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ISC Class XII Prelims 2026 : Mathematics (The Doon School, Dehradun)

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The Doon School, Dehradun

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PRE-BOARD - I NOVEMBER 2025 Name of the Candidate : Sch. No./House: Form : SC Day & Date Subject Paper Initials of Master(s) Resource Booklet / Inserts / Data Booklet (if Any) : MATHEMATICS : THEORY : ANC, SRT, MKS : Graph Papers Session Duration No. of Students Maximum Marks : Monday, 17th November 2025 : FIRST : 3 HOURS : 41 : 80 INSTRUCTION TO CANDIDATES The Question paper consists of three sections A, B and C. Candidates are required to attempt all questions from Section A and all questions from either Section B OR Section C Section A: Internal choice has been provided in two questions of two marks each, two questions of four marks each and two questions of six marks each. Section B: Internal choice has been provided in one question of two marks and one question of four marks. Section C: Internal choice has been provided in one question of two marks and one question of four marks. All working, including rough work, should be done on the same sheet as, and adjacent to, the rest of the answer. The intended marks for questions or parts of the questions are given in brackets [ ]. A calculator may be used, and graph papers are provided. This paper consists 15 printed sides. Candidates are allowed an additional 15 minutes for reading the paper only. They must NOT start writing during this time. 1|Page SECTION A 65 MARKS Question 1. [15] In subparts (i) to (x) choose the correct options and in subpart (xi) to (xv), answer the questions as instructed. (i) Let A = {1, 2, 3} and R be a relation on A defined by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}. The relation R is: (a) Reflexive and Symmetric (b) Symmetric and Transitive (c) Reflexive and Transitive (d) Only Reflexive 1 (ii) 2 d 2 y dy 4 The order and the degree of the differential equation = y + is dx 2 dx (a) 4 and 2 (b) 2 and 4 (c) 1 and 4 (d) 2 and 2 (iii) Statement 1: Let R be a relation on the set R(real numbers) defined by R = {( x, y ) : x y} . R is reflexive and transitive. Statement 2: a relation R on a set A is an equivalence relation if and only if it is a reflexive, symmetric, and transitive. Choose the correct option: (a) Statement 1 is true, statement 2 is true; statement 2 is the correct explanation for statement 1. (b) Statement 1 is true, statement 2 is true; statement 2 is not the correct explanation for statement 1. (c) Statement 1 is true, statement 2 is false. (d) Statement 1 is false, statement 2 is true. 2|Page (iv) The derivative of the function is: (v) (a) (b) (c) (d) If tan 1 x = (a) 3|Page 5 (b) 3 5 (c) 2 5 (d) 4 5 10 for some x R then the value of cot 1 x is: x2 + x + 3 ( x 2)( x + 1) dx using partial fraction as follows: (vi) A student solves the problem Step 1: Let x2 + x + 3 A B = + ( x 2)( x + 1) ( x 2) ( x + 1) Step 2: x 2 + x + 3 = A( x + 1) + B ( x 2) Step 3: at x = 1 , B = 1 and at x = 2 , A = 3 Step 4: x2 + x + 3 dx dx ( x 2)( x + 1) dx = 3 ( x 2) ( x + 1) Step 5: = 3log( x 2) log( x + 1) + C Identify at which steps he has done wrong: (a) Step 1 and step 3 (b) Step 2 and step 3 (c) Step 1 only (d) Step 3 and step 5 cos x sin x 1 0 (vii) Assertion (A): Given that A = and A ( adjA) = k , then the value of k is 1 . sin x cos x 0 1 Reason (R): sin x [ 1,1] and cos x [ 1,1] , x [0, 2 ] Choose the correct option: (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation for Assertion (A). (b) Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation for Assertion (A). (c) Assertion (A) is true, Reason (R) is false. (d) Assertion (A) is false, Reason (R) is true. (viii) In which one of the following intervals is the function f ( x) = ( x 2 + 2 x 1) increasing (a) [ 1, ) (b) ( , 1] (c) [ 2, 1] (d) ( , 2] 4|Page (ix) From the table given, the value of (a) g ( x) f ( x ) g ( x) 1 2 5 1 2 2 3 3 3 4 3 4 1 5 6 1 4 1 2 0 2b 2 If the matrix 3 1 3 is a symmetric matrix, then the value of a and b are: 3a 3 1 2 3 (a) a = , b = 3 2 3 2 (b) a = , b = 2 3 3 2 (c) a = , b = 2 3 2 3 (d) a = , b = 3 2 5|Page f ( x) 1 2 (d) (x) x 1 4 (b) (c) d g ( x) at x = 1 is dx f ( x) (xi) Refer to the figures for the question to follow: I. II. III. IV. While discussing in the group, four students claimed as follows: Harsh: As the function in Figure I