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ISC Class XII Prelims 2026 : Mathematics (Delhi Public School (DPS), Newtown, Kolkata)

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Adhrit Layek
Delhi Public School (DPS), Newtown, Kolkata

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DELHI PUBLIC SCHOOL NEWTOWN SESSION 2025-2026 PRE BOARD EXAMINATION CLASS: XII SUBJECT: MATHEMATICS [SET A] FULL MARKS: 80 TIME: 3 HOURS Instructions to Candidates 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. You are allowed an additional fifteen minutes for only reading the paper. You must NOT start writing during reading time. The question paper has ten printed pages. The Question Paper is divided into three sections and has 22 questions in all. Section A is compulsory and has fourteen questions. You are required to attempt all questions either from Section B or Section C. Section B and Section C have four questions each. Internal choices have been provided in two questions of 2 marks, two questions of 4 marks and two questions of 6 marks in Section A Internal choices have been provided in one question of 2 marks and one question of 4 marks each in Section B and Section C. While attempting Multiple Choice Questions in Section A, B and C, you are required to write only ONE option as the answer. The intended marks for questions or parts of questions are given in the brackets []. All workings, including rough work, should be done on the same page as, and adjacent to the rest of the answer. Mathematical tables and graph papers are provided. SECTION A - 65 MARKS Question 1 In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed [1] (i) If A =[ ] is a square matrix of order such that = (a) Symmetric matrix (b) Null matrix (c) Skew- symmetric matrix (ii) If m and n respectively, are the order and the degree of the differential equation (iii) (d) Diagonal matrix (( ) ) = , then m+n is: (a) 1 (b) 2 (c) 3 (d) 4 If x = t2 and y = t3, then (a) (c) [1] [1] is equal to (b) (d) A/1 (iv) Assertion: The value of the determinant of a matrix of order 3 is given by = + + ; where is a cofactor of [1] Reason: is the sum the product of elements of any row (or column) with their corresponding cofactors (a) Both Assertion and Reason are true and Reason is the correct explanation for Assertion. (b) Both Assertion and Reason are true but Reason is not the correct explanation for Assertion. (c) Assertion is true and Reason is false. (d) Assertion is false and Reason is true. (v) One ticket is selected at random from 50 tickets numbered 00, 01, 02, .. [1] ...49. Then, the probability that the sum of the digits on the tickets is 8, given that the product of the digits is zero, equals (a) (c) (vi) (vii) (b) (d) If = [ [1] ], the is equal to ] (b) [ ] (d) [ (a) [ (c) [ ] ] Observe the following graph of a function. Statement 1: A function which is differentiable at a point may not be continuous at that point. Statement 2: The above graph is an example of a function that is continuous but not differentiable at x = 2. Which of the following is correct? (a) Statement 1 is true and Statement 2 is false. (b) Statement 2 is true and Statement 1 is false. (c) Both the statements are true. (d) Both the statements are false. A/2 [1] [1] (viii) Evaluate (a) Infinity ( ) (b) 1 (c) -1 (ix) (d) 0 Statement 1: The relation R on the set of natural number N given by [1] = {( , ): + = } is an equivalence relation. Statement 2: A relation on set is said to be equivalence if it is reflexive, symmetric, and transitive. Which one of the following is correct? (a) Statement 1 is true and Statement 2 is false. (b) Statement 2 is true and Statement 1 is false. (c) Both the statements are true. (d) Both the statements are false. (x) + Assertion: If = ( ); Reason: + < < , then =1 [1] = ( ) (a) Both Assertion and Reason are true and Reason is the correct explanation for Assertion. (b) Both Assertion and Reason are true but Reason is not the correct explanation for Assertion. (c) Assertion is true and Reason is false. (d) Assertion is false and Reason is true. (xi) Two events A and B are such that P(A) = 0.7, P(B) = 0.4 and P(A/B) = 0.3. Then the probability that neither A nor B occurs is equal to (a) 0.02 (b) 0.98 (c) 0.12 (xii) [1] (d) 0.16 Which one of the following graphs is a function of x? Give reason. (xiii) Let A is a square matrix of order 3 and |A| = 4, then find the value of | | + | |. A/3 [1] [1] (xiv) (xv) If A = {1, 2, 3} and B = { a, b}, then find the total number of relations from [1] A to B. Two balls are drawn from an urn containig 2 white 3 red and 4 black balls one [1] by one without replacement. What is the probability both the balls are red? Question 2 [2] If = + + +. . . . . . . . . . . , prove