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ISC Class XII Prelims 2026 : Mathematics (Convent of Jesus and Mary School (CJM), Ranaghat, Nadia)

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Set B Pre board 1 (December 2025) Subject Mathematics Class XII 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 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 x 15 = 15) 4 2 1 3 i. If A = [ ]3 x 4 is matrix given by A = [ 5 7 9 6 ] then a23 + a24 will be 21 15 18 25 equal to the element: a. a14 b. a44 c. a13 d. a32 ii. An integrating factor of the differential equation x + y log x = x (log ) 2 iii. 1 2 log 2 , (x > 0) is: a. log b. ( )log c. ( ) d. A number is selected at random from the set {1, 2, 3, .., 20}. What is the probability that the number is divisible by 3, given that it is even? 1 3 7 a. 10 b. 10 c. 10 d. None of these iv. v. vi. vii. 1 The graph of the function f(x) = [ ], where [x] is greatest integer function is shown below. Then, a. f(x) is continuous at x = 1 b. f(x) is differentiable at x = 2 c. f(x) is differentiable at x = 3 d. f(x) is not differentiable for all x I. The events E and F are independent. If P(E) = 0.3 and P (E F) = 0.5, then P(E/F) P(F/E) is equal to: 1 2 3 1 a. 7 b. 7 c. 35 d. 70 U and V are two non-singular matrices of order n and k is any scalar. Assertion (A): det (kU) = kn x det (U) Reason (R): If W is a matrix obtained by multiplying any one row or column of V by the scalar k, then det (W) = k x det (V). 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. 6 2 2 |? If | | = k, then what is the value of |3 3 a. 6 b. 2k c. 3k d. 6k 1 1 Assertion (A): The value of cos 1 (2) + 2sin 1 (2) is viii. a. b. c. d. ix. 2 3 . Reason (R): Principal values branch of cos 1 is [0, ] and sin 1 is [ 2 , 2 ]. 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. Shown below is the graph of a function, f(x). Statement 1: The function f(x) decreases as x moves from 6 to 3. Statement 2: The function f(x) increases as x moves from 0 to 3. 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. 1 1 xiii. For the curve + = 1, at (4 , 4) is: a. 2 b. 0 c. 1 d. 2 Statement 1: The relation R defined in the set A = [1, 2, 3, 4, 5, 6, 7] by R = {(a, b): both a and b are either odd or even}. Then, R is an equivalence relation. Statement 2: A relation R on a set A is said to be an equivalence relation if it is simultaneously reflexive, symmetric and transitive. 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. Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? If g is described by g(x) = x + , then what value should be assigned to and ? Which one of the following graphs is a function of x? xiv. Show that (A A ) is skew-symmetric, if A = [ xv. Evaluate 23 2 dx x. xi. xii. 2 4 ]. 3 5 Question 2 Evaluate: [2] 8 10 2 + 10 Question 3 [2] , 1 , then prove that f is not differentiable at x = 1. , < 1 OR i. If ( ) = { ii. If ( + 1) = 1, then show that dy dx = Question 4 Find value of k, if 1 [ (2 1 [2] 3 )] 2 = 3 Question 5 [2] i. A stone is dropped into a quiet lake and waves move in circles at a speed of 3.5 cm/s. At the instant when the radius of the circular wave is 9 cm, how fast is the enclosed area increasing? OR 1 Let I be an interval disjoint from ( 1, 1). Prove that the function f(x) = + is strictly increasing on I. Question 6 Solve the differential equation: 3 tan + (1 ) sec 2 = 0 [2] Question 7 Solve the equation for x: 1 + 1 (1 ) = 1 , 0 [4] Question 8 [4] i. Sand is pouring from a pipe at the rate of 12 cm /s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm? OR ii. 2 2 If 1 ( 2 + 2 ) = 1 , = . Question 9 [4] i. The search committee has been given the task to short listed three candidates for the position of a Principal for the Vardhman college. The chances of short listing of three candidates Rahul, Harpreet and Manohar are in the proportion 4: 2: 3 respectively. The probability that Rahul, if selected, will introduce co-education in the Vardhman College is 0.3. The probability of Harpreet and Manohar doing the same are 0.5 and 0.8. a) What is probability of shortlisting Rahul? b) What is the probability that there will be co-education in the Vardhman college? c) What is the probability that Harpreet introduces co-education in the Vardhman College? d) What is the probability that Manohar introduces co-education in the Vardhman College? OR ii. Radhika was planning to conduct a test based on Multiple Choice Questions (MCQs) with four choices and she decided to take the feedback about the efficacy of the same from her colleague. An examinee either guesses or copies or knows the answer to MCQs. 