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ISC Class XII Sample / Model Paper 2026 : Mathematics : Half yearly exam

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Raghvendra Prajapati
Vikas Bharti School, Gorakhpur

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VIKAS BHARTI SCHOOL BHARTIPURAM, GORAKHPUR Half yearly exam (2025-26) th Form: SC/Class 12 Subject: Mathematics MM: 80 / Time: 2:30 hrs ___________________________________________________________________ Instructions for Candidates: (i). This question papers consists of three sections A, B and C. (ii). Candidates are required to attempt all questions from Section-A and all questions from either Section-B or Section-C. However, internal choices have been provided in which students has to attempt any one. (iii). The intended marks for questions or parts of questions are given in brackets. _____________________________________________________________________________________ SECTION - A (65 MARKS) 1. In sub-parts (i) to (x), choose the correct option and in sub-parts (xi) to (xv), answer the questions as instructed. [15 X 1 = 15] (i). Let R be the relation in the set of natural numbers N, given by R = {(x, y): x = y + 3, y >5}, choose the correct answer from the following: (ii). (a) (7, 4) R (b) (4, 7) R The simplified value of sin (a) (b) -1 1 3 (1, 2) (b) [-1, 1] (8, 5) R. (c) 1 (d) -1 (c) (d) -1 0 (-1, 1) (9, 6) R [cot {cos (tan 1)}] -1 (iii). What is the Domain of the function f(x)=cos (a) (c) (2x-3) 2 . 3 (d) [1, 2] (iv). If A is a 2 X 3 matrix, such that AB and AB both are defined, then order of the matrix B is (v). (a) 2X2 (b) 2X1 (c) 3X2 (d) 3X3 If A is a square matrix of order 3 then which of the following is not true? (b) |A| = |A'| |kA| = k3|A| (c) Minor of an element of |A| can never be equal to cofactor of the same (d) Order of minors and of cofactors of elements of |A| is same. (a) element. (vi). If A is a non-singular matrix, then (a (b) |A| |A'| |AA'| |A2| (c) (d) |A-1| |A|-1 |A| + |A'| 0 For function f(x) = |sinx| (vii). (a) (b) (c) f is differentiable at everywhere. f is continuous at everywhere but not differentiable at x = n , n Z. f is continuous at everywhere but not differentiable at x = (2n + 1) , n 2 Z. (d) None of these. 2 2 (viii). If the tangent to the curve x = t - 1, y = t - t is parallel to x-axis, then (b) (a) t=0 (c) t= t=2 1 2 (ix). The value of (d) t = - dy dx if y = |x-1| + |x - 4|, at x = 3 is 1 2 (a) -2 (c) (x). (b) 2 0 (d) 4. ( ) Which of the following function is decreasing on interval 0, (a) sin 2x (c) cos x (d) cos 3x. (b) -1 2 tan x -1 (xi). Differentiate tan x w.r.t cot x. | -1| = _________. (xii). If A is an invertible matrix of order 3 X 3 then A { 2x-1 (xiii). Examine the continuity of the function given as f(x) = 3 (xiv). If [25 04] = P + Q 2 x x<0 x 0 . , where P is a symmetric matrix and Q is Skew-symmetric matrix, then find matrix Q. (xv). Find the range of the function f(x) = acos x + 2sin x, where a is a positive constant. 2. (i). For what value of k the function f(x) = k(x+sin x) + k, is increasing? [2] OR (ii) (3) If f : R R is defined by f(x) = x|x|, show that f(x) is one - one and onto. Prove that d -1 sin x = dx 1 2 1-x . [2] OR 2 2 If x = at and y = 2at, then find dy 2. dx (4) [ x 0x] and A If A = 2x -1 [ = 1 -1 ] 0 , then find the value of x. 2 [2] (5) Using determinant, find the equation of the line passing through the points (1, 2) and [2] (3, 6). (6) -1 -1 If f = {(1, 5), (2, 3), (3, 0), (4, -2)}, then write f (x) and find the range of f (x). [2] (7) -1 Prove that cos 12 -1 3 -1 56 +sin =sin . 13 5 65 [4] (8) 3 3 Find the equations of the tangent and normal to the curve x = asin , y = acos [4] at = . 4 (9) The two side of an isosceles triangle with fixed base a are decreasing at a rate of [4] 3 cm/sec. How fast its area decreasing when the two sides are equal to the base? (10) Find the volume of the largest cone that can be inscribed in a sphere of radius r. [4] (11) Solve the following system of equation using matrix: [6] 2 3 10 + + = 4, x y z (12) If [6] 2 2 1-x + 1-y = a(x-y) 4 6 5 - + = 1, x y z then show that 6 9 20 + = 2. x y z dy = dx 2 1-y 2. 1-x (13) A water tank has the shape of an inverted right circular cone with its axis vertical [6] and vertex lowermost. Its semi-vertical angle is tan -1 () 1 . Water is poured into it at 2 a constant rate of 5 cubic metre per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4 m. OR Show that the maximum value of function, f(x) = x + 1 is less than its minimum x value. Use the first derivative test. (14) Show that the relation R defined by (a, b) R (c, d) iff ad(b + c) = bc(a + d) is an [6] equivalence relation. { OR n+1 if n is odd 2 Let f:N N is a function defined by, f(x) = , show that f is a n if n is even 2 bijection. SECTION - B (15 MARKS) (15) (5x1=5) (i) The value of for which the vectors 3 i - 6 j + k and 2 i - 4 j + k are parallel. (ii) (a) 2/3 (c) 5/2 (b) 3/2 (d) 2/5 If points A (60 i + 3 j ), B(40 i 8 j ) and C( a i - 52 j ) are collinear, then a is equal to (iii) (a) 40 (c) 20 (b) -40 (d) -20. Find the value of i . ( j x k ) + j . ( i x k ) + k . ( i x j ). (iv) Find the unit vector in the direction of the sum of the vectors a = 2 i + 2 j - 5 k and b = 2 i + j + 3 k . Find a vector of magnitude 3 2 units which makes an angle of /4, /2 with (v) y and z-axes, respectively. (16) Let a , b and c be three vectors such that | a |= 3, | b |= 4 , | c | = 5 and each one of [4] them being perpendicular to the sum of the other two, find | a + b + c |. 2 (17) Find the area enclosed by the parabola 4y = 3x and the line 2y = 3x + 12. [6]. SECTION - C (15 MARKS) Q.15. (5x1=5) (i) (ii) 2 th If R(x) = 36x + 3x + 5, then actual revenue from selling 50 item is (a) 333 (c) 330 (b) 300 (d) None of these. 1 3 2 x + x - 15x + 3, then marginal cost function is 3 1 2 3 2 (c) x + x - 15 + x - 2x 3 x 2 (d) None of these. x + 2x - 15 If C(x) = (a) (b) 2 (iii) Find the break-even point, given C(x) = x + 40 and R(x) = 10x - 0.2x . (iv) 2 The demand function of a monopolistic market is given by p = 100 - x - x . Find the revenue function. (v) 2 if C(x) = ax + bx + c, represents the total cost function. Then find the slope of the marginal cost function. 2 Q.16. If A(x) = 0.05x + 16 + 100 is the manufacturer average cost function, what is x th the marginal cost when 50 unit are produced. [4] [2] Q.17. The manufacturing cost of an item consist of 900 as overhead, the material cost as 3 [6] ( ) 2 x per item and the labor cost as for x items produced. Find: 100 (i) Average cost. (ii) Average variable cost. (iii) Average fixed cost in term of x. (iv) The values of average cost, average variable cost and average fixed cost at the level of output of 50 items. ***********

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