Mathematics Past Paper PDF Class XII 2022-2023

Sample Question Paper Class XII Session 2022-23 Mathematics (Code-041) Time Allowed: 3 Hours Maximum Marks: 80 General Instructions :...

Sample Question Paper Class XII Session 2022-23 Mathematics (Code-041) Time Allowed: 3 Hours Maximum Marks: 80 General Instructions : 1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions. 2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each. 3. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each. 4. Section C has 6 Short Answer (SA)-type questions of 3 marks each. 5. Section D has 4 Long Answer (LA)-type questions of 5 marks each. 6. Section E has 3 source based/case based/passage based/integrated units of assessment (4 marks each) with sub parts. SECTION A (Multiple Choice Questions) Each question carries 1 mark Q1. If A =[aij] is a skew-symmetric matrix of order n, then (a) 𝑎 = ∀ 𝑖, 𝑗 (b) 𝑎 ≠ 0 ∀ 𝑖, 𝑗 (c)𝑎 = 0, 𝑤ℎ𝑒𝑟𝑒 𝑖 = 𝑗 (d) 𝑎 ≠ 0 𝑤ℎ𝑒𝑟𝑒 𝑖 = 𝑗 Q2. If A is a square matrix of order 3, |𝐴′| = −3, then |𝐴𝐴′| = (a) 9 (b) -9 (c) 3 (d) -3 Q3. The area of a triangle with vertices A, B, C is given by (a) 𝐴𝐵⃗ × 𝐴𝐶⃗ (b) 𝐴𝐵⃗ × 𝐴𝐶⃗ (b) 𝐴𝐶⃗ × 𝐴𝐵⃗ (d) 𝐴𝐶⃗ × 𝐴𝐵⃗ , 𝑖𝑓 𝑥 ≠ 0 Q4. The value of ‘k’ for which the function f(x) = is continuous at x = 0 is 𝑘, 𝑖𝑓 𝑥 = 0 (a) 0 (b) -1 (c) 1. (d) 2 Q5. If 𝑓 (𝑥) = 𝑥 + , then 𝑓(𝑥) is (a) 𝑥 + log |𝑥| + 𝐶 (b) + log |𝑥| + 𝐶 (c) + log |𝑥| + 𝐶 (d) − log |𝑥| + 𝐶 Q6. If m and n, respectively, are the order and the degree of the differential equation = 0, then m + n = (a) 1 (b) 2 (c) 3 (d) 4 Q7. The solution set of the inequality 3x + 5y < 4 is (a) an open half-plane not containing the origin. (b) an open half-plane containing the origin. (c) the whole XY-plane not containing the line 3x + 5y = 4. (d) a closed half plane containing the origin. Page 1 Q8. The scalar projection of the vector 3𝚤̂ − 𝚥̂ − 2𝑘 𝑜𝑛 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝚤̂ + 2𝚥̂ − 3𝑘 is (a) (b) (c) (d) √ Q9. The value of ∫ dx is (a) log4 (b) 𝑙𝑜𝑔 (c) 𝑙𝑜𝑔2 (d) 𝑙𝑜𝑔 Q10. If A, B are non-singular square matrices of the same order, then (𝐴𝐵 ) = (a)𝐴 𝐵 (b)𝐴 𝐵 (c)𝐵𝐴 (d) 𝐴𝐵 Q11. The corner points of the shaded unbounded feasible region of an LPP are (0, 4), (0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the objective function Z = 4x + 6y occurs at (a)(0.6, 1.6) 𝑜𝑛𝑙𝑦 (b) (3, 0) only (c) (0.6, 1.6) and (3, 0) only (d) at every point of the line-segment joining the points (0.6, 1.6) and (3, 0) 2 4 2𝑥 4 Q12. If = , 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 value(s) of ‘x’ is/are 5 1 6 𝑥 (a) 3 (b) √3 (c) -√3 (d) √3, −√3 Q13. If A is a square matrix of order 3 and |A| = 5, then |𝑎𝑑𝑗𝐴| = (a) 5 (b) 25 (c) 125 (d) Q14. Given two independent events A and B such that P(A) =0.3, P(B) = 0.6 and P(𝐴 ∩ 𝐵 ) is (a) 0.9 (b) 0.18 (c) 0.28 (d) 0.1 Q15. The general solution of the differential equation 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0 𝑖𝑠 (a) 𝑥𝑦 = 𝐶 (b) 𝑥 = 𝐶𝑦 (c) 𝑦 = 𝐶𝑥 (d) 𝑦 = 𝐶𝑥 Q16. If 𝑦 = 𝑠𝑖𝑛 𝑥, then (1 − 𝑥 )𝑦 𝑖𝑠 equal to (a) 𝑥𝑦 (b) 𝑥𝑦 (c) 𝑥𝑦 (d) 𝑥 Page 2 Q17. If two vectors 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗ are such that |𝑎⃗| = 2 , 𝑏⃗ = 3 𝑎𝑛𝑑 𝑎⃗. 