Maths SQP 2024-2025 PDF

SAMPLE QUESTION PAPER (2024 - 25) CLASS- XII SUBJECT: Mathematics (041) Time: 3 Hours Maximum Marks: 80 General Instructions:...

SAMPLE QUESTION PAPER (2024 - 25) CLASS- XII SUBJECT: Mathematics (041) Time: 3 Hours Maximum Marks: 80 General Instructions: Read the following instructions very carefully and strictly follow them: (i) This Question paper contains 38 questions. All questions are compulsory. (ii) This Question paper is divided into five Sections - A, B, C, D and E. (iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and Questions no. 19 and 20 are Assertion-Reason based questions of 1 mark each. (iv) In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2 marks each. (v) In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each. (vi) In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each. (vii) In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each. (viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 3 questions in Section C, 2 questions in Section D and one subpart each in 2 questions of Section E. (ix) Use of calculators is not allowed. SECTION-A 1  20  20 (This section comprises of multiple choice questions (MCQs) of 1 mark each) Select the correct option (Question 1 - Question 18): 𝟐𝟎𝟐𝟓 𝟎 𝟎 Q.1. If for a square matrix A, 𝑨. (𝒂𝒅𝒋𝑨) = [ 𝟎 𝟐𝟎𝟐𝟓 𝟎 ], then the value of A  adj A is 𝟎 𝟎 𝟐𝟎𝟐𝟓 equal to: (C)  2025   45 (D) 2025   2025  2 2 (A) 1 (B) 2025  1 Q.2. Assume X , Y , Z , W and P are matrices of order 2  n, 3  k , 2  p, n  3 and p  k , respectively. Then 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.3. The interval in which the function f defined by 𝑓(𝑥) = 𝑒 𝑥 is strictly increasing, is (A) [1, ∞) (B)   ,0  (C) (−∞, ∞) (D) (0,∞) Class-XII/Sample Paper/2024-25/Mathematics/Page 1 of 9  𝟐 Q.4. If A and B are non-singular matrices of same order with 𝒅𝒆𝒕(𝑨) = 𝟓, then 𝒅𝒆𝒕(𝑩−𝟏 𝑨𝑩) is equal to 2 4 (A) 5 (B) 5 (C) 5 (D) 5 5 𝒅𝒚 Q.5. The value of ' n ' , such that the differential equation 𝒙𝒏 𝒅𝒙 = 𝒚(𝒍𝒐𝒈𝒚 − 𝒍𝒐𝒈𝒙 + 𝟏); (𝐰𝐡𝐞𝐫𝐞 𝒙, 𝒚 ∈ 𝑹+ ) is homogeneous, is (A) 0 (B) 1 (C) 2 (D) 3 Q.6. If the points (𝒙𝟏 , 𝒚𝟏 ), (𝒙𝟐 , 𝒚𝟐 ) and (𝒙𝟏 + 𝒙𝟐 , 𝒚𝟏 + 𝒚𝟐 )are collinear, then 𝒙𝟏 𝒚𝟐 is equal to (A) x2 y1 (B) x1 y1 (C) x2 y2 (D) x1 x2 0 1 c  Q.7. If A   1 a  b  is a skew-symmetric matrix then the value of a  b  c   2 3 0  (A)1 (B) 2 (C) 3 (D) 4 Q.8. For any two events A and B , if P A    1 2   2 3 1 , P B  and P  A  B   , then 𝑃 (𝐴⁄ ̅ ) equals: 4 ̅ 𝐵 3 8 5 1 (A) (B) (C) (D) 8 9 8 4 Q.9. The value of 𝛼 if the angle between 𝑝⃗ = 2𝛼 2 𝑖̂ − 3𝛼𝑗̂ + 𝑘̂ and 𝑞⃗ = 𝑖̂ + 𝑗̂ + 𝛼𝑘̂ is obtuse, is (A) 𝑅 − [0, 1] (B) (0, 1) (C) [0, ∞) (D) [1, ∞) Q.10. If |𝑎⃗| = 3, |𝑏⃗⃗| = 4 and |𝑎⃗ + 𝑏⃗⃗| =5, then |𝑎⃗ − 𝑏⃗⃗| = (A) 3 (B) 4 (C) 5 (D) 8 Q.11. For the linear programming problem (LPP), the objective function is Z  4 x  3 y and the feasible region determined by a set of constraints is shown in the graph: Class-XII/Sample Paper/2024-25/Mathematics/Page 2 of 9  (Note: The figure is not to scale.) Which of the following statements is true? (A) Maximum value of Z is at R  40,0 . (B) Maximum value of Z is at Q  30, 20 . (C) Value of Z at R  40,0  is less than the value at P  0,40 . (D) The value of Z at Q  30, 20  is less than the value at R  40,0 . 