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ISC Class XII Prelims 2025 : Mathematics (The Doon School, Dehradun)

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Akash Yanglem
The Doon School, Dehradun

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MATHEMATICS (SC FORM ISC)-PAPER 1 Maximum Marks: 80 Time Allotted: Three Hours Reading Time: Additional Fifteen minutes Instructions to Candidates You are allowed an additional 15 minutes for only reading the paper. You must NOT start writing during reading time. The question paper has 12 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. Instruction to Supervising Examiner Kindly read aloud the instructions given above to all the candidates present inthe examination hall. Page 1 of 16 SECTION A - 65 MARKS Question 1 In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed. (i) The graph of y = f ( x) is given below: I. f has a jump discontinuity at x = 4. II. f has a removable discontinuity at x = 0. III. f has a jump discontinuity at x = 0. IV. f has an infinite discontinuity at x = 4. Based on the above graph, which of the above is correct? a. I and II only b. I and IV only c. II and III only d. III and IV only. (ii) If A is a symmetric matrix, B is a skew-symmetric matrix, A + B is non-singular and C= ( A + B) 1 ( A B ) .If C T ( A + B ) C =k ( A + B) , then the value of k is [1] a. b. c. d. (iii) [1] 0 1 2 3 The probability that Aseem is late for school tomorrow is 0.05. The probability that Anand is late for school tomorrow is 0.15. Purnima says that the probability that Aseem and Anand will both be late for school 0.0075 . tomorrow is 0.0075 because 0.05 0.15 = What assumption has Purnima made? [1] Page 2 of 16 a. b. c. d. (iv) The events are dependent The events are independent The events are mutually exclusive The events are exhaustive The figure shows a sketch of a curve C with parametric equations x = 1 8cos2t , y = 9sint 0 t All points on C satisfy = y function f= ( x) 2 9 x + 7 . The curve C has equation y = f ( x) where f is the 4 9 x + 7 a x b . The value of a and b are 4 [1] 9 a. a= 7, b = b. a=7, b = 9 9 c. a= 7, b = d. a=7, b = 9 (v) The derivative of the function a. b. [1] 1 1 x2 1 1 x2 Page 3 of 16 1 c. d. (vi) 1 + x2 1 1 + x2 a Let f ( x ) =2 x + x , x . The value of a f ( x ) dx where a > 0 is a. [1] a2 2 b. 2a 2 c. a 2 d. a 2 dy y tan x = sec x is a linear differential equation. dx dy Reason: It is of the type + Py = Q where P & Q are the function of x only which is dx the standard form of linear differential equation [1] a. Both Assertion and Reason are true, and Reason is the correct explanation of Assertion. b. Both are true but Reason is not the correct explanation of Assertion. c. Assertion is true, Reason is false. d. Assertion is false, Reason is true. (vii) Assertion: (viii) Statement I: The maximum number of equivalence relations on the set A = {1, 2,3, 4} is 15. Statement II: Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b a, b T . Then R is an equivalence relation. [1] a. Both Statement I and Statement II are true. b. Both Statement I and Statement II are false. c. Statement I is true, Statement II is false. d. Statement I is false, Statement II is true. (ix) The value of tan 1 3 sec 1 ( 2 ) is a. b. 2 3 2 3 c. d. [1] 3 3 Page 4 of 16 (x) Urn X contains 4 red and 2 green balls and urn Y contains 5 red and 1 green ball. An urn is chosen at random and 2 balls are drawn from it without replacement. If A is the event of two red balls being drawn and B is the event of at least one red ball being drawn from urn X , the value of P ( A B ) is [1] 1 7 2 b. 7 3 c. 7 4 d. 7 a. (xi) (xii) (xiii) (xiv) a 5 [1] Show that the matrix A = is non-singular for all values of a . 2 a + 4 x 3 Explain why the function y = is one-one for all values of x , except x = 2. [1] x 2 A bag contains slips of paper with letters written on them as follows: A, A, B, B, B, C , C , D, D, D, D, E. If you draw 3 slips, what is the probability that the letters will spell out (in order) the word BAD ? [1] An organization conducted bike race under two different categories - Boys and girls. There were 28 participants in all. 