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CBSE Class 12 Pre Board 2026 : Mathematics

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Shambhu Dutta
Convent of Jesus and Mary School (CJM), Ranaghat, Nadia

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ISC 2025 EXAMINATION Guess Paper Mathematics Time Allowed: 3 hours Maximum Marks: 80 General Instructions: This Question Paper consists of three sections A, B and C. Candidates are required to attempt all questions from Section A and all questions EITHER from Section B OR Section C. Section A: Internal choice has been provided in two questions of two marks each, two questions of four marks each and two questions of six marks each. Section B: Internal choice has been provided in one question of two marks and one question of four marks. Section C: Internal choice has been provided in one question of two marks and one question of four marks. All working, including rough work, should be done on the same sheet as, and adjacent to the rest of the answer. The intended marks for questions or parts of questions are given in brackets [ ]. Mathematical tables and graph papers are provided. SECTION A - 65 MARKS 1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as [15] instructed. (a) If the value of 3rd order determinant is 5, then the value of determinant formed by replacing its element by its co-factor is: a) 5 b) 1/5 c)125 (b) d) 25 [1] 1/2 [1] cos x log( 1+x 1 x )dx is equal to 1/2 a) b) 0 5 c) 1 (c) (d) If tan 1 x + tan 1y = a) 3 c) 2 4 5 b) d) 5 Find the particular solution for 2xy + y c) y = If P(A) = 2x 1 log|x| 2x 1+log|x| 4 5 [1] then cot 1 x + cot 1 y equals 5 a) y = (e) d) 1 2 2 2x dy dx 5 = 0; y = 2 (x 0, x e) b) y = (x 0, x e) d) y = , and P(A B) = 7 10 then P(B|A) is equal to when x = 1 3x 1 log|x| 5x 1+log|x| (x 0, x e) (x 0, x e) [1] [1] a) 7 8 c) (f) (g) 1 10 b) 17 d) 1 20 8 Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is a) 2n + 1 b) nP2 c) 2n - 2 d) 2n - 1 If y = a) 1/x e e dy then dx [1] ? = [1] 1/x b) e d) 2e 1/x log x 2 x c) (h) (1/x 1) e If y = sec-1 ( a) c) (i) 1 x 1 2 ) 2x 1 then dy dx = 1/x log x [1] ? 2 b) 2 (1 x ) 2 d) 2 (1+x ) 2 1 x2 2 1 x2 Which of the following is not correct? a) |kA| = k3|A|, where A = [a ij ] 3 3 [1] b) If A is a skew-symmetric matrix of odd order, then |A| = 0 c) a + b e + f 2. a c+ d = e g+ h b c + f g d) |A| = |AT|, where A = [a d h ij ] 3 3 (j) Which one of the following is true? (a) sin (cos 1 x) = cos (sin 1 x) (b) sec (tan 1 x) = tan (sec 1 x) (c) cos (tan 1 x) = tanx tan (sin 1 x) = sin (tan 1 )x (d) [1] (k) Let f : R R : f(x) = x-2 + 1. Find : f-1{10} [1] (l) Solve for x given that [ 2 3 1 1 ][ x ] = [ 4 1 [1] ] 3 (m) Prove that the greatest integer function f: R (n) If P(A) = 0.8, P (B) = 0.5 and P(B|A) = 0.4, find P(A|B) [1] (o) Given two independent events A and B such that P (A) = 0.3, P (B) = 0.6. Find P(A and B). [1] Show that the function f(x) = { 1 + x, if x 2 5 x, if x > 2 R , given by f(x) = [x], is neither one-one nor onto. [1] [2] is not differentiable at x = 2. OR Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3. 3. Evaluate the Integral: ( 4. Find the absolute maximum value and the absolute minimum value of the function f(x) = ( 1 tan x x+log cos x [2] ) dx 1 2 - x)2 + x3 in the [2] given interval [ -2, 25]. 5. Evaluate: [2] 5 (|x| + |x + 1| + |x - 5|) dx 1 OR Evaluate: sin3 x cos x dx 6. If f(x) = 7. Prove that tan { 8. Evaluate: 9. 2x 2 (1+x ) then show that f(tan ) = sin 2 4 + 1 2 1 2e 2x +3e x 1 cos ( a b )} + tan{ 4 [2] 1 2 1 cos ( a b )} = 2b a . [4] dx +1 If x = a cos3 and y = a sin3 , find [4] d 2 y 2 dx . Also, find its value at = [4] 6 OR Find 10. dy dx when y = (sin x)cos x + (cos x)sin x Read the text carefully and answer the questions: [4] There are two antiaircraft guns, named as A and B. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time. (a) How is Bayes' theorem different from conditional probability? (b) Write the rule of Total Probability. (c) What is the probability that the shell fired from exactly one of them hit the plane? (d) If it is know that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B? OR Read the text carefully and answer the questions: [4] A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is 0.65. The probability that many are not present and still the work gets completed on time is 0.35. The probability that work will be completed on time when all workers are present is 0.80. Let: E1: represent the event when many workers were not present for the job; E2: represent the event when all workers were present; and E: represent completing the construction work on time. (a) What is the probability that all the workers are present for the job? (b) What is the probability that construction will be completed on time? (c) What is the probability that many workers are not present given that the construction work is completed on time? (d) What is the probability that all workers were present given that the construction job was completed on time? 11. Read the text carefully and answer the questions: [6] A manufacture produces three stationery products Pencil, Eraser and Sharpener which he sells in two markets. Annual sales are indicated below: Market Products (in numbers) Pencil Eraser Sharpener I 10,000 2,000 18,000 II 6,000 20,000 8,000 If the unit Sale price of Pencil, Eraser and Sharpener are 2.50, 1.50 and 1.00 respectively,Based on the information given above, answer the following questions: (a) What is the total revenue collected from Market-I? (b) What is the total revenue collected from Market-II? (c) What is the gross profit from both markets considering the unit costs of the three commodities as 2.00, 1.00, and 50 paise respectively? 