𝒅𝒙 Q.12. ∫ 𝟏 equals 𝟑 𝟒 𝒙 (𝟏+𝒙 )𝟐 1 1 (A) − 2𝑥 2 √1 + 𝑥 4 + 𝑐 (B) 2𝑥 √1 + 𝑥 4 + 𝑐 1 1 (C) − 4𝑥 √1 + 𝑥 4 + 𝑐 (D) 4𝑥 2 √1 + 𝑥 4 + 𝑐 𝟐𝝅 Q.13. ∫𝟎 𝒄𝒐𝒔𝒆𝒄𝟕 𝒙 𝒅𝒙 = (A) 0 (B) 1 (C) 4 (D) 2 ′ Q.14. What is the general solution of the differential equation ey = x? (A)𝑦 = 𝑥𝑙𝑜𝑔𝑥 + 𝑐 (B) 𝑦 = 𝑥𝑙𝑜𝑔𝑥 − 𝑥 + 𝑐 (C) 𝑦 = 𝑥𝑙𝑜𝑔𝑥 + 𝑥 + 𝑐 (D) 𝑦 = 𝑥 + 𝑐 Q.15. The graph drawn below depicts (A) y = 𝑠𝑖𝑛−1 𝑥 (B) y = 𝑐𝑜𝑠 −1 𝑥 (C) y = 𝑐𝑜𝑠 𝑒 𝑐 −1 𝑥 (D) y = 𝑐𝑜𝑡 −1 𝑥 Q.16. A linear programming problem (LPP) along with the graph of its constraints is shown below. Class-XII/Sample Paper/2024-25/Mathematics/Page 3 of 9  The corresponding objective function is: Z  18 x  10 y , which has to be minimized. The smallest value of the objective function Z is 134 and is obtained at the corner point  3,8  , (Note: The figure is not to scale.) The optimal solution of the above linear programming problem __________. (A) does not exist as the feasible region is unbounded. (B) does not exist as the inequality 18 x  10 y  134 does not have any point in common with the feasible region. (C) exists as the inequality 18 x  10 y  134 has infinitely many points in common with the feasible region. (D) exists as the inequality 18 x  10 y  134 does not have any point in common with the feasible region. Q.17. The function 𝑓: 𝑅 → 𝑍 defined by f  x    x ; where .  denotes the greatest integer function, is (A) Continuous at x  2.5 but not differentiable at x  2.5 (B) Not Continuous at x  2.5 but differentiable at x  2.5 (C) Not Continuous at x  2.5 and not differentiable at x  2.5 (D) Continuous as well as differentiable at x  2.5 Q.18. A student observes an open-air Honeybee nest on the branch of a tree, whose plane figure is parabolic shape given by x 2  4 y. Then the area (in sq units) of the region bounded by parabola x 2  4 y and the line y  4 is 32 64 128 256 (A) 3 (B) 3 (C) 3 (D) 3 ASSERTION-REASON BASED QUESTIONS (Question numbers 19 and 20 are Assertion-Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the options (A), (B), (C) and (D) as given below.) Class-XII/Sample Paper/2024-25/Mathematics/Page 4 of 9  (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. Q.19. Assertion (A): Consider the function defined as 𝑓(𝑥) = |𝑥| + |𝑥 − 1|, 𝑥 ∈ 𝑅. Then f  x  is not differentiable at x  0 and x  1. Reason (R): Suppose f be defined and continuous on  a , b  and c   a , b  , then f  x  is not 𝑓(𝑐+ℎ)−𝑓(𝑐) 𝑓(𝑐+ℎ)−𝑓(𝑐) differentiable at 𝑥 = 𝑐 if lim− ≠ lim+. ℎ→0 ℎ ℎ→0 ℎ 𝜋 Q.20. Assertion (A): The function 𝑓: 𝑅 − {(2𝑛 + 1) 2 : 𝑛 ∈ 𝑍 } → (−∞, −1] ∪ [1, ∞) defined by f  x   sec x is not one-one function in its domain. Reason (R): The line y  2 meets the graph of the function at more than one point. SECTION B  2  5  10 (This section comprises of 5 very short answer (VSA) type questions of 2 marks each.) 𝜋 Q.21. If 𝑐𝑜𝑡 −1 (3𝑥 + 5) > 4 , then find the range of the values of x. Q.22. The cost (in rupees) of producing x items in factory, each day is given by 𝐶(𝑥) = 0.00013𝑥 3 + 0.002 𝑥 2 + 5𝑥 + 2200 Find the marginal cost when 150 items are produced. Q.23. (a) Find the derivative of tan 1 x with respect to log x ; (where x   1,   ). OR 𝜋 Q.23. (b) Differentiate the following function with respect to x : (𝑐𝑜𝑠 𝑥)𝑥 ; (where𝑥 ∈ (0, 2 )). Q.24. (a) If vectors 𝑎⃗ = 2ı̂ + 2ȷ̂ + 3k̂, 𝑏⃗⃗ = − ı̂ + 2ȷ̂ + k̂ and 𝑐⃗ = 3ı̂ + ȷ̂ are such that 𝑏⃗⃗ + λ𝑐⃗ is perpendicular to 𝑎⃗ , then find the value of λ. OR Q.24. (b) A person standing at O  0 , 0 , 0  is watching an aeroplane which is at the coordinate point A  4 , 0 , 3 . At the same time he saw a bird at the coordinate point B  0 , 0 ,1. Find the angles which ⃗⃗⃗⃗⃗⃗ makes with the x,y and z axes. 