4−𝜋)2, 200+50𝜋, (d) (4+𝜋)2 sq. m, , (c), , 8