that ( ) = OR Find the values of a and b if = + + has its extreme values at [2] x = -1 and x = 2. (i) (ii) Question 3 Find the point on the curve = + at which the equation of the tangent is = . Question 4 Show that = is the general solution of the differential equation: + = . [2] [2] Question 5 (i) Evaluate: [2] [2] OR (ii) Evaluate: [ ] Question 6 If f ( x ) = 10 x 10 x is invertible, find f 10 x + 10 x 1 ( x) . Question 7 Solve: ( ) = [2] [4] Question 8 If = , prove that ( ) = Question 9 (i) Solve the following differential equation: [ ( )] + [ ( ) ( )] = (ii) OR Solve the following differential equation: = ( ) ; ( ) = A/4 [4] [4] [4] Question 10 [4] (i) An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company s statistics show that an accident-prone person will have an accident at sometime within a fixed one-year period with probability 0.6, whereas this probability is 0.2 for a person who is not accident prone. The company knows that 20 percent of the population is accident prone. Based on the given information, answer the following questions. (a) What is the probability that a new policyholder will have an accident within a year of purchasing a policy? (b) Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone? OR (ii) In a test an examinee either guesses or copies or knows the answer to a multiple- [4] choice question with four choices. The probability that he makes a guess is 1/3 and the probability that he copies the answer is 1/6. The probability that his answer is correct, given that he copied it is 1/8. Find the probability that he knew the answer to the questions, given that it correctly answered it. Question 11 [6] An insurance company agent has the following record of policies sold in the month of April, May and June 2022 for three different policies-Policy A, Policy B and Policy C. He is paid a fixed commission per policy sold but the commission varies for the policies A, B and C. Month (i) (ii) (iii) Number of policies Sold Total commission earned in the month (In ) Policy A Policy B Policy C April 8 4 6 7850 May 9 9 6 9600 June 12 9 12 15000 Translate the problem into a system of equations. Solve the system of equation by using matrix method. Hence find the fixed commission payable on policies A, B and C per unit using matrix method. Question 12 Evaluate: ( + [6] ) dx OR Evaluate: ( [6] . ) dx A/5 Question 13 (i) [6] Mr Sam, who is an architect, designs a building for a small company. The design of window on the ground floor is proposed to be different than other floors. The window is in the shape of a rectangle which is surmounted by a semi-circular opening. This window is having a perimeter of 30 m as shown in the figure. Based on the above information answer the following: (a) If 2x and 2y represents the length and breadth of the rectangular portion of the window, then find the relation between the variables. (b) Express the combined area (A) of the rectangular region and semi-circular region of the window as function of x. (c) The owner of this small company is interested in maximizing the area of the whole window so that maximum light input is possible. For this to happen, what should the dimensions of the rectangular portion of the window? (d) Hence find the maximum area in terms of . (ii) OR Mala watches a painting which has its bottom edge 2 meters (m) above eye level and its top edge is 3 m above eye level as shown in the diagram. Based on the above information answer the questions that follow. (a) Given and as shown in the diagram, find tan and tan( + ). (b) Find in terms of only (c) Find so that ( )=0 (d) Hence find the distance Mala should stand from the wall to maximize her viewing angle of the painting. A/6 [6] Question 14 [6] At a school, three sports coaches Football, Basketball, and Cricket independently decide whether to hold an extra after-school practice session on a given day. Based on previous data: The Football coach holds extra practice with probability 0.6. The Basketball coach holds extra practice session with probability 0.5. The Cricket coach holds extra practice session with probability 0.7. Let denote the number of sports conducting extra practice session on any given day. Hence, { , , , }. The principal wants to ensure that students are not overburdened. According to policy, if the average number of practices per day exceeds 1.5, coaches should coordinate their schedules to reduce the load on students. Otherwise, the schedule is considered acceptable. (a) (b) (c) (d) Find the probability for each possible value of . Construct the probability distribution table for . Calculate the mean (expected value) of . Should the coaches improve their coordination? Or is the current plan sufficient