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. [1] Let B = {b1 , b2 , b3 } and G = { g1 , g 2 } , where B represents the set of Boys selected and G the set of Girls selected for the final race. How many relations are possible from B to G ? (xv) State true or false: Adj ( kA ) = k n ( AdjA ) . If false, write the correct statement. [1] Question 2 (i) Let f ( x) = cos x , find f 19 ( x) , where f 19 ( x) is the 19th derivative of f ( x) . (ii) OR It is given that f ( x ) = (4 x 1) + x , for x . Show that f is an increasing function.[2] [2] 3 Question 3 Find x2 + 4 x + 2 x + 1 dx [2] Page 5 of 16 Question 4 ( x) Let f = 4 x + 5 for x 1.25 . Consider another function g . Let R be a point on the graph of g . y 3x + 6 . The x -coordinate of R is 1 . The equation of the tangent to the graph at R is = Let h= ( x ) f ( x ) g ( x ) . Find the equation of the tangent to the graph of h at the point where x = 1 . [2] Question 5 (i) Use the substitution u = 1 + lnx to find lnx x(1 + lnx) 2 [2] dx OR (ii) x Evaluate ln dx 2 [2] Question 6 If 0 x 1 , show that arcsin ( 2 x 1) 2arcsin x = [2] 2 Question 7 1 Prove that: cot tan 1 x + tan 1 + cos 1 1 2 x 2 + cos 1 2 x 2 1 = , x > 0 x ( Question 8 Evaluate: cos d d sin 7sin + 10 2 ) ( ) [4] [4] Question 9 (i) The diagram shows a water trough. Water is being poured into this trough at 2.4 cubic metres per minute. a. Find an expression for the volume of water in the trough in terms of its depth. b. Find the rate at which the water level is rising when the depth is 0.5 metres. c. Find the rate at which the exposed surface area of the water is increasing after 1 minute. [4] Page 6 of 16 OR (ii) Given y = x sin 1 x show that d2y 2 x2 = dx 2 (1 x 2 ) 1 x 2 [4] Question 10 (i) Lisa enters a chess tournament in which the result of every match is either win, lose or draw. The probability that she wins the tournament if she wins her first match is 60%. The probability that she wins the tournament if she draws her first match is 50%. The probability that she wins the tournament if she loses her first match is 10%. There is a 50% chance that she wins her first match and a 30% chance that she draws her first match. Given that she wins the tournament, find the probability that she drew her first match. [4] OR (ii) A box contains 3 decks of cards. The first deck is a standard deck (52 cards, no jokers), the second deck is a defective deck where 10 of the 52 cards are missing (randomly chosen), and the third deck is a special deck where all 4 kings are removed. A deck is selected at random, and then a card is drawn at random from the selected deck. What is the probability that a king is drawn from the selected deck? [4] Question 11 In a small town, three friends Hrithik, Manjot and Ankur decide to compare the number of hours they spend on three activities: working, studying, and relaxing in a day. After tracking their time, they notice the following patterns: Hrithik spends the same amount of time working as Bob spends studying and relaxing combined. Manjot spends twice the amount of time working as Alice does. Ankur spends twice the amount of time studying as Bob spends relaxing, and they differ by 1 hour. Answer the following question: a. b. c. Translate the problem into a system of equations. Solve the system of equation by using matrix method. How many hours does each friend spend working, studying, and relaxing?