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ISC Class XII Notes 2026 : Other

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Set A 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) 0 2 + ] is skew- symmetric, then the value is of i. If the matrix = [3 is + 4 0 a. -2 b. 0 c. 1 d. 2 2 ii. Let ( ) = | 4|. Which of the following is true about ( ) at = 2 ? a. (2) = 4 b. (2) does not exist. c. (2) = 0 d. ( ) is not continuous at x = 2 iii. The solution of the differential equation + = 0 represent a family of a. straight lines b. parabolas c. circles d. ellipse iv. 1 (B) P( ) is equal to (ii) (C) P(A ) is equal to (iii) (D) The probability that the question is solved is (iv) v. vi. 1 The probability of solving a specific question independently by A and B is 3 and 5, respectively. Both try to solve the question independently. Match Column I with Column II and choose the correct option: Column I Column II 4 (A) P( ) is equal to (i) 5 7 15 2 5 1 15 a. A (iv), B (iii), C (ii), D (i) b. A (ii), B (iii), C (iv), D (i) c. A (ii), B (iv), C (i), D (iii) d. A (iii), B (i), C (iv), D (ii) Given below are the graphs of f(x) along the statements. Which one of the following statements is NOT correct about the graphs? Consider the graph of the function f(x) shown below: 1 Statement 1: The function f(x) is increasing in (2, 2). Statement 2: The function f(x) is strictly increasing in 1 ( , 1). 2 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. vii. viii. ix. Assertion (A): The system of linear equations 5x + y = 4, 2x + k = 3 has a unique 2 solution, if k is not equal to . 5 Reason (R): A system of linear equations has a unique solution only when the determinant of its coefficient matrix is non-zero. 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. If A denotes the set of continuous functions and B denotes the set of differentiable functions, then which of the following depicts the correct relation between set A and B? Consider the following statements: 1 Statement 1: f R R given by f(x) = 2 , is neither injective nor surjective. 3 x. xi. Statement 2: f R R given by f(x) = 9 , is neither injective nor surjective. 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. Identify the function shown in the graph: a. 1 b. 1 (2 ) c. 1 (2 ) d. 2 1 1 7 Assertion (A): The value of 1 1 + 1 ( 2 ) is 12 . 1 Reason (R): 1 1 = 4 and 1 ( 2 ) = 6 Which one 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. xii. xiii. xiv. Find the value of the determinant given below, without expanding it at any stage 1 ( + ) | 1 ( + )| 1 ( + ) State the reason why the relation R {( , ): 2 } on the set R of real number is not reflexive. While solving an inverse trigonometric problem on the blackboard. Satish wrote the following as part of his solution: cos 1 xv. 5 = 3 His solution stopped him and said that he must have made a mistake in the solution. How did the teacher recognise that Satish had made a mistake? Justify your answer. A discrete random variable X has the following probability distribution: 1 2 3 4 5 0.1 0.2 0.3 0.25 0.15 ( = ) Verify if this is a valid probability distribution. Find P (( 3). Question 2 Find the slope of tangent to the curve = 2 + 3 8, = 2 2 2 5 at the point (2, 1). [2] Question 3 i. Solve the differential equation: ( 2 + 3 + 2 ) 2 = 0. [2] OR ii. 1 Evaluate: ( + 1)( 2) Question 4 Solve 1 ( 1 ) = 3 [2] Question 5 i. If = , then find . [2] ii. OR Given that + = 12 and y decreases at the rate of 1 cm/s, find the rate of change of 1 1 1 x when x = 5 cm and y = 1 cm. Question 6 [2] 5 Prove that: 0 4 + 4 4 + 4 + 9 4 Question 7 If 1 + 1 = Question 8 2 = 5 2 [4] 2 , 2 + 1 = 5 [4] i. ii. Show that the function ( ) = 1 ( + ), > 0 is always an increasing function in (0, 4 ). OR If = 2 and y = 2 prove that = Question 9 [4] i. A city has three main internet service providers. Provider X: serves 50% of the population Provider Y: serves 30% of the population Provider Z: serves 20% of the population On a particular day, the probability of experiencing internet downtime is: 0.02 for provider X 0.05 for provider Y 0.10 for provider Z A random customer reports internet downtime on that day. Based on this, answer the following: (a) Represent the situation using probability notation. Define all events clearly. (b) What is the probability that this customer uses provided Z? (c) Which provider is most likely responsible for the downtime? justify numerically. OR ii. School organisers debate competition there are two groups of participants. Group A: 6 students (4 boys and 2 girls) Group B: 4 students (1 boy and 3 girls) One student is chosen at random from each group to represent in the final round. Answer the following questions: (a) Find the probability that both selected students are girls. (b) If the student chosen from group A is a boy find the probability that the student from group B is also boy. (c) Are the events, student chosen from group A is a girl and student chosen from group B is a girl independent? Justify. Question 10 2 +7 Evaluate: 2 2 [4] Question 11 [6] 4 that: 0 (1+ ) = i. Prove ii. OR Solve the differential equation: = ( 3 + 2 ) , given that: = 0, = 1. Question 12 [6] i. The owner of a school wants to lay out a running track of 400 metres enclosing a football field, the shape of which is rectangular with the semi-circle, at each end. (a) If x and y denote the length and breadth of the rectangular region, then find the relation between the variables. (b) What is the area of the rectangular region A expressed as