is both continuous and differentiable, we can say that every continuous function is differentiable . Anugya: As the graph is not in one piece, the function on fig II is discontinuous . Agastya: The function in fig III is continuous . Abhimanyu: The function in fig IV is discontinuous . Check whether Agastya and Abhimanyu s claims are correct. Justify your answer. (xii) 4 Find the value of sin cot 1 cot 3 (xiii) . A die is rolled. If the outcomes is an even number, what is the probability that it is a prime number? sin 2 x cos x dx (xiv) Evaluate: (xv) The surface area of a spherical balloon is increasing at the rate of 2 cm2/sec. When the radius is 6 cm, find the rate of change of its radius. Question 2 (i) 2 x 3; 3 x 2 If f ( x) = x + 1; 2 x 0 [2] Check the differentiability of f ( x) at x = 2 OR (ii) 6|Page 3x 8; if x 5 If the function f ( x) = is continuous at x = 5 , then find the value of ' k ' . if x 5 2k ; Question 3 [2] 3 5 Let f : R R be defined by f ( x) = 3x + 2 is invertible. Find f 1 ( x ) 5x 3 Question 4 [2] Without expanding at any stage, find the value of: 10 a b + c 10 b c+a 10 c a+b Question 5 (i) If [2] x 1 3 1 .5 dx = k.5 + C , then find the value of k (show all working). x2 x2 OR 9 (ii) Evaluate: x 7 dx 2 Question 6 [2] Given two events A and B such that P ( A B ) = 0.25 and P( A B) = 0.12 . Find the value of P( A B ) . Question 7 1 Given tan 1 x cot 1 x = tan 1 , solve for x , for x 0 . 3 7|Page [4] Question 8 [4] The figure below shows the curve y = 2 x 3 + 3 x 2 11x 6 (a) Find dy dx [1] The curve crosses the x axis at the points P, Q and R (2, 0). The tangent to the curve at R is a straight line L1 . (b) Find the equation of L1 . [1] (c) Find the x coordinate of the minimum point of the curve. [2] OR ( ) 2 1 If y = sec x , show that x 2 ( x 2 1) Question 9 Evaluate: 8|Page d2y dy + (2 x 3 x ) =2 2 dx dx [4] x 2 x+7 dx + 4x + 7 Question 10 (i) In a carpet factory, machines A, B, and C manufacture 60%, 25% and 15% of the total output, respectively. Of the total output, 1%, 2% and 1% are defective carpets, respectively for machines A, B and C. A carpet is drawn at random from the total production and is found to be defective. From which machine is the defective carpet most likely to have been manufactured? Show all working. OR (ii) A contractor has taken a contract for a construction job. The probabilities are 0.35 if there will be no strike, 0.80 if the construction job will be completed on time if there is no strike and 0.32 that the construction job will be completed on time if there is a strike. (a) What is the probability that there is a strike? (b) What is the probability that the construction job will be completed on time? (c) What is the probability that the construction job will not be completed on time? (d) If the construction work was completed on time, then what is the probability that there was a strike? Question 11 [6] In a thrilling T20 cricket match, the Chennai Super Kings needed precisely six runs to win off the last ball of the final over. The batsman on the crease hit the last ball high into the air. The path of the ball follows a Parabolic trajectory in a vertical plane, which the equation can model: y = ax 2 + bx + c Where x is the horizontal distance (in units), and y is the height (in units) of the ball from the ground. During its flight, the ball was observed to pass through the following three points (10, 8), (20, 16) and (30, 18). The boundary rope is located at a horizontal distance of 50 units from the batsman. Can you conclude that Chennai Super Kings Won the match? (i.e Did the ball clear the 50 units boundary?) Solve this using the matrix method 9|Page Question 12 (i) (a) Find [6] d log(2 x 1) dx (b) Show that the solution to the homogeneous differential equation x condition y = 4 at x = 1 is y = dy + y = xy with the dx ( x + 3) 2 4x OR log(1 + x) dx = (log 2) 2 (1 + x ) 8 0 1 (ii) Prove that: Question 13 (i) [6] A person has manufactured a water tank in the shape of a closed right circular cylinder. The volume of the cylinder is 539 cubic units. If the height and radius of the cylinder be h and r 2 respectively (a) Express h in terms of radius r and given volume. (b) Let the total surface area of the closed cylinder tank be S, Express S in terms of radius r . (c) If the total surface area of the tank is minimum, then prove that the radius r is equal to units. (d) Find the height of the tank. 10 | P a g e 7 2 OR (ii) The