[6] Question 12 (i) Solve the following differential equation dy ( x + 1)( x + 2 ) + y = x + 1 dx giving your answer in the form y = f ( x ) . [6] Page 7 of 16 OR (ii) Evaluate x 1 + sin sinx dx [6] Question 13 (i) Two shops, A and B linked by a straight road, R1 , are 4 kilometres apart. A second straight road, R 2 , bisects R1 at right angles. One kilometre along R 2 from where it bisects R1 a third shop, C , can be found. A bus stop is to be placed on R 2 , somewhere between shop C and R1 . Where should the bus stop be placed so that the sum of the direct distances from the shops to the bus stop is a minimum? [6] OR (ii) A roof gutter is to be made from a long flat sheet of tin 21 cm wide by turning up sides of 7 cm so that it has a trapezoidal cross-section as shown in the diagram. Find the value of that will maximize the carrying capacity of the gutter. [6] Question 14 Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered 0,1, 2 , and 3 respectively. The dice are rolled and the numbers face down on the two dice are recorded. The random variable R is the score on the red die and the random variable B is the score on the blue die. a. Find P( R = 3 and B = 0) . The random variable T is R multiplied by B . b. Complete the diagram below to represent the sample space that shows all the possible values of T . c. The table below represents the probability distribution of the random variable T . Page 8 of 16 t P (T = t ) 0 1 2 3 4 6 9 a b 1/ 8 1/ 8 c 1/ 8 d Find the values of a, b, c and d . d. Find the values of E (T ) [6] SECTION B - 15 MARKS Question 15 In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed. (i) 3 1 Given vectors a = 2 and b = 4 , the value of scalar k such that a + kb is parallel to 2 7 0 [1] vector 10 is 23 a. 1 2 1 3 1 c. 4 1 d. 5 b. (ii) 3y +1 2 z A vector equation of the line with Cartesian equation x = 2, = 4 5 2 0 1 + 4 a. r = 3 3 2 5 [1] Page 9 of 16 2 0 1 + 4 b. r = 3 3 2 5 2 0 1 + 4 c. r = 3 3 2 5 2 0 1 + 4 d. r = 3 3 2 5 (iii) (iv) (v) x +1 y +1 2 z 12 . A line given by = = is parallel to the plane 6 x + 4 y + Gz = 3 2 2 The value of G is [1] a. 10 b. 11 c. 12 d. 13 Explain why a b a = 0. ( [1] Find the equation of the plane passing through the point ( 2,1, 3) with the normal vector n = [3, 2,1] . [1] Question 16 (i) ) Show that the lines r1 = 2i + j + t i + 2 j + 4k and r2 = i + 3 j + 2k + s 2i 5 j + 3k do ( ) ( not intersect. (ii) OR A straight line L , has vector equation ) [2] 5 5 r= 0 + sin , , R 0 cos x p, p R . The plane, p , has equation= Show that the angle between L and p is independent of both and p . [2] Page 10 of 16 Question 17 (i) Three towns are joined by straight roads. Dehradun is the state capital and is considered as the 'origin'. Clock tower is 3 km east and 9 km north of Dehradun and Rajpur is 5 km east and 5 km south of Clock tower. Considering i as a 1 km vector pointing east and j a 1 km vector pointing north: a. Find the position vector of Clock tower relative to Dehradun. b. Find the position vector of Rajpur relative to Dehradun. A bus stop ( S) is situated two thirds of the way along the road from Dehradun to Clock Tower. c. Find the vectors OS and BS . d. Prove that the bus stop is the closest point to Rajpur on the Dehradun to Clock Tower Road. [4] (ii) OR In the trapezium shown, BE : BC = 1: 3 , Show that 3 AC DE= 2 ( 4m 2 n 2 ) where = AB m= , DC 2 AB and DA = n [4] Question 18 The finite region R , shown shaded in Figure below, is bounded by the curve with equation= y 4 x 2 + 3 , the y -axis and the line with equation y = 23 . Show that the exact area of R is k 5 where k is a rational constant to be found. [4] Page 11 of 16 SECTION C - 15 MARKS Question 19 In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed. (i) In the figure below, the blue straight line is the cost function, and the red straight line is the revenue function, then the profit function is [1] 3 x+2 2 3 b. = y x 2 2 3 c. y = x+2 2 3 d. y = x 2 2 a. = y (ii) From the given feasible region in a Linear programming problem, the correct equation of a constraint is [1] 200 150 100 50 50 100 150 200 a. x 150 b. y 120 (iii) c. y x + 200 d. y x + 200 Given the following scatter plot representing the relationship between the variables X and Y , determine the closest value of the correlation coefficient r . [1] Page 12 of 16 11 10 9 Variable y 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 Variable x a. b. c. d. r =0 r = 0.5 r = 0.8 r = 0.9 (iv) A store sells 3000 celebration-themed wigs at $20 each every month. For each $1 increase in price, they would sell 100 less wigs. What is the price that would maximize the store s revenue? [1] (v) A cost function is defined by C ( x )= 5 + f ( x ) where f ( x ) is a linear function. Find the marginal cost function. [1] Question 20 (i) The total cost (in dollars) to produce x units of a good is given by the function: C= ( x ) 5.3x + 40, 000 a. What is the total cost to produce 6900? b. How many units can be produced with a total of $77, 630 ? (ii) [2] OR The cost per item, in euros, to produce x items is C ( x ) = x3 3 x 2 9 x + 30 Find the number of items that must be produced to minimize the cost per item. [2] Question 21 = bxy 0.85, = byx 0.89 and the standard deviation of X = 6 , find the value of ' r ' (i) a. Given and y . b. The following data relate to the marks in Company Law and Statistics in a certain year: Mean Marks in Statistics: 39.5 Mean Marks in Company Law: 47.6 Standard Deviation of Marks in Statistics: 1 0.8 Standard Deviation of Marks in Company Law: 16.9 Page 13 of 16 ' r ' between marks in Statistics and Company Law: 0.42 Determine the two equations of regression. Also calculate expected marks in Company Law of a candidate who obtained 50 marks in Statistics. (ii) [4] OR A company is analyzing the relationship between two variables: the number of hours an employee spends on training (denoted by X ) and their monthly performance score (denoted by Y ) in a performance evaluation. After conducting a study, the company's analysts determined that the relationship between X and Y is governed by the following two regression equations: 3 X 2Y = 5 (Regression of X on Y ) X 4Y = 7 (Regression of Y on X ) Calculate the following: a. The regression coefficients of both the regressions. b. The coefficient of correlation between the hours of training and performance scores. [4] Question 22 Fly High Airlines sells business class and tourist class seats for its charter flights. To charter a plane, at least 5 business class tickets must be sold and at least 9 tourist class tickets must be sold. The plane does not hold more than 30 passengers. Fly-High makes $40 profit for each business class ticket sold and $45 profit for each tourist class ticket sold. In order for Fly-High Airlines to maximize profits, how many tourist class seats should they sell?[4] ************************************************************************************** Page 14 of 16 DISTRIBUTION OF MARKS FOR THIS THEORY PAPER S.No. UNIT QUESTION NO WEIGHTAGE SECTION A (65 MARKS) 1 Relations and Functions Q 1(iv), Q 1(viii), Q 1(ix), Q 1(xii), Q 1(xiv), Q6, Q7. All Question 1 is 1 mark, Q6 is 2 marks and Q7 is 4 marks. 10 (This paper contains 11 marks). 2 Algebra Q 1(ii), Q 1(xi), Q 1(xv), Q11.All Question 1 is 1 mark and Q11 is 6 marks. 10 (This paper contains 9 marks). 3 Calculus Q 1(i), Q 1(v), Q 1(vi), Q 1(vii), Q2, Q3, Q4, Q5, Q8, Q9, Q12, Q13. All Question 1 is 1 mark, Q2-Q5 is 2 marks, Q8-9 is 4 marks and Q12-13 is 6 marks. 32 4 Probability Q 1(iii), Q 1(x), Q 1(xiii), Q10, Q14. All Question 1 is 1 mark, Q10 is 4 marks and Q14 is 6 marks. 13 SECTION B (15 MARKS) 5 Vectors Q15 (i), Q15 (iv), Q17. All Question 15 is 1 mark and Q17 is 4 marks. 5(This paper contains 6 marks). 6 Three - Dimensional Geometry Q15 (ii), Q15 (iii), Q15 (v), Q16. All Question 15 is 1 mark and Q16 is 2 marks. 6(This paper contains 5 marks). 7 Applications of Integrals Q18. Q18 is 4 marks. 4 Page 15 of 16 SECTION C (15 MARKS) 8 Application of Calculus Q19 (i), Q19 (iv), Q19(v), Q20. All Question 19 is 1 mark and Q20 is 2 marks. 5 9 Linear Regression Q19 (iii), Q16. All Question 19 is 1 mark and Q21 is 4 marks. 6(This paper contains 5 marks). 10 Linear Programming Q19 (ii), Q22. All Question 19 is 1 mark and Q22 is 4 marks. 4(This paper contains 5 marks). TOTAL 80 Page 16 of 16

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