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ISC Class XII Sample / Model Paper 2026 : Mathematics

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Sinjon Barik
Delhi Public School (DPS) Megacity, Kolkata

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INDIAN SCHOOL CERTIFICATE (CLASS XII) EXAMINATION MATHEMATICS Maximum Marks: 80 Time Allowed: Three Hours (Candidates are allowed an additional 15 minutes for only reading the paper. They must NOT start writing during this time.) General Instructions: This Question Paper consists of four compulsory sections: Section A (20 Marks), Section B (17 Marks), Section C (20 Marks), and Section D (23 Marks) in accordance with the revised ISC 2027 format. Candidates are required to attempt all questions across Sections A, B, C, and D (Linear Regression and Application of Calculus in Commerce & Economics have been excluded as per the 2027 curriculum). Section A consists of objective and very short-answer questions of 1 mark each (20 Marks). Section B consists of short-answer questions of 2 marks and 3 marks (17 Marks). Section C consists of medium-answer and Case-Study questions of 4 marks each (20 Marks). Section D consists of long-answer questions of 5 marks and 6 marks (23 Marks). Internal choices have been provided in selected questions across Sections B, C, and D. SECTION A (20 Marks) (Answer all questions in this section) Question 1 [20 1 = 20] In sub-parts (i) to (xii), choose the correct options and in sub-parts (xiii) to (xx), answer the questions as instructed. (i) Let f : R R be defined by f (x) = 3x 4. Then the function f is: (a) One-one but not onto (b) Onto but not one-one (c) Both one-one and onto (Bijective) (d) Neither one-one nor onto (ii) The principal value of cos 1 cos 7 6 is: (a) (b) (c) (d) 7 6 5 6 6 6 (iii) If A is an invertible matrix of order 2 2 such that |A| = 5, then the value of |A 1 | is: (a) 5 (b) 25 (c) (d) 1 5 1 25 1 cos sin (iv) If A = , then A + AT = I2 when is equal to: sin cos (a) (b) 6 3 (c) (d) 4 2 (v) If y = esin(x ) , then dy dx (a) 2x cos(x2 )esin(x (b) cos(x2 )esin(x (c) 2xecos(x is equal to: 2) 2) 2) (d) 2x sin(x2 )esin(x 2) Z e (vi) The value of the definite integral e 2024 e x cos(2024x) dx is: arctan(2024x) (a) e (b) 2024 (c) 0 (d) 1 Z (vii) The value of the definite integral (a) (b) (c) (d) x2 p dx is: 1 + sin x + 1 + sin2 x 3x2 2 3 3 2 3 3 3 6 (viii) The order and degree of the differential equation d2 y dx2 r + 1+ dy dx 3 = 0 are respectively: (a) 2, 1 (b) 2, 2 (c) 2, 3 (d) 3, 2 (ix) If A and B are two events such that P (A) = 0.4, P (B) = 0.8 and P (B|A) = 0.6, then P (A|B) is equal to: (a) 0.24 (b) 0.30 (c) 0.48 (d) 0.75 (x) Assertion (A): The function f (x) = [x] (greatest integer function) is discontinuous at x = 2. Reason (R): The left-hand limit limx 2 [x] = 1 and the right-hand limit limx 2+ [x] = 2 are not equal. (a) Both (A) and (R) are true and (R) is the correct explanation of (A). 2 (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. (xi) If |~a| = 3, |~b| = 4 and |~a ~b| = 6, then the acute angle between ~a and ~b is: (a) (b) (c) (d) 6 4 3 2 (xii) The direction ratios of the normal to the plane 3x 4y + 12z = 7 are: (a) 3, 4, 12 (b) 3, 4, 12 (c) 3, 4, 12 (d) 3, 4, 7 (xiii) Find the principal value of tan 1 ( 3) sec 1 ( 2). Z x e (1 + x ln x) (xiv) Evaluate: dx. x (xv) Write the integrating factor of the differential equation dy dx + y tan x = sec2 x. (xvi) A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once? (xvii) Find the rate of change of the area of a circle with respect to its radius r when r = 5 cm. (xviii) Find the value of for which the vectors ~a = 3 i 2 j + k and ~b = 2 i + 6 j + k are mutually perpendicular. (xix) Find the intercepts cut off by the plane 2x + y z = 5 on the coordinate axes. (xx) Write the Cartesian equation of the line passing through the point ( 2, 4, 5) and parallel y 4 z+8 to the line x+3 3 = 5 = 6 . SECTION B (17 Marks) (Answer