temperature of a person in Fahrenheit at x days during an intestinal illness is given by f ( x) = 0.1x 2 + mx + 98.6 , 0 x 12 ( m being a constant). (a) If the function differentiable in the interval (0, 12)? Justify your answer. (b) If 6 is the critical point of the function, then find the value of the constant m . (c) 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]. (d) Also, find the corresponding local maximum /local minimum and the absolute maximum /absolute minimum values of the function. Question 14 [6] In a school, three subject teachers, English, Maths and Account, sometimes give surprise tests on the same day. Based on the past records The English teacher gives a test 90% of the time. The Math teacher gives a test 80% of the time. The Account teacher gives a test 70% of the time. Each teacher decides independently. If the average number of surprise tests is less than 2.3, then the teachers should coordinate better to increase the performance of the students. Otherwise, no action is needed. Let X be the number of surprise tests a student gets on a given day. So X {0,1, 2,3} . (a) Find the probability for each possible number of surprise tests. (b) Use the probabilities to build a distribution table. (c) Calculate the average number of surprise tests per day. (d) Based on your calculation, decide: should the teacher coordinate better? or is the current plan acceptable? 11 | P a g e (v) Find the equation of the plane perpendicular to the z axis and passing through the point (1,3, 2) . 12 | P a g e SECTION 15 MARKS Question 19 [5] In subpart (i) and (ii), choose the correct option, and in subpart (iii) to (v), answer the questions as instructed. (i) The average cost function associated with producing and marketing x units of an item is given by AC = 3x 11 + (a) 2 (b) 5 (c) 3 (d) 1 13 | P a g e 10 . Then the marginal cost (MC) at x = 2 is x (ii) Given byx = 0.95, x = 5 and y = 12. Choose the correct option: I. x = 0.95 y + 6.0 is the regression line x on y II. x = 0.95 y 6.0 is the regression line x on y III regression lines passing through ( x , y ) IV regression lines do not pass through ( x , y ) Which of the following is correct? (iii) (a) Only I and IV are correct (b) Only III is correct (c) Only I and III are correct (d) Only II and III are correct The total cost function for a production and marketing activity is given by C( x ) = 3 2 x 7 x + 27. Find the level of output (number of units produced) for which 4 MC = AC . (iv) The revenue function of a monopolist is given by R( x ) = 120 x 2 + 300 x . Find the average revenue at x = 10 . (v) If two lines of regression are 3x 2 y = 5 and x 4 y = 7 , then find the value of the correlation coefficient between x and y . Question 20 (i) [2] A real estate company is going to build a new residential complex. The land they have purchased can hold at most 500 apartments. Also, if they make x apartments, then the monthly maintenance cost for the whole complex would be as follows: Fixed cost = 4000 Variable cost = (14 x 0.04 x 2 ) How many apartments should the complex have to minimise the maintenance cost? 14 | P a g e OR (ii) The demand function of a monopoly is given by x = 100 4 p. Find the quantity at which the marginal revenue will be zero. Question 21 (i) [2] The coefficient of correlation between the values denoted by x and y is 0.5. The mean of x is 3 and that of y is 5. Their standard deviation are 5 and 4, respectively. Find (a) the two lines of regression (b) the expected value of y when x = 14 (c) the expected value of x when y = 9 OR (ii) Consider the following data x 1 2 3 6 y 6 5 4 1 Using the least squares method, find the regression line y on x and hence find the expected value of y when x = 4 . Question 22 [4] A firm deals with two kinds of fruit juices pineapple and orange juice. These are mixed, and two mixtures are sold as soft drinks A and B. One tin of A requires 4 litres of pineapple and 1 litre of orange juice. One tin of B requires 2 litres of pineapple and 3 litres of orange juice. The firm has only 46 litres of apple juice and 24 litres of orange juice. Each tin of A and B is sold at a profit of 4 and 3, respectively. How many tins of each type should the firm produce to maximise the profit? Make it an LPP and solve it graphically. ___________________________________________________________________________________ 15 | P a g e

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