𝑏⃗ = 4, 𝑡ℎ𝑒𝑛 𝑎⃗ − 2𝑏⃗ is equal to (a) √2 (b) 2√6 (c) 24 (d) 2√2 Q18. P is a point on the line joining the points 𝐴(0,5, −2) and 𝐵(3, −1,2). If the x-coordinate of P is 6, then its z-coordinate is (a) 10 (b) 6 (c) -6 (d) -10 ASSERTION-REASON BASED QUESTIONS In the following questions, a statement of assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices. (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true. Q19. Assertion (A): The domain of the function 𝑠𝑒𝑐 2𝑥 is −∞, − ∪ [ , ∞) Reason (R): 𝑠𝑒𝑐 (−2) = − Q20. Assertion (A): The acute angle between the line 𝑟̅ = 𝚤̂ + 𝚥̂ + 2𝑘 + 𝜆(𝚤̂ − 𝚥̂) and the x-axis is Reason(R): The acute angle 𝜃 between the lines 𝑟̅ = 𝑥 𝚤̂ + 𝑦 𝚥̂ + 𝑧 𝑘 + 𝜆 𝑎 𝚤̂ + 𝑏 𝚥̂ + 𝑐 𝑘 and | | 𝑟̅ = 𝑥 𝚤̂ + 𝑦 𝚥̂ + 𝑧 𝑘 + 𝜇 𝑎 𝚤̂ + 𝑏 𝚥̂ + 𝑐 𝑘 is given by 𝑐𝑜𝑠𝜃 = SECTION B This section comprises of very short answer type-questions (VSA) of 2 marks each Q21. Find the value of 𝑠𝑖𝑛 [𝑠𝑖𝑛 ] OR Prove that the function f is surjective, where 𝑓: 𝑁 → 𝑁 such that 𝑛+1 , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 𝑓(𝑛) = 2 𝑛 , 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 2 Is the function injective? Justify your answer. Q22. A man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening? Q23. If 𝑎⃗ = 𝚤̂ − 𝚥̂ + 7𝑘 𝑎𝑛𝑑 𝑏⃗ = 5𝚤̂ − 𝚥̂ + 𝜆𝑘, then find the value of 𝜆 so that the vectors 𝑎⃗ + 𝑏⃗ 𝑎𝑛𝑑 𝑎⃗ − 𝑏⃗ are orthogonal. 𝑶𝑹 Page 3  Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6𝑥 − 12 = 3𝑦 + 9 = 2𝑧 − 2 Q24. If 𝑦√1 − 𝑥 + 𝑥 1 − 𝑦 = 1 , 𝑡ℎ𝑒𝑛 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 = − Q25. Find |𝑥⃗| if (𝑥⃗ − 𝑎⃗). (𝑥⃗ + 𝑎⃗) = 12, where 𝑎⃗ is a unit vector. SECTION C (This section comprises of short answer type questions (SA) of 3 marks each) Q26. Find: ∫ √ Q27. Three friends go for coffee. They decide who will pay the bill, by each tossing a coin and then letting the “odd person” pay. There is no odd person if all three tosses produce the same result. If there is no odd person in the first round, they make a second round of tosses and they continue to do so until there is an odd person. What is the probability that exactly three rounds of tosses are made? OR Find the mean number of defective items in a sample of two items drawn one-by-one without replacement from an urn containing 6 items, which include 2 defective items. Assume that the items are identical in shape and size. Q28. Evaluate: ∫ √ OR Evaluate: ∫ |𝑥 − 1| 𝑑𝑥 Q29. Solve the differential equation: 𝑦𝑑𝑥 + (𝑥 − 𝑦 )𝑑𝑦 = 0 OR Solve the differential equation: 𝑥𝑑𝑦 − 𝑦𝑑𝑥 = 𝑥 + 𝑦 𝑑𝑥 Q30. Solve the following Linear Programming Problem graphically: Maximize Z = 400x + 300y subject to 𝑥 + 𝑦 ≤ 200, 𝑥 ≤ 40, 𝑥 ≥ 20, 𝑦 ≥ 0 Q31. Find ∫ ( ) 𝑑𝑥 SECTION D (This section comprises of long answer-type questions (LA) of 5 marks each) Q32. Make