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ISC Class XII Board Exam 2026 : Mathematics

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Vaibhav Sharma
City Montessori School CMS Rajajipuram Campus I, Lucknow
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MATHEMATICS Code No. 041 SAMPLE QUESTION PAPER CLASS - XII (2025-26) Maximum Marks: 80 Time: 3 hours General Instructions: Read the following instructions very carefully and strictly follow them: 1. 2. 3. 4. 5. 6. 7. 8. 9. This Question paper contains 38 questions. All questions are compulsory. This Question paper is divided into five Sections - A, B, C, D and E. In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) with only one correct option and Questions no. 19 and 20 are Assertion-Reason based questions of 1 mark each. In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2 marks each. In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each. In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each. In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each. There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 3 questions in Section C, 2 questions in Section D and one subpart each in 2 questions of Section E. Use of calculator is not allowed. SECTION-A This section comprises of multiple choice questions (MCQs) of 1 mark each. Select the correct option (Question 1 - Question 18) Q.No. 1. Questions Marks Identify the function shown in the graph (A) sin 1 (B) sin 1(2 ) 1 (C) sin 1 (2) (D) 2 sin 1 For Visually Impaired: 1. 1 1 Inverse Trigonometric Function, whose domain is [ 3 , 3] , is (A) cos 1 (B) cos 1 (3) (C) cos 1 (3 ) (D) 3 cos 1 *Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e. 2025-26. Page 1 of 10 2. If for three matrices = [ ] 4 , B = [ ] 3 C = [ ] products and 1 both are defined and are square matrices of same order, then value of , , and are: (A) = = 3 = = 4 (C) = = 4 = = 3 (B) = 2, = 3 = = 4 (D) = 4, = 2 = = 3 3. 0 2 + ] is skew-symmetric, then value of If the matrix = [3 is . + 4 0 (A) 2 (B) 0 (C) 1 (D) 2 1 4. If is a square matrix of order 4 and | | = 27, then ( ) is equal to (A) 3 (B) 9 (C) 3 (D) 9 1 5. 3 0 The inverse of the matrix [0 2 0 0 1 0 0 (A) [0 2 5 0 3 0 0 2 0 0 5] 1 [0 6. 7. 0 0 0 1 (B) (B) 0 1 [0 0 0 2 1 5] 3 0 0 (D) [ 0 2 0 ] 0 0 5 cos 67 Value of the determinant | sin 23 (A) 0 1 3 3 0] 0 1 (C) 0 0] is 5 1 sin 67 | is cos 23 (C) 2 If a function defined by ( ) = { 1 3 2 (D) 1 1 + 1, cos , > is continuous at = , then the value of is (A) 8. 1 (B) 4 + 2 (D) 2 1 1 2 (C) 4 1 2 (D) 4 + 1 2 1 A function ( ) = 10 2 2 is increasing on the interval 1 (A) ( , 4] 10. (C) 0 If ( ) = tan 1 , then (1)is equal to (A) 4 9. 1 (B) 1 (B) ( , 4) 1 (C) [ 4 , ) 1 1 (D)[ 4 , 4] The solution of the differential equation + = 0 represents a family of (A) straight lines (B) parabolas (C) Circles (D) Ellipses 1 *Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e. 2025-26. Page 2 of 10 11. (A) (C) 12. 1 10 6 14 units (D) + 2 + 2 ( ) ( ) 1 1 (B) 20 1 (C) 1 (D) 4 5 (B) 14 6 units (C) 8 14 units (D) 5 6 (B) 34 units (C) 5 units (B) + 1 units 1 (D) 5 2 units If = 3 + 2 + 4 , = + 3 and = 6 + 2 are three given vectors, then (2 . ) ( . ) + ( . ) is same as the vector (A) 16. (B) The distance of the point with position vector 3 + 4 + 5 from the y-axis is (A) 4 units 15. ) A bird flies through a distance in a straight line given by the vector + 2 + . A man standing beside a straight metro rail track given by = (3 + ) + (2 1) + 3 is observing the bird. The projected length of its flight on the metro track is (A) 14. ( 2 ( ) 2 + If 3 4 ( 4 ) cos( 4 ) = 5 ( 4 ) + C, then is equal to (A) 13. 1 If ( + ) = ( ), then ( ) is equal to (C) 1 (D) A student of class XII studying Mathematics comes across an incomplete question in a book. 1 Maximise = 3 + 2 + 1 Subject to the constraints 0, 0, 3 + 4 12, He/ She notices the below shown graph for the said LPP problem, and finds that a constraint is missing in it: Help him/her choose the required constraint from the graph. The missing constraint is (A) + 2 2 (C) 2 + 2 (B) 2 + 2 (D) + 2 2 *Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e. 2025-26. Page 3 of 10 16. For Visually Impaired: If = + + , where , , > 0, attains its maximum value at two of its corner points (4,0) and (0,3) of the feasible region determined by the system of linear inequalities, then (A) 4 = 3 17. (B) 3 = 4 (C) 4 + = 3 (D) 3 + = 4 The feasible region of a linear programming problem is bounded but the objective function attains its minimum value at more than one point. One of the points is (5,0). 