12. Find the equation of a curve passing through the point (1, 1). If the tangent drawn at any point P(x, y) on the [6] curve meets the co-ordinate axes at A and B such that P is the mid-point of AB. OR Find the particular solution of the following differential equation : (x2 + xy)dy = (x2 + y2)dx, given that y=0,when x=1. 13. Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius 10 cm is a square of side 10 2 cm [6] . OR Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r 3 14. Read the text carefully and answer the questions: Ram and Shyam play a game with a coin. Ram stakes 1.00 and throws the coin 4 times. If he throws 4 heads, he gets his stake and 3.00 from Shyam. If he throws only three heads and they are consecutive, he gets his stake and 2.00 from Shyam. If he throws only 2 heads and they are consecutive, he gets his stake and 1.00 from Shyam. In all other cases, Shyam takes the stake money. Is this game fair? Provide reasons for your answer. [6] SECTION B - 15 MARKS 15. In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as [5] instructed. (a) ^ ^ ^ ^ The value of p for which the vectors 2 i + p j + k and 4 i ^ j + 26k are perpendicular to each other, [1] ^ 6 is: a) b) 3 17 3 c) -3 d) 17 3 (b) Find the distance between the planes 2x - y + 2z = 5 and 5x - 2.5y + 5z = 20. [1] (c) Find the value of p for which the vectors [1] (d) Find the coordinates of the foot of the perpendicular drawn from the origin to 2x + 3y + 4z 12 = 0 a) ( c) ( (e) 27 29 24 29 , , 36 29 36 29 , , 48 29 48 29 and ^ ^ ^ 3 i + 2j + 9k ) b) ( ) d) ( 24 29 24 29 , , ^ ^ ^ i 2pj + 3k 39 29 36 29 , , 48 29 49 29 are parallel. [1] ) ) Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal [1] ^ vector is 2^i 3^j + 6k . 16. Find a unit vector in the direction of the vector a = ^ ^ ^ i + 2j + 3k [2] . OR Show that (a b) (a + b) = 17. 2 (a b) , a and b Show that the distance d from point P to the line l having equation r = a + b is given by d = |b P Q| | b| where Q [4] is any point on the line l. OR Show that the lines 18. x 1 y 2 = 2 = z+3 3 and x 2 2 y 6 = 3 = z 3 4 intersect and find their point of intersection. Find the area under the given curves and given lines: y = x2, x = 1, x = 2 and x - axis. [4] SECTION C - 15 MARKS 19. In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as [5] instructed. (a) If C(x) and R(x) are respectively Cost function and Revenue function, then profit function P(x) is [1] given by (b) a) P(x) = R(x) C(x) b) P(x) = R(x) c) P(x) = C(x) + R(x) d) P(x) = R(x) - C(x) The value of objective function Z = 2x + 3y at comer point (3, 2) is a) 9 b) 5 c) 15 d) 12 [1] (c) Find the coefficient of correlation from the regression lines: x - 2y + 3 = 0 and 4x - 5y + 1 = 0 (d) The total cost and total revenue functions of a commodity are given by C(x) = x + 40 and R(x) = 10x - [1] 0.2x2. Find the break-even point. [1] (e) The demand function for a certain commodity is given by p = 1000 - 15x - x, 0 < x < 25. What is the [1] price per unit and the total revenue from the sale of 2 units? 20. A monopolist has a demand function x = 106 - 2p and the average cost function AC = 5 + x 50 , where p is the [2] price per unit output and x is the number of units of output. If the total revenue is R = px, determine the most profitable output and the maximum profit. OR The total cost function is given by C = x + 2x3 - 3.5x2, find the marginal average cost function (MAC). Also, find the points where the MAC curve cuts the x-axis and y-axis. 21. The following table shows the mean and standard deviation of the marks of Mathematics and Physics scored by [4] the students in a school: Mathematics Physics Mean 84 81 Standard Deviation 7 4 The correlation coefficient between the given marks is 0.86. Estimate the likely marks in Physics if the marks in Mathematics are 92. [4] 22. A manufacturing company produces two types of cell phones, Android and iOS. The company has resources to make at the most 300 sets a week. It takes 1800 to make an Android set and 2700 to make an iOS set. The company cannot spend more than 648000 a week to make cell phones. The company makes a profit of 510 per Android and 675 per iOS set. If x and y denote, respectively, the number of Android sets and iOS sets made each week, then formulate this problem as a Linear Programming Problem (LPP) given that the objective is to maximize the profit. Based on it, answer the questions that follow. (a) What will be the maximum profit function on x and y sets? b) What will be the values of your objective function in the feasible region? c) At what point the maximum profit will occur? (d)What s the weekly cost (in ) of manufacturing the sets?

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