as it stands? Justify your answer. SECTION-B Question 15 In subparts (i) to (iii) choose the correct options and in subparts (iv) and (v), answer the questions as instructed. (i) Consider the following statements and choose the correct option: Statement 1: The vector area of triangle ABC with vertices having position + respectively is ( , and + ). vectors Statement 2: The condition for the collinearity of three vectors is + + ) = . ( Which of the following is correct? (a) Statement 1 is true and Statement 2 is false. (b) Statement 2 is true and Statement 1 is false. (c) Both the statements are true. (d) Both the statements are false. (ii) If the plane + + = is perpendicular to the plane + + = , hen the value of is (a) 11 (b) -11 (c) 12 (d) 0 (iii) The direction cosine of the line joining the points (4, 3, -5) and (-2, 1, -8) are (a) ( , (c) ( , , ) (b) ( , , ) (d) ( A/7 , , ) , ) [1] [1] [1] (iv) If and are unit vectors enclosing an angle between them, then show that: ( ) = (v) [1] |. | In the figure given below. If the coordinates of point P are (a, b, c), then write the perpendicular distances of P from XY, YZ and ZX planes respectively Question 16 (i) Three birds are sitting on a tree at positions A(4,6,8) , B (6, 7 ,7) and C(5,6, 9) [1] [2] Using the concept of vector algebra. (a) (b) (ii) and . Find the vectors and . Also find the angle between the vectors OR and , are three vectors of magnitude 3, 4 and 5 respectively. If each [2] Let one is perpendicular to the sum of the other two vectors, then prove that + + | = 5 . | Question 17 (i) Find the image of the point (1, 6, 3) in the line = = [4] OR (ii) Obtain the equation of the plane through the points (2, 1, -1) and (-1, 3, 4) and [4] perpendicular to the plane x 2y +4z = 10 Question 18 Find the area enclosed between the curves = + and = A/8 [4] SECTION-C Question 19 In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed. (i) If x is the number of units, then (a) ( ) = ( ) (b) (c) (d) (ii) (iii) (iv) (v) ( ) = [1] ( ) ( ) = ( ) ( ) = ( ) Consider the following statements and choose the correct option: Statement 1: If correlation coefficient r = 0, then the regression lines coincide each other.. Statement 2: If correlation coefficient r = , then the regression lines are perpendicular to each other. Which of the following is correct? (a) Statement 1 is true and Statement 2 is false. (b) Statement 2 is true and Statement 1 is false. (c) Both the statements are true. (d) Both the statements are false. = , = , = , = and correlation coefficient If = . , find the regression coefficient of y on x. [1] [1] [1] The average revenue function is given by = . Find the marginal revenue function. A company starts producing pens and finds that production cost of each pen is [1] 10 with the fixed cost of production is 4500. If each pen is sold for 25, find the break-even point. Question 20 (i) [2] In a real estate business, companies buy, sell and rent properties. A real estate company of Kolkata is going to build a new residential complex. The land they have purchased can hold at the most 550 apartments. Also, if they make x apartments, then the monthly maintenance cost for the whole complex would be as follows: Fixed cost = 60000 Variable cost = (180 x 0.5 x2 ) A/9 How many apartments should the complex have in order to maximise the maintenance costs? OR (ii) If the cost function of a product is given by ( ) = - 45x2 900x, where x is [2] the number of units produced, determine the number of units that should be produced to minimize the marginal cost. Question 21 An analyst for a certain company was studying the relationship between travel expenses [4] (y) for 102 sales trip and the duration in days (x) of these trips. He found that relationship between y and x is linear. A summary of the data is given below: = , = , = , = , = (a) Estimate the two lines of regression from the above data. (b) A given trip has to take seven days. How much money should the salesman be allowed so that he will not run short of money? Question 22 (i) Two tailors A and B earn 150 and 200 per day respectively. A can stitch 6 shirts and 4 pants while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce at-least 60 shirts and 32 pants at the minimum labour cost? Also find the minimum cost. OR (ii) The feasible region for an L.P.P. is shown in the figure given below: Based on the above graph, answer the following questions. (a) Write down all the constraints (inequations) of the given Linear Programming Problem (LPP), including the non-negativity constraints. (b) Find the co-ordinates of the point C. (c) Find the maximum value of the objective function = 50 + 60 . A/10 [4] [4]

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