1 Then one of the other possible points at which the objective function attains its minimum value is (A) (2,9) (B) (6,6) (C) (4,7) (D) (0,0) For Visually Impaired: The graph of the inequality 3 + 5 < 10 is the (A) Entire plane (B) Open Half plane that doesn t contain origin (C) Open Half plane that contains origin, but not the points of the line 3 + 5 = 10 (D) Half plane that contains origin and the points of the line 3 + 5 = 10 18. A person observed the first 4 digits of your 6-digit PIN. What is the probability that the person can guess your PIN? (A) 1 81 (B) 1 100 (C) 1 90 1 (D) 1 *Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e. 2025-26. Page 4 of 10 ASSERTION-REASON BASED QUESTIONS (Question numbers 19 and 20 are Assertion-Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the options (A), (B), (C) and (D) as given below.) (A) Both (A) and (R) are true and (R) is the correct explanation of (A). (B) Both (A) and (R) are true but (R) is not the correct explanation of (A). (C) (A) is true but (R) is false. (D) (A) is false but (R) is true. 19. 3 Assertion (A): Value of the expression sin 1 ( 2 ) + tan 1 1 sec 1 ( 2) is 4 . 1 Reason (R): Principal value branch of sin 1 is [ 2 , 2 ] and that of s 1 is [0, ] { 2 }. 20. Assertion(A): Given two non-zero vectors and . If is another non-zero vector such that ( + ) = 0 . Then is perpendicular to . Reason (R): The vector ( + ) is perpendicular to the plane of and 1 SECTION B This section comprises of 5 very short answer (VSA) type questions of 2 marks each. 21A 2 Evaluate tan (tan 1( 1) + 3 ) OR 21B 22 23A Find the domain of cos 1 (3 2) If = log tan ( 4 + 2), then prove that = ( 3) Find: ( 1)3 2 2 OR 23B Find out the area of shaded region in the enclosed figure. *Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e. 2025-26. Page 5 of 10 23 B For Visually Impaired: Find out the area of the region enclosed by the curve 2 = , = 3 and -axis in the first quadrant. 24. If ( + ) = ( ) ( ) for all , R and (5)= 2, (0) = 3, then using the definition of derivatives, find (5). 2 25. + 3 represent the two sides and , The two vectors + + and 3 respectively of a , where is the origin. The point lies on such that is a median. Find the area of the parallelogram formed by the two adjacent sides as and . 2 SECTION C This section comprises of 6 short answer (SA) type questions of 3 marks each. 26A. log If = prove that = (log( ))2 and hence find its value at = . 3 OR 26B. If = ( sin ), = (1 cos ) find 2 2 . 27 A spherical ball of ice melts in such a way that the rate at which its volume decreases at any instant is directly proportional to its surface area. Prove that the radius of the ice ball decreases at a constant rate. 28A Sketch the graph = | + 1| . Evaluate 4| + 1| . What does the value of this 2 3 3 integral represent on the graph? OR 28B Using integration find the area of the region {( , ) 2 4 0, 0} For Visually Impaired: 28A 2 Define the function = | + 1| . Evaluate 4| + 1| . What does the value of this integral represent? OR 28B Using integration find the area enclosed within the curve: 25 2 + 16 2 = 400 29A Find the distance of the point (2, 1,3) from the line = (2 + 2 ) + (3 + 6 + 2 ) measured parallel to the z-axis. 3 OR 29B Find the point of intersection of the line = (3 + ) + ( + + ) and the line through (2, 1,1) parallel to the z-axis. How far is this point from the z-axis? *Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e. 2025-26. Page 6 of 10 30. Solve graphically: Maximise = 2 + subject to + 1200 + 600 2 3 0, 0 . 30 For Visually Impaired: The objective function = 3 + 2 of a linear programming problem under some constraints is to be maximized and minimized. The corner points of the feasible region are (600,0), (1200,0), (800, 400) and (400, 200). Find the point at which is maximum and the point at which is minimum. Also, find the corresponding maximum and minimum values of .) 31. Two students Mehul and Rashi are seeking admission in a college. The probability that Mehul is selected is 0.4 and the probability of selection of exactly one of the them is 0.5. Chances of selection of them is independent of each other. Find the chances of selection of Rashi. Also find the probability of selection of at least one of them. 3 SECTION D This section comprises of 4 long answer (LA) type questions of 5 marks each 32. 