a rough sketch of the region {(𝑥, 𝑦): 0 ≤ 𝑦 ≤ 𝑥 , 0 ≤ 𝑦 ≤ 𝑥, 0 ≤ 𝑥 ≤ 2} and find the area of the region using integration. Q33. Define the relation R in the set 𝑁 × 𝑁 as follows: For (a, b), (c, d) ∈ 𝑁 × 𝑁, (a, b) R (c, d) iff ad = bc. Prove that R is an equivalence relation in 𝑁 × 𝑁. OR Page 4  Given a non-empty empty set X, define the relation R in P(X) as follows: For A, B ∈ 𝑃(𝑋), (𝐴, 𝐵) ∈ 𝑅 iff 𝐴 ⊂ 𝐵. Prove that R is reflexive, transitive and not symmetric. Q34. An insect is crawling along the line 𝑟̅ = 6𝚤̂ + 2𝚥̂ + 2𝑘 + 𝜆 𝚤̂ − 2𝚥̂ + 2𝑘 and another insect is crawling along the line 𝑟̅ = −4𝚤̂ − 𝑘 + 𝜇 3𝚤̂ − 2𝚥̂ − 2𝑘. At what points on the lines should they reach so that the distance between them is the shortest? Find the shortest possible distance between them. OR The equations of motion of a rocket are: ar 𝑥 = 2𝑡, 𝑦 = −4𝑡, 𝑧 = 4𝑡, where the time t is given in seconds, and the coordinates of a moving point in km. What is the path of the rocket? At what distances will the rocket be from the starting point O(0, (0, 0, 0) and from the following line in 10 seconds? 𝑟⃗ = 20𝚤̂ − 10𝚥̂ + 40𝑘 + 𝜇((10𝚤̂ − 20𝚥̂ + 10𝑘 ) 2 −3 5 Q35. If A = 3 2 −4 , find 𝐴. Use 𝐴 to solve the following system of equations 1 1 −2 2𝑥 − 3𝑦 + 5𝑧 = 11, 3𝑥 + 2 2𝑦 − 4𝑧 = −5, 𝑥 + 𝑦 − 2𝑧 = −3 SECTION E (This This section comprises of 3 case-study/passage-based questions of 4 marks each with two sub-parts. First two case study questions have three sub-parts sub parts (i), (ii), (iii) of marks 1, 1, 2 respectively. The third case study question has two sub sub-parts of 2 marks each.) Q36. Case-Study 1: Read the following passage and answer the questions given below. The temperature of a person during an intestinal illness is given by 𝑓(𝑥) = −0.1𝑥 + 𝑚𝑥 + 98 98.6,0 ≤ 𝑥 ≤ 12, m being a constant, where f(x) is the temperature in °F at x days. (i) Is the function differentiable in the interval (0, 12)? Justify your answer. (ii) If 6 is the critical point of thee function, then find the value of the constant m.  (iii) Find the intervals in which the function is strictly increasing/strictly decreasing. OR (iii) Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute maximum/absolute minimum values of the function. Q37. Case-Study 2: Read the following passage and answer the questions given below. In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of + = 1. (i) If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x. (ii) Find the critical point of the function. (iii) Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area. OR (iii) Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area. Page 6 Q38. Case-Study 3: Read the following passage and answer the questions given below. There are two antiaircraft guns, named as A and B. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time. (i) What is the probability that the shell fired from exactly one of them hit the plane? (ii) If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B? Page 7