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

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PRE-BOARD - I NOVEMBER 2024 Mathematics (SC ISC) Duration: 3-hours + 15 Minutes for reading DAY & DATE: Saturday 16.11. 2024 (1st Session) INITIALS OF TEACHER(S): ANC, RLR M.M: 80 Answers to this Paper must be written on the paper provided separately. (Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time.). This Question Paper consists of three sections A, B and C. Candidates are required to attempt all questions from Section A and all questions EITHER from 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 questions are given in brackets [ ]. Mathematical tables and graph papers are provided. SECTION A - 65 MARKS Question 1 [15] In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed 1 2 2 (i) A matrix A = and A xA = I 2 then the value of x is: 2 3 (ii) (a) -4 (b) -2 (c) 4 (d) 2 The value of ( x + 1)e x dx is (a) ex + c (b) xe x + c . (c) e2x + c (d) x+c (iii) The domain of the function cos 1 1 x is (a) (b) (c) (d) [0, 1] [-1, 1] [ ,1] [1, ] (iv) Assertion: The order and degree of the differential equation: 5 3 2 d3y dy 2 d y + 3 xy + y 3 2 = 0 is 2, 3 dx dx dx Reason: Order of a differential equation is the order of the highest order derivative present in the differential equation and the degree of the differential equation is the highest exponent of the highest order derivative present in the differential equation if each term involving derivatives of a differential equation is a polynomial (or can be expressed as polynomial) Which of the following is correct? (a) (b) (c) (d) Both Assertion and Reason are true, and Reason is the correct explanation for Assertion. Both Assertion and Reason are true, but Reason is not the correct explanation for Assertion. Assertion is true and Reason is false. Assertion is false and Reason is true. (v) X speaks truth in 60% cases and Y speaks truth in 70% cases. The probability that they agree with each other is: (a) 42 . 100 (b) 12 . 100 (c) 50 . 100 (d) 54 . 100 l m n 10l (vi) If p q r = a , then what is the value of 5 p 5x x y z (a) a 10 (b) 2a (c) 10a (d) 5a 2m 2n q r is: y z (vii) Consider the graph y = x . Statement 1: The above graph is continuous at x = 0 Statement 2: The above graph is differentiable at x = 0 Which of the following is correct? (a) (b) (c) (d) Both the statements are true. Both the statements are false. Statement 1 is true, and Statement 2 is false. Statement 2 is true, and Statement 1 is false. (viii) (ix) If y = 3sin x 4sin 3 x ,then (a) 0 (b) -3 (c) 3 (d) 1 dy at x = is dx 3 Let R be a relation on set A = {a, b.c} given by R = {(a, a),(b, b),(c, c),(a, b),(b, a),(a, c)} Statement1: Ris an equivalence relation on set A. Statement 2: R is not a symmetric relation on A. Which one of the following is correct? (a) Both the statements are true and statement 2 is correct explanation of Statement 1. (b) Both the statements are true and statement 2 is not correct explanation of Statement 1. (c) Statement 1 is true, and Statement 2 is false. (d) Statement 1 is false and statement 2 is true. (x) In a third order matrix aij denotes the element of the ith row and the jth column. Matrix A 3 for i j has element aij = 0 for i = j 3 for i j Assertion: Matrix A is not invertible. Reason: Determinant A = 0 Which of the following is correct? (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. A B (xi) Two events A and B are such that P = 0.4 , P = 0.25 , P( A B) = 0.12 . The B A value of P( B) is: (a) 0 12 (b) 0 30 (c) 0 36 (d) 0 48 (xii) The value of the determinant of a matrix A of order 3 is 4. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of C 2 ? (xiii) If a relation R on the set of natural numbers be defined as R = {( x, y) : x, y N , x + y = 10} then classify the relation. (xiv) The given function : is not onto function. Give reason. (xv) Out of two friends A and B the probability that A tells the truth is 90% while B tells the truth is 80%. Find the probability that both agree. Question 2 [2] dy ex , find . dx 1 + sin x (i) If y = (ii) OR Find the interval in which the following function is strictly increasing or strictly decreasing: f ( x) = 2 x2 + 10 6 x . Question 3 k If x 2 0 [2] 1 dx = , find k. +4 8 Question 4 [2] Find the equation to the tangent to the curve y 2 = 4ax at point ( at 2 , 2at ). Question 5 (i) Evaluate [2] 1 dx . 9 + 16 x 2 OR (ii) 2 1 Evaluate e x 3 + 2 dx . x x Question 6 [2] Find the value of cos 1[sin(cos 1 x)] = 3 . . Question 7 Solve: [4] sin 1 (6 x) + sin 1 (6 3x) = 2 Question 8 . [4] 1 x Evaluate tan 1 dx . 1+ x Question 9 (i) (ii) [4] A man 1.6m in height walks at the rate of 30 meters per minute away from a lamp which is 4 meters above the ground. How fast is the man s shadow lengthening? OR d2y dy If log y = tan 1 x prove that (1 + x 2 ) 2 + (2 x 1) = 0 . dx dx Question 10 (i) [4] Three friends go to the Founder s Day fete. They planned that one of them will pay for the burger. They decided that each of them will pick up one card one after the other from a deck of 52 cards and then replace after their chance. The person getting a card of a colour different from the other two friends will pay for the burger. If all three pick card of same colour, then they will a card again until they get a different result. (a) What is the probability that all three friends get the same result (Card of same colour) in one round? (b) What is the probability that they will get a different result in one round of picking cards? (c) What is the probability that they will need exactly 6 rounds of picking cards to determine who will pay? OR (ii) There are three boxes containing coloured balls. Box I contain 2 white and 3 black balls. Box II contains 4 white and 1 black ball and box III contains 3 white and 4 black balls. A dice having three red, two yellow and one green face is thrown to select the box. If red face turns up, we pick up box I, if yellow face turns up, we pick up box II, otherwise we pick up box III. Then we draw a ball from the selected box. If the ball drawn is white, what is the probability that the dice has turned up with a red face? Question 11 [6] For a social service project in the local community, three schools contributed study material for free distribution at a slum school. School P purchased 30 pens, 20 calculators and 10 notebooks for 8200. School Q purchased 20 pens, 10 calculators and 20 notebooks for 5800. School R purchased 20 pens, 20 calculators and 20 notebooks for 8800. Answer the following questions. (a) Translate the problem into system of equations. (b) Solve the system of equations by using matrix method. (c) Hence find the cost of 1 pen, 1 calculator and 1 notebook. Question 12 [6] (i) Solve the differential equation ( x2 + y 2 )dy = xydx , if y (1) = 1 and y( x0 ) = e . Find the value of x0 . OR 2 (ii) x sin x cos x dx . sin 4 x + cos 4 x 0 Evaluate Question 13 (i) [6] A cone of given volume has radius r, altitude h and slant height l. (a) Find the volume and curved surface area of the cone. (b) If the curved surface area is to be minimum, then find the ratio of its altitude and radius OR (ii) A cylinder of height h and radius R is inscribed in a sphere of radius 3 3 cm. (a) Find the volume of the cylinder. (b) Find in terms of the maximum volume of such a cylinder. Question 14 [6] Jason had a long drive back home and he was hungry. He picked up a basket of 30 fruits from a roadside vendor. He wanted to eat 2 fruits though he knew that 10 were rotten in that lot. (i) If X denotes the random variable Unspoiled fruits then what are the possible values X can take? (ii) Find the probability distribution of this random variable X. (iii) Find the mean of the probability distribution. SECTION B-15 MARKS Question 15 [5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer questions as instructed (i) Consider the following statements and consider the correct option: Statement I: If a and b are unit vectors such that a + b = 3 , then the angle between a and b is . 