𝐵𝐴 Q.25. The two co-initial adjacent sides of a parallelogram are 2ı̂ − 4ȷ̂ − 5k̂ and 2ı̂ + 2ȷ̂ + 3k̂. Find its diagonals and use them to find the area of the parallelogram. Class-XII/Sample Paper/2024-25/Mathematics/Page 5 of 9  SECTION C  3  6  18 (This section comprises of 6 short answer (SA) type questions of 3 marks each.) Q.26. A kite is flying at a height of 3 metres and 5 metres of string is out. If the kite is moving away horizontally at the rate of 200 cm/s, find the rate at which the string is being released. Q.27. According to a psychologist, the ability of a person to understand spatial concepts is given by 1 A t, where t is the age in years, t   5,18. Show that the rate of increase of the ability to 3 understand spatial concepts decreases with age in between 5 and 18. Q.28. (a) An ant is moving along the vector ⃗⃗⃗ 𝑙1 = 𝑖̂ − 2𝑗̂ + 3𝑘̂. Few sugar crystals are kept along the vector ⃗𝑙⃗⃗⃗2 = 3𝑖̂ − 2𝑗̂ + 𝑘̂ which is inclined at an angle  with the vector ⃗⃗⃗ 𝑙1. Then find the angle . Also find the scalar projection of ⃗⃗⃗ 𝑙1 𝑜𝑛 ⃗⃗⃗⃗ 𝑙2. OR Q.28. (b) Find the vector and the cartesian equation of the line that passes through (−1, 2, 7) and is perpendicular to the lines 𝑟⃗ = 2ı̂ + ȷ̂ − 3k̂ + λ(ı̂ + 2ȷ̂ + 5k̂) and 𝑟⃗ = 3ı̂ + 3ȷ̂ − 7k̂ + μ(3ı̂ − 2ȷ̂ + 5k̂). 𝟏 𝟏 Q.29. (a) Evaluate: ∫ {𝒍𝒐𝒈 𝒙 − (𝒍𝒐𝒈 𝒙)𝟐 } 𝒅𝒙; (where𝒙 > 𝟏). OR 𝟏 Q.29. (b) Evaluate : ∫𝟎 𝒙(𝟏 − 𝒙)𝒏 𝒅𝒙; (𝐰𝐡𝐞𝐫𝐞 𝒏 ∈ 𝑵). Q.30. Consider the following Linear Programming Problem: Minimise Z  x  2 y Subject to 2 x  y  3, x  2 y  6, x , y  0. Show graphically that the minimum of Z occurs at more than two points Q.31. (a) The probability that it rains today is 0.4. If it rains today, the probability that it will rain tomorrow is 0.8. If it does not rain today, the probability that it will rain tomorrow is 0.7. If 𝑃1 : denotes the probability that it does not rain today. 𝑃2 : denotes the probability that it will not rain tomorrow, if it rains today. 𝑃3 : denotes the probability that it will rain tomorrow, if it does not rain today. 𝑃4 : denotes the probability that it will not rain tomorrow, if it does not rain today. (i) Find the value of P1  P4  P2  P3.  2 Marks  (ii) Calculate the probability of raining tomorrow. 1 Mark  OR Q.31. (b) A random variable X can take all non – negative integral values and the probability that X takes Class-XII/Sample Paper/2024-25/Mathematics/Page 6 of 9  the value r is proportional to 5−𝑟. Find P  X  3 . SECTION D  5  4  20 (This section comprises of 4 long answer (LA) type questions of 5 marks each) 𝜋 𝜋 Q.32. Draw the rough sketch of the curve 𝑦 = 20 𝑐𝑜𝑠 2 𝑥; (where 6 ≤ 𝑥 ≤ 3 ). Using integration, find the area of the region bounded by the curve y = 20 cos2x from the ordinates 𝜋 𝜋 𝑥= to 𝑥 = and the x  axis. 6 3 Q.33. The equation of the path traversed by the ball headed by the footballer is 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐; (𝑤ℎ𝑒𝑟𝑒 0 ≤ 𝑥 ≤ 14 𝑎𝑛𝑑 𝑎, 𝑏, 𝑐 ∈ 𝑅 𝑎𝑛𝑑 𝑎 ≠ 0) with respect to a XY-coordinate system in the vertical plane. The ball passes through the points  2,15  ,  4, 25  and 14,15 . Determine the values of a, b and c by solving the system of linear equations in a, b and c, using matrix method. Also find the equation of the path traversed by the ball. Q.34. (a) If 𝑓: 𝑅 → 𝑅 is defined by 𝑓(𝑥) = |𝑥|3 , show that 𝑓"(𝑥) exists for all real x and find it. OR 3 𝑑𝑦 2 2 [1+( ) ] 𝑑𝑥 Q.34. (b) If (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑐 2 , for some c  0, prove that 𝑑2 𝑦 is a constant independent 𝑑𝑥2 of a and b. Q.35. (a) Find the shortest distance