CBSE Class 12 Mathematics Guess Paper 2027

If integral 3ax/(b² + c² x²) dx = A log |b² + c² x²| + K, then the value of A is (A) 3a (B) 3a/(2b²) (C) 3a/(b² c²) (D) 3a/(2c²)

The area of the shaded region bounded by the curves y² = x, x = 4 and the x-axis is given by (A) integral₀⁴ x dx (B) integral₀² y² dy (C) 2 integral₀⁴ sqrt(x) dx (D) integral₀⁴ sqrt(x) dx

A function f(x) = 10 - x - 2x² is increasing on the interval (A) (-infinity, -1/4] (B) (-infinity, 1/4) (C) [-1/4, infinity) (D) [-1/4, 1/4]

If A is a square matrix of order 4 and |adj A| = 27, then A (adj A) is equal to (A) 3 (B) 9 (C) 3 I (D) 9 I

Assertion (A): Let Z be the set of integers. A function f: Z → Z defined as f(x) = 3x - 5, ∀ x ∈ Z is a bijective. Reason (R): A function is a bijective if it is both surjective and injective. (A) Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

If f(x) = |x| + |x - 1|, then which of the following is correct? (A) f(x) is both continuous and differentiable, at x = 0 and x = 1. (B) f(x) is differentiable but not continuous, at x = 0 and x = 1. (C) f(x) is continuous but not differentiable, at x = 0 and x = 1. (D) f(x) is neither continuous nor differentiable, at x = 0 and x = 1.

The feasible region of a linear programming problem with objective function Z = 5x + 7y is shown below [feasible region is the shaded polygon with corner points (0, 0), (0, 2), (3, 4) and (7, 0)]. The maximum value of Z - minimum value of Z is (A) 8 (B) 29 (C) 35 (D) 43

If for three matrices A = [a_ij]_m×4, B = [b_ij]_n×3 and C = [c_ij]_p×q, products AB and AC both are defined and are square matrices of same order, then value of m, n, p and q are: (A) m = q = 3 and n = p = 4 (B) m = 2, q = 3 and n = p = 4 (C) m = q = 4 and n = p = 3 (D) m = 4, p = 2 and n = q = 3

Identify the function shown in the graph [graph of an inverse trigonometric function with range from -pi/2 to pi/2 and domain shown between -1 and 1]. (A) sin^-1 x (B) sin^-1(2x) (C) sin^-1(x/2) (D) 2 sin^-1 x

If a + b + c = 0, |a| = sqrt(37), |b| = 3 and |c| = 4, then angle between b and c is (A) pi/6 (B) pi/4 (C) pi/3 (D) pi/2

The area bounded by the curve y = x|x|, x-axis and the ordinates x = -1 and x = 1 is given by (A) 0 (B) 1/3 (C) 2/3 (D) 4/3

If a function defined by f(x) = { kx + 1, x <= pi ; cos x, x > pi } is continuous at x = pi, then the value of k is (A) pi (B) -1/pi (C) 0 (D) -2/pi

The value of p for which vectors i^ + 2j^ + 3k^ and 2i^ - pj^ + k^ are perpendicular to each other is (A) 0 (B) 1 (C) 5/2 (D) -5/2

What is the general solution of the differential equation e^y' = x? (A) y = xlogx + c (B) y = xlogx - x + c (C) y = xlogx + x + c (D) y = x + c

The order and degree of the differential equation d/dx(e^y) = 0 respectively are (A) 0, 1 (B) 1, 1 (C) 2, 1 (D) 1, not defined

If the matrix A = [[0, r, -2], [3, p, t], [q, -4, 0]] is skew-symmetric, then value of (q+t)/(p+r) is.... (A) -2 (B) 0 (C) 1 (D) 2

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). 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)

The least value of f(x) = x³ - 12x, x in [0, 3] is (A) -16 (B) -9 (C) 0 (D) 16

If the direction cosines of a line are sqrt(3) k, sqrt(3) k, sqrt(3) k, then the value of k is : (A) +/- 1 (B) +/- sqrt(3) (C) +/- 3 (D) +/- 1/3

If 2 cos^-1 x = y, then (A) 0 <= y <= pi (B) -pi <= y <= pi (C) 0 <= y <= 2pi (D) -pi <= y <= 0

A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of 1 mm³/min. Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is 10 mm, if the semi-vertical angle of conical bottle is pi/6.

Evaluate: tan(sin^-1 1 - cos^-1(-1/2))

Find: integral (x-3)/(x-1)³ e^x dx

If y = log tan(pi/4 + x/2), then prove that dy/dx - sec x = 0

If sqrt(3)(x² + y²) = 4xy, then find dy/dx at (1/2, sqrt(3)/2).

Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.

(a) A relation R on set A = {1, 2, 3, 4, 5} is defined as R = {(x, y) : |x² - y²| < 8}. Check whether the relation R is reflexive, symmetric and transitive. OR (b) A function f is defined from R -> R as f(x) = ax + b, such that f(1) = 1 and f(2) = 3. Find function f(x). Hence, check whether function f(x) is one-one and onto or not.

Solve the following linear programming problem graphically: Minimize Z = 13x - 15y Subject to constraints x + y <= 7, 2x - 3y + 6 >= 0, x >= 0, y >= 0

Find: integral x²/((x² + 9)(x² + 16)) dx

Evaluate: integral from 0 to 1 of x(1 - x)ⁿ dx; (where n in N).

Find the