3 6 1 1 For two matrices = [ 2 5 1] and = [0 2 4 1 2 and hence solve the system of equations: 2 1 1 1], find the product 0 3 5 3 6 = 3 2 5 + 2 = 0 2 + 4 + = 5 33A Evaluate: 1 log(1+ ) 0 1+ 2 5 OR 33B 34A (3 sin 2) cos Find 5 2 4 sin Solve the differential equation: + ( ) = (sin + ) 5 OR 34B Find the particular solution of the differential equation: 2 +( 2 ) = 0 given that (0) = 1 *Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e. 2025-26. Page 7 of 10 35. The two lines 1 3 = , + 1 = 0 and 2 = +1 2 = + 2 intersect at a point 5 whose y-coordinate is 1. Find the co-ordinates of their point of intersection. Find the vector equation of a line perpendicular to both the given lines and passing through this point of intersection. SECTION- E This section comprises of 3 case-study/passage-based questions of 4 marks each with subparts. The first two case study questions have three subparts (I), (II), (III) of marks 1, 1, 2 respectively. The third case study question has two subparts of 2 marks each 36. Case Study -1 4 A city s traffic management department is planning to optimize traffic flow by analyzing the connectivity between various traffic signals. The city has five major spots labelled , , , , . The department has collected the following data regarding one-way traffic flow between spots: 1. Traffic flows from to , to , and to . 2. Traffic flows from to and to . 3. Traffic flows from to . 4. Traffic flows from to and to . The department wants to represent and analyze this data using relations and functions. Use the given data to answer the following questions: I. Is the traffic flow reflexive? Justify. [1] II. Is the traffic flow transitive? Justify. [1] III A. Represent the relation describing the traffic flow as a set of ordered pairs. Also state the domain and range of the relation. OR III B. Does the traffic flow represent a function? Justify your answer. [2] *Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e. 2025-26. Page 8 of 10 37. Case Study -2 4 LED bulbs are energy-efficient because they use significantly less electricity than traditional bulbs while producing the same amount of light. They convert more energy into light rather than heat, reducing waste. Additionally, their long lifespan means fewer replacements, saving resources and money over time. A company manufactures a new type of energy-efficient LED bulb. The cost of production and the revenue generated by selling x bulbs (in an hour) are modelled as ( ) = 0.5 2 10 + 150 and ( ) = 0.3 2 + 20 respectively, where ( ) and ( ) are both in . To maximize the profit, the company needs to analyze these functions using calculus. Use the given models to answer the following questions: I. Derive the profit function ( ) [1] II. Find the critical points of ( ). [1] III A. Determine whether the critical points correspond to a maximum or a minimum profit by using the second derivative test. OR III B. Identify the possible practical value of (i.e., the number of bulbs that can realistically be produced and sold) that can maximize the profit, if the resources available and the expenditure on machines allows to produce minimum 10 but not more than 18 bulbs per hour. Also calculate the maximum profit. [2] *Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e. 2025-26. Page 9 of 10 38. Case Study -3 4 Excessive use of screens can result in vision problems, obesity, sleep disorders, anxiety, low retention problems and can impede social and emotional comprehension and expression. It is essential to be mindful of the amount of time we spend on screens and to reduce our screen-time by taking regular breaks, setting time limits, and engaging in non-screen-based activities. In a class of students of the age group 14 to 17, the students were categorised into three groups according to a feedback form filled by them. The first group constituted of the students who spent more than 4 hours per day on the mobile screen or the gaming screens, while the second group spent 2 to 4 hours /day on the same activities. The third group spent less than 2 hours /day on the same. The first group with the high screen time is 60% of all the students, whereas the second group with moderate screen time is 30% and the third group with low screen time is only 10% of the total number of students. It was observed that 80% students of first group faced severe anxiety and low retention issues, with 70% of second group, and 30% of third group having the same symptoms. I. What is the total percentage of students who suffer from anxiety and low retention issues in the class? [2] II. A student is selected at random, and he is found to suffer from anxiety and low retention issues. What is the probability that he/she spends screen time more than 4 hours per day? [2] *Please note that the assessment scheme of the Academic Session 2024-25 will continue in the current session i.e. 2025-26. Page 10 of 10

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