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 coped it, is 1/8. a) What is the probability that the examinee knows the answer? b) What is the probability that he answers correctly on the basis of guess? c) What is the probability that the examinee answers MCQs correctly? d) What is the probability that examinee knew the answers to the MCQs, given that he answered correctly? Question 10 1 Evaluate: [6( )2 +7 +2] [4] Question 11 [6] 3 2 i. Evaluate: 1 | | ii. OR Solve the differential equation for a particular solution: = + , when y = 0 and = 1 Question 12 [6] i. A gardener working in a company wants to fence off a flower bed having area A with a wire of length of 20 m in the form of a sector of circle. a) The area A of the flower bed as a function of radius (x) will be. b) If the gardener wants to have flower bed with greatest possible area, then radius is equal to. c) What is the maximum area of flower bed in the garden? d) What is the ratio between area of the flower bed to that of the circular flower bed having same radius? OR ii. Consider the scenario, let ABCD be a trapezium. The length of three sides are 10 cm each as shown in the figure. Based on the above information, answers the following questions. a) Find the area of trapezium in terms of . b) From the obtained equation in terms of , find the critical points. c) Find the point at which the area of trapezium is maximum. d) Find the maximum area of trapezium. Question 13 [6] Two farmers Ramkishan and Gurucharan Singh cultivates only three varieties of rice namely Basmati, Parmal and Naura. The sales (in ) of these varieties of rice by both the farmers in the month of September and October are given by in matrices from A and B. September sales (in ) Basmati 10000 A=[ 50000 Permal Naura 20000 30000 30000 ] 10000 Ramkishan Gurucharan Singh October sales (in ) Basmati Permal Naura 5000 10000 6000 Ramkishan B=[ ] 20000 10000 10000 Gurucharan Singh On the basis of above information, answer the following questions. (a) Find the combined sales in September and October for each farmer in each variety. (b) Find the decrease in sales from September to October. (c) If both farmers received 2% profit on gross sales, then find the profit for each farmer and for each variety sold in October. Question 14 [6] At a hospital, three different diagnostic tests are sometimes prescribed on the same day: Blood Test, X-Ray and MRI. Past data shows: The Blood Test is prescribed 40% of the time. The X-Ray is prescribed 55% of the time. The MRI is prescribed 30% of the time. Each test is prescribed independently. If the average number of tests per patient is greater than 1.4, doctors are advised to review their testing policy; otherwise, the current practice is considered balanced. Let X be the number of diagnostic tests a patient undergoes on a given day. Clearly, X {0, 1, 2, 3} (a) Find the probability for each possible number of diagnostic tests. (b) Construct the probability distribution table for X. (c) Calculate the expected number of tests per patient. (d) Decide whether the doctors need to review their testing policy. Section B (15 marks) Question 15 (1 x 5 = 5) In subparts (i) to (iii) choose the correct options and in subparts (iv) and (v), answer the questions as instructed. i. If | |= 4 and plane 3 2, then the range of | | is a. [0, 8] b. [ 12, 8] c. [0, 12] d. [8, 12] +1 2 1 +3 4 ii. Statement I: The angle between the lines 5 = 2 = 2 2 = 3 , = 5 iii. iv. v. is 2 Statement II: The given two lines are not parallel. Choose the correct statement. a. Both statements are