3 a b Statement II: Angle between two vectors a and b is given by sin = . a b Which of the following is correct: (a) Both the statements are true, and statement II is correct explanation of Statement I. (b) Both the statements are true, and statement II is not correct the explanation of Statement I. (c) Statement I is true, and Statement II is false. (d) Statement I is false, and statement II is true. (ii) If OACB is a parallelogram with OC = p and AB = q , then OA is equal to (a) p + q . (b) p q . 1 ( p q) . (c) 2 1 (q p) . (d) 2 x +1 y 2 z + 2 = = (iii) The length of perpendicular from the point P (1, 1, 2 ) on the line is: 2 3 4 (a) 6 units. 21 units. (c) 29 units. (d) 0 units. (b) (iv) Find the direction ratios of the line perpendicular to the lines x 3 y +7 z 2 = = and 2 3 1 x + 2 y +3 z 5 = = . 1 2 2 (v) Find the area of a parallelogram whose adjacent sides are given by vectors a = i j + 3k and b = 2i 7 j + k . Question 16 [2] (i) If a = i + 2 j + k , b = 2i + j and c = 3i 4 j 5k , then find a unit vector perpendicular to both of the vectors (a b ) and (c b ) . OR (ii) Find the angle between the vectors a + b and a b if a = 2i j + 3k and b = 3i + j 2k . Question 17 [4] (i) Find the value of for which the four points with position vectors are 2i + 5 j + k , j 4k , 3i + j + 8k and 4i + 3 j + 4k are coplanar. OR 3 (ii) Find the length of perpendicular from the point 1, , 2 to the plane 2 2x 2 y + 4z + 5 = 0 . Question 18 [4] Two curves are given by equation y = 6 x x 2 and y = x 2 2 x . (i) Find the point of intersection of the curves. (ii) Find the area of the region bounded by the curves. SECTION C-15 MARKS Question 19 [5] In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer questions as instructed. (i) Which relation is correct below the breakeven point in relation to revenue function R(x) and cost function C(x). (a) R( x) = C ( x) . (b) R( x) C ( x) . (c) R( x) C ( x) . (d) R( x) C ( x) = 0 . (ii) Read the following statements and choose the correct option: I. Overlapping lines imply no correlation. II. Greater angle between the lines implies greater correlation between two series. III. If lines are perpendicular to each other there is no correlation. IV. Coinciding lines implies perfect correlation. Which of the following is correct (a) Only I and IV are correct. (b) Only II and IV are correct. (c) Only III and IV a correct. (d) Only II and III are correct. (iii) The coefficient of regression of y on x is 1.6 and coefficient of regression of x on y is 0.4. If the mean of x is 23 and mean of y is 17, find the coefficient of correlation (r). x3 (iv) The total cost function for x units of a commodity is C ( x) = + 3x 2 7 x + 16 . Find 3 the marginal cost when x = 2. (v) A company manufactures certain goods and finds that the daily cost of producing x items is given by C ( x) = 1000 + 30 x . Find the minimum number that must be produced and sold daily if each item is sold for 40. Question 20 [2] (i) The cost function of producing and marketing x units of an item is given by C ( x) = 2 x 2 11x + 50 . Find the range of values of the output x for which the average cost is increasing. OR p (ii) A monopolist s demand function is x = 60 . At what level of output will the 5 marginal revenue be zero? Question 21 [4] (i) For five pair of observations for correlated variables x and y following results are obtained: x = 15, y = 25, x 2 = 55, xy = 83 . Estimate the value of y when the value of x is 6. OR (ii) The random variables have regression line 4 x + 3 y + 7 = 0 and 3x + 4 y + 8 = 0 . Calculate: (a) The coefficient of correlation. (b) Mean value of x and y. Question 22 A linear programming problem is given Z = 60 x + 15 y subject to the constraints: x + y 50, 3x + y 90, x 0, y 0 . (i) Solve graphically to find the corner points of the feasible region. (ii) Maximize Z = 60 x + 15 y . [4]

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