all questions in this section) Question 2 [2] (i) Discuss the continuity of the function f (x) at x = 0, where: ( sin 5x + cos x, x 6= 0 f (x) = 8 3x x=0 3, OR (ii) If x = a( sin ) and y = a(1 + cos ), find dy dx 3 at = 3 . Question 3 [2] Find the equation of the tangent to the curve y = x3 3x2 9x + 7 at the points where the tangent is parallel to the x-axis. Question 4 [2] A problem in Mathematics is given to three students A, B, and C whose chances of solving it independently are 12 , 13 , and 14 respectively. Find the probability that the problem will be solved. Question 5 [2] (i) Find the area of the parallelogram whose adjacent sides are determined by the vectors ~a = i j + 3k and ~b = 2 i 7 j + k. OR (ii) If ~a, ~b, ~c are unit vectors such that ~a + ~b + ~c = ~0, find the value of ~a ~b + ~b ~c + ~c ~a. Question 6 [3] Z (i) Evaluate the improper integral: 1 dx , where 0 < < . (x cos ) x2 1 OR (ii) Solve the differential equation: (x2 y 2 )dx + 2xy dy = 0, given that y = 1 when x = 1. Question 7 [3] Solve for x: tan 1 (2x) + tan 1 (3x) = 4 , where x > 0. Question 8 If y = e [3] a cos 1 x, where 1 x 1, prove that (1 d2 y x2 ) dx 2 dy x dx a2 y = 0. SECTION C (20 Marks) (Answer all questions in this section) Question 9 [4] 3 Z (i) If p = 3 ln 5 ln 4 u2 cot 1 (u3 ) du, then find the value of (4e6p ). cot 1 (u3 ) + cot 1 (ln 20 u3 ) OR Z (ii) Evaluate the indefinite integral: x2 (1 ln x) ln4 x x4 Question 10 dx. [4] (i) In a factory which manufactures bolts, machines A, B, and C manufacture respectively 25%, 35%, and 40% of the bolts. Of their outputs, 5%, 4%, and 2% are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by machine B? 4 OR (ii) A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes and hence find its variance. Question 11 [4] (i) Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XY -plane. Also find the angle which this line makes with the XY -plane. OR (ii) Find the vector and Cartesian equations of the plane passing through the line of intersec = 1 and ~r (2 i + 3 j k) + 4 = 0 and parallel to the x-axis. tion of the planes ~r ( i + j + k) Question 12 [4] Using integration, find the area of the region bounded by the curve x2 = 4y and the straight line x = 4y 2. Question 13 [4] Case Study: A logistics company is designing a closed right circular cylindrical storage tank of fixed volume V = 16 cubic meters to store liquid nitrogen. Let r meters be the radius of the circular base and h meters be the height of the cylinder. The cost of manufacturing the tank is directly proportional to its total surface area S. Based on the above information, answer the following questions: (i) Express the height h of the cylindrical tank in terms of its radius r. [1] (ii) Write the total surface area S(r) of the closed tank as a function of r alone. [1] (iii) Using calculus, find the radius r and height h for which the total surface area S is minimum. [2] SECTION D (23 Marks) (Answer all questions in this section) Question 14 Z /2 Evaluate: 0 [5] sin3 x dx. (cos4 x + 3 cos2 x + 1) tan 1 (sec x + cos x) Question 15 1 1 1 If A = 2 1 3 , find A 1 and hence solve the following system of linear equations: 1 1 1 [6] x + 2y + z = 4 x + y + z = 0 x 3y + z = 2 Question 16 [6] 5 (i) Solve the differential equation: dy cos x (tan 2x)y = cos4 x, dx 2 3 3 where |x| < , given that y = . 4 6 8 OR Z (ii) Evaluate: 0 /2 x sin x cos x dx. sin4 x + cos4 x Question 17 [6] (i) Prove that the semi-vertical angle of the right circular cone of given slant height and max imum volume is tan 1 2. OR (ii) A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum? END OF QUESTION PAPER Reference Answer Key for Your Integrals & DE: Ra Ra R 3 3 Q1(vii): (b) 3 (Using a f (x) dx = 0 [f (x) + f ( x)] dx = 0 x2 dx = 3 ) | Q6(i): sin x x 1 1 ln x 1 Q9(ii): 14 ln ln tan + C | Q14: ln | Q16(i): y cos 2x = sin 2x + ln x+x 2 x 4 4 tan 1 2 Q1(vi): (c) 0 Q9(i): 5 | | 6 3 8

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