between the lines l1 and l2 whose vector equations are 𝑟⃗ = (−𝑖̂ − 𝑗̂ − 𝑘̂) + 𝜆(7𝑖̂ − 6𝑗̂ + 𝑘̂) and 𝑟⃗ = (3𝑖̂ + 5𝑗̂ + 7𝑘̂) + 𝜇(𝑖̂ − 2𝑗̂ + 𝑘̂) where  and  are parameters. OR Q.35. (b) Find the image of the point 1,2, 1 with respect to the line 𝑥−3 𝑦+1 𝑧−1 = =. Also find the 1 2 3 equation of the line joining the given point and its image. SECTION- E  4  3  12 (This section comprises of 3 case-study/passage-based questions of 4 marks each with subparts. The first two case study questions have three subparts (i), (ii), (iii) of marks 1, 1, 2 respectively. The third case study question has two subparts of 2 marks each) Case Study-1 Q.36. Ramesh, the owner of a sweet selling shop, purchased some rectangular card board sheets of dimension 25 cm by 40 cm to make container packets without top. Let x cm be the length of the side of the square to be cut out from each corner to give that sheet the shape of the container by folding up the flaps. Based on the above information answer the following questions. Class-XII/Sample Paper/2024-25/Mathematics/Page 7 of 9  (i) Express the volume (V) of each container as function of x only. 1 Mark  1 Mark  𝑑𝑉 (ii) Find 𝑑𝑥 (iii) (a) For what value of x , the volume of each container is maximum?  2 Marks  OR  2 Marks  65 (iii) (b) Check whether V has a point of inflection at x = or not? 6 Case Study-2 Q.37. An organization conducted bike race under 2 different categories-boys and girls. In all, there were 250 participants. Among all of them finally three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B  b1 , b2 , b3  , G   g1 , g2  where B represents the set of boys selected and G the set of girls who were selected for the final race. Ravi decides to explore these sets for various types of relations and functions. On the basis of the above information, answer the following questions: (i) Ravi wishes to form all the relations possible from B to G. How many such relations are possible? 1 Mark  (ii) Write the smallest equivalence relation on G. 1 Mark  (iii) (a) Ravi defines a relation from B to B as 𝑹𝟏 = {(𝒃𝟏 , 𝒃𝟐 ), (𝒃𝟐 , 𝒃𝟏 )}. Write the minimum ordered pairs to be added in 𝑹𝟏 so that it becomes (A) reflexive but not symmetric, (B) reflexive and symmetric but not transitive.  2 Marks  OR (iii) (b) If the track of the final race (for the biker b1 ) follows the curve 𝑥 2 = 4𝑦; (where0 ≤ 𝑥 ≤ 20√2&0 ≤ 𝑦 ≤ 200), then state whether the track represents a one-one and onto function or not. (Justify).  2 Marks  Case Study- 3 Q.38. Arka bought two cages of birds: Cage-I contains 5 parrots and 1 owl and Cage –II contains 6 parrots. One day Arka forgot to lock both cages and two birds flew from Cage-I to Cage-II (simultaneously). Then two birds flew back from cage-II to cage-I(simultaneously). Assume that all the birds have equal chances of flying. On the basis of the above information, answer the following questions:- Class-XII/Sample Paper/2024-25/Mathematics/Page 8 of 9 (i) When two birds flew from Cage-I to Cage-II and two birds flew back from Cage-II to Cage-I then find the probability that the owl is still in Cage-I.  2 Marks  (ii) When two birds flew from Cage-I to Cage-II and two birds flew back from Cage-II to Cage-I, the owl is still seen in Cage-I, what is the probability that one parrot and the owl flew from Cage-I to Cage-II?  2 Marks  ******************************************************************************* Class-XII/Sample Paper/2024-25/Mathematics/Page 9 of 9