a function of x? ii. (c) What is the maximum value of area A? (d) What is the area generated if the area of the whole floor is maximised? OR A publisher wants to publish a book consisting of 700 pages. Each printed page is to have a total area of 80 sq cm with a margin of 1 cm at the top and on each side and a margin of 1.5 cm at the bottom. (a) If x and y denote the length and breadth of the page and if printed area of the page is A, then determine the relation between A and x. (b) Find the length of the page for which the printed area is maximum. (c) What is the maximum printed area of the page? (d) What should be the dimension of the printed page of the book? Question 13 [6] Garima owns a cafe that serves burgers, samosas and juice bottles. On a particular day, 100 items in total were sold. The number of burgers sold was thrice the number of samosas sold. Further, the number of samosas sold was 10 more than the number of juice bottles sold. Based on the above information, answer the following questions. (a) Form a set of simultaneous equations for the above information. (b) Solve the set of equations formed in (a) by matrix method (c) Hence, find the number of items sold in each category. Question 14 [6] In a raffle draw, 1000 raffle tickets are sold for 1 each. Each has an equal chance of winning. First prize is 300, second prize is 200 and third prize is 100. Let X denote the net gain from the purchase of one ticket. (a) Construct the probability distribution of X. (b) Find the probability of winning any money in the purchase of one ticket. (c) Find the expected value of X and interpret its meaning. 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 the plane 2 + + 3 = 5 is perpendicular to the plane + + 3 = 9, then the value of is a. 11 b. 11 c. 12 d. 0 ii. Statement I: The angle between the planes 2 + = 6 and + + 2 = 7 is 3 Statement II: The angle between the planes 1 + 1 + 1 = 1 and 2 + 2 + 1 2 + 1 2 + 1 2 2 = 2 is given by cos = 1 2 + 1 2 + 1 2 2 2 + 2 2 + 2 2 a. Both statements are true and statement II is the correct explanation of statement I. iii. iv. v. 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| |= 3| | and ( + ). ( ) = 32, then value of | |is a. 2 b. 3 c. 4 d. 12 Show that points whose position vectors are ( 2 + 3 + 5 ), ( + 2 + 3 ), (7 ) are collinear. Vectors , , and are given by = + + , = 2 + 3 , = 3 + 5 2 and = . Prove that and are parallel and find the ratio of their lengths. Question 16 [2] i. Let , and are three vectors of positive magnitude such that = , = . a. Prove that , and are mutually at right angles. b. Prove that | |= 1, | |= | | OR ii. Find the angle between the vectors = + and = + . Question 17 [4] Consider the curves y = x + 2, y = x, x = 0, x = 3. i. Make the rough sketch of the parabola y = x + 2 ii. Determine the region enclosed by given curves. Do mention the corner points of the region while plotting. iii. Find the area of the region bounded by the given curves and the lines. Question 18 i. Show that the lines other. ii. 1 3 = +1 2 = 1 5 2 4 = 1 3 = +1 2 [4] do not intersect each OR Find the equation of plane passing through the points ( 1, 1, 2) and perpendicular to the Planes 3x + 2y 3z = 1 and 5x 4y + z = 5. 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. i. By using the data = 5, = 13 and = 2.5, the regression equation y on x is a. y = 2.5x 0.5 b. y = 0.5x + 2.5 c. y = 0.5x 2.5 d. y = 2.5x + 0.5 ii. Let p(x) = 50 3x be a monopolist demand function Then, read the following statements and choose the correct option Statement I: The revenue function R(x) is 50 3 Statement II: The marginal revenue at = 0 is 50. a. Both statements are true and statement II is the correct explanation of statement I. iii. 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. In a factory, it is found that the number of units (x) produced in a day depends upon 5 the number of workers (n) is obtained by the relation = +5. The demand function 2 iv. v. of product is = + . Determine the marginal revenue when n = 44. Given the cost function and revenue function, respectively as C(x) = 2x + 40 and R(x) = 11 0.2 , then find the break-even point. For the given lines of regression 2x 3y = 6 and 5x 4y = 20. Find regression coefficients and . Question 20 i. ii. [2] Given the total cost for x unit of commodity as C(x) = 3 + 3x 7x + 16. Find the marginal cost and the average cost. OR Given the total cost for x unit of commodity as C(x) = + bx cx + d, where a > 0, b < 0, c > 0, show that average variable cost and marginal cost curves intersect at minimum average variable cost. Question 21 [4] i. Solve the following linear programming problem graphically Maximise Z = x + y Subject to constraints are x y 1, x + y 0 and x, y 0 OR ii. A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contain 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and atmost 300 units of cholesterol. a) Formulate the given problem as LPP assuming the amount of vitamin A is to be minimised. b) Plot the graph of each inequalities. c) Identify the feasible reason and the corner points. d) How many packets of each food must be used to minimise the amount of vitamin A in the diet and what is the minimum amount of vitamin A? Question 22 [4] Given the observations are (10, 5), (10, 3), (11, 2), (11, 0), (12, 1), (15, 6), (16, 4), (11, 2). I. Find and II. Predict the value of y corresponding to the value 14 of x. III. Predict the value of x, when the value of y is 3.

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