true and statement II is the correct explanation of statement I. b. Both statements are true and statement II is not the correct explanation of statement I. c. Statement I is true and statement II is false. d. Statement II is true and statement I is false. The vector equation of the plane passing through the points (a, b, c) and parallel to the plane . ( + + )= 2 is a. . ( + + )= a + b + c b. . ( + )= a + b + c c. . ( )= a + b + c d. . ( + + )= a + c Find the distance of a point (3, 4, 8) from a plane . (6 + 3 2 )= 4. Let , and be three vectors such that | |= 1, | |= 2 and | |= 3. If the projection of along is equal to the projection of along ; and , are perpendicular to each other, then find |3 2 +3 | Question 16 [2] i. Let = + + and = , then find a vector such that = and . = 3. OR ii. Find a vector of magnitude 9, which is perpendicular to both the vectors 4 + 3 and 2 + 2 . Question 17 [4] Find the area bounded by y = x , the x-axis and the lines x = 1, x = 1. Question 18 [4] i. A line passes through (2, 1, 3) and is perpendicular to the line = ( + ) + (2 2 + ) = (2 3 ) + ( + 2 + 2 ). Obtain its equation in Cartesian and vector form. OR ii. Find the equation of plane passing through the point (1, 1, 1) and perpendicular to the Planes x + 2y + 3z 7 = 0 and 2x 3y + 4z = 0. Section C (15 marks) Question 19 (1 x 5 = 5) In subparts (i) and (ii), choose the correct option and in subparts (iii) to (v), answer the questions as instructed. 9 i. If = 10, = 20 and = 20 , then the equation of regression line of x on y is 9 a. = 20 + 1 b. = 20 + 1 ii. iii. iv. v. c. = 50 + 5 d. None of these If the cost function ( ) of producing quantities of a product is given by ( ) = 500 2 + 2500 + 5000 and if each unit of product is sold at 6000, then Statement I: The profit function is 3500 500 2 5000. Statement II: The break-even points are 5, 3. a. Both statements are true and statement II is the correct explanation of statement I. b. Both statements are true and statement II is not the correct explanation of statement I. c. Statement I is true and statement II is false. d. Statement II is true and statement I is false. If two regression lines 1 : 4x + 3y + 7 = 0 and 2 : 3x + 4y + 8 = 0 then determine i. the regression line of y on x. ii. the regression line of x on y. A firm has cost function, C = 2 7 + 111 and demand function, = 70 , then find the total revenue in terms of . 17 2 The demand function is = 4 , where is the number of units demanded and is price per unit. Find the revenue function R in terms of . Question 20 [2] i. A cable TV operator has 5000 subscribers, each of them pay 100 per month. The operator proposes to increase the subscription and he found that for every increase of 0.50, 10 subscribers will discontinue the service. Find what increase in the subscription rate will increase maximum revenue and what will be the maximum revenue? OR ii. The demand function for a manufacturer s product is = 70 5 , where is the number of units and is the price per unit. At what value of will there be maximum revenue? what is the maximum revenue? Question 21 [4] i. The objective function Z = 5 + 7 , subject to constraints are 2 + 8, + 2 10 and , 0 a) Draw the graph of all the constraints. b) Check whether the feasible reason is bounded or not. c) Find the corner points of the feasible region. d) Find the values of Z at the corner points and the minimum value of Z. OR ii. If a young man rides his motorcycle at 30 km/h, then he had to spend 4 per km on petrol. If he rides at a faster speed of 45 km/h, then the petrol cost increases at 6 per km. He has 120 to spend on petrol and wishes to find what is the maximum distance, he can travel within 1 hr. a) Express this as a linear programming problem. b) Draw the graph of all the constraints. c) Find the feasible region and its corner point. d) What is the maximum distance that he can travel. Question 22 Find the regression coefficient of x on y and y on x for the data X 1 2 3 4 5 Y 7 6 5 4 3 and hence find an estimate of variate y for x = 3.5. [4]

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