NBSE Question Papers Mathematics for Class 12

Class 12   Code No. 041/12/1 Series NBSE/XII/2023 Roll No.  Candidates must write the Code No. on the title page of the Answer 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 Answer sheet. l Please check that this question paper contains 38 questions. l 15 minutes time has been allotted to read this question paper. MATHEMATICS Time Allowed : 3 Hours Maximum Marks : 80 General Instructions: 1. This question paper contains two parts A and B. Each part is compulsory. Part–A carries 24 marks and Part–B carries 56 marks. 2. Part–A has Objective Type Questions and Part–B has Descriptive Type Questions. 3. Both Part A and Part B have choices. Part-A: 1. It consists of two Sections–I and II. 2. Section–I comprises of 16 very short answer type questions. 3. Section–II contains 2 case studies. Each case study comprises of 5 case-based MCQs. An examinee is to attempt any 4 out of 5 MCQs. 4. Internal choice is provided in 5 questions of Section I. You have to attempt only one of the alternatives in all such questions. Part-B: 1. It consists of three Sections–III, IV and V. 2. Section–III comprises of 10 questions of 2 marks each. 3. Section–IV comprises of 7 questions of 3 marks each. 4. Section–V comprises of 3 questions of 5 marks each. 5. Internal choice is provided in 3 questions of Section–III, 2 questions of Section–IV and 3 questions of Section–V. You have to attempt only one of the alternatives in all such questions. NBSE 2023 1 P.T.O.

Part–A Section – I All questions are compulsory. In case of internal choices attempt any one. Q1. Let A = {1, 2, 3}. Then number of equivalence relation containing (1, 2) is 1 (a) 1 (b) 2 (c) 3 (d) 4 Q2. A function f from the set of natural numbers to integers 1  n − 1, when n is odd  2 is even f(n) =  is  − n , when n  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 Q3. A relation R is defined in set A = {1, 3, 5, 7, 9} such that: R = {(a, b) : a + b ≤ 10}, a ∈ A, b ∈ A}. Write relation R in roster form. 1 OR Take a set A = {2, 3, 5} and relation R is defined in set A as: R = {(a, b) : a ≥ b and a, b ∈ A}. Is the relation R reflexive? Give reason. Q4. Are the following matrices equal? 1 A = 1 7 3 ,  1 7 7 2 4 0 B = 2 4 – 12×3 2×3 Q5. If A = 4 x + 2 is symmetric, then find the value of x. 1 2x − 3 x + 1 OR Write a 2 × 2 matrix which is both symmetric and skew symmetric. 1 3 –2 1 Q6. If Δ = 4 – 5 6 , write the co-factor of a32 (the element of third row and 2nd column). 35 2 Q7. Evaluate: ∫ (x + 2)3 dx  1 OR ∫2 Evaluate: x dx −1 Q8. Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant. 1 Q9. Solve: dy = x 1  1 dx x2 + OR NBSE 2023 2

Write the sum of the order and degree of the D.E. d  dy  3  =0 dx  dx    →→ 1 Q10. For what value of λ the vectors a = 2i + λ j + k and b = i − 2 j + 2k are perpendicular to each other? Q11. Write the magnitude of position vector → = xi + y j + zk . 1 | p| →→ Q12. Find the work done in moving an object along a vector d = 8i + 2 j − 5k if the applied force is F = 7i − k . 1 Q13. Change the equations of the line in vector form: x + 3 = 3 − 2y = 5z + 4 . 1 2 3 1 →∧∧∧ 1 Q14. Change into Cartesian form of equations of line: r = (3λ + 2) i + (5 − 2λ) j + (1 − 4λ) k . Q15. Let E and F be events with P(E) = 3 , P(F) = 3 and P(E ∩ F) = 1 . Are E and F independent? 1 5 10 5 Q16. Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that both balls are red.  1 Section-II Both the Case study-based questions are compulsory. Attempt any 4 sub-parts from each question (17–18). Each sub-part carries 1 mark. Q17. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Based on the above information, answer the following: (i) The volume of tank (V) is (b) xy 1 (a) x2y (d) xy2 1 (c) x + y P.T.O. (ii) Total Surface Area (S) of tank in terms of x is (a) x2 + 4V (b) x2 + 4V x NBSE 2023 3

(c) x + 4V (d) x2 – 4V x x (iii) For maximum or minimum value of surface area S, dS is 1 dx 1 (a) 2x + 4V (b) x – 4V x2 x2 (c) x + 4V (d) 2x – 4V x2 x2 (iv) From dS = 0, relation between length (x) and volume (V) is dx (a) x2 = 2V (b) x3 = 2V (c) x = 2V (d) x = 2V2 (v) For minimum surface area (S), relation between length (x) and height (y) is 1 (a) x = 2 + y (b) x = 2 – y (c) x = y (d) x = 2y Q18. A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result of 0.5% of the healthy person tested (i.e., if a healthy person is tested, then with probability 0.005, the test will imply he has the disease). Based on the above information answer the following: (i) Probability that a person has a disease is 1 1 (a) 0.001 (b) 0.01 (c) 0.1 (d) 0.0001 1 (ii) Probability that a person does not have the disease is (a) 0.09 (b) 0.999 (c) 0.092 (d) 0.091 (iii) Probability that a person having a disease and blood test positive is (a) 0.91 (b) 9.9 (c) 0.99 (d) 9.2 NBSE 2023 4

(iv) Probability that a person not having a disease but has positive blood test is 1 1 (a) 0.004 (b) 0.002 (c) 0.003 (d) 0.005 (v) Find the probability that a person has the disease given that his test result is positive.  (a) 129 (b) 118 (c) 198 (d) 108 1179 1190 1197 1107 Part–B Section–III All questions are compulsory. In case of internal choices attempt any one. Q19. Evaluate cos cos −1  − 1  + sin −1  − 1    2  7   7   2  1 −1 5 2 Q20. Show that the matrix A = −51 13 is a symmetric matrix. 2 1 OR 6 –3 2 Find the value of Δ = 2 – 1 2 –10 5 2 Q21. Examine the continuity of the function f (x) = x2 – 25  x + 5 ( ) Q22. Find the intervals in which the function: π on [0, π] f (x) = sin x + 4 (a) strictly increases (S↑) (b) strictly decreases (S↓) 2 Q23. Evaluate: ∫ dx  2 (2 − x) (x2 + 3) OR ∫π 1 + cos x dx π/2 1 − cos x Evaluate: Q24. Find the area of the region {(x, y) : x2 ≤ y ≤ x}. 2 2 Q25. Solve dy = 2x3 – x, given y = 1, when x = 0. 2 dx Q26. Find the magnitude of the vector → × → → = 2iˆ + kˆ, → =iˆ + ˆj + kˆ  a b if a b → Q27. Find the equation of the line passing through the point i + j − 3k and perpendicular to the lines r = i + λ(2 → 2 i + j − 3k )  and  r = (2 i + j + k ) + μ( i + j + k ). Q28. Given P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.3, then, find P(A′|B′). 2 NBSE 2023 5 P.T.O.

OR Events A and B are such that P(A) = 1 , P(B) = 7 and P(not A or not B) = 1 2 12 4 State whether A and B are independent or not. Section-IV All questions are compulsory. In case of internal choices attempt any one. Q29. Let A = {1, 2, 3} and define a relation R on A as follows: R = {(1, 1), (2, 2), (3, 3)} Prove that R is an equivalence relation. 3 Q30. If y = x sin y, prove that x dy = 1– y y or dy = x (1 – y y) . 3 dx x cos dx x cos Q31. Is f (x) = | x – 1 | + | x – 2 | differentiable at x = 2? 3 OR If x = cos θ + θ sin θ, y = sin θ – θ cos θ, then find d2 y dx2 Q32. Find the intervals on which the following function is (i) increasing (ii) decreasing : f (x) = x3 – 12x2 + 36x + 17. 3 Q33. Evaluate: ∫ (x + 8 dx . 3 2)( x 2 + 4) 3 Q34. Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the first quadrant. 3 OR Find the area of the ellipse x2 + y2 = 1. a2 b2 Q35. Find the particular solution of the differential equation ddyx – 3y cot x = sin 2x; y = 2 when x = π 2 Section-V All questions are compulsory. In case of internal choices attempt any one. Q36. Using matrices, solve the following system of equations: x + 2y – 3z = 6; 3x + 2y – 2z = 3; 2x – y + z = 2 5 OR 1 −2 0 2 3 If A = 0 1 1 , find A–1. −2 Using A–1, solve the system of linear equations : NBSE 2023 6

x – 2y = 10 2x + y + 3z = 8 – 2y + z = 7. Q37. Find the equation of a line passing through a point A(1, 2, 3) and parallel to the line 2x −1 = 3y + 2 = 2 − z . 5 1 2 1 OR Find a point on the line: x −1 = y −1 = z − 2 at a distance of 3 units from a point (3, 2, 2). 1 2 −1 Q38. Solve the following linear programming problem (L.P.P.) graphically. 5 Maximize Z = 5x + 3y subject to constraints; 3x + 2y ≤ 20 2x + y ≤ 12 x, y ≥ 0 OR The corner points of the feasible region determined by the system of linear constraints are as shown below: Let F = 4x + 6y be the objective function. Find the minimum value of F and also the corresponding point at which minimum value occurs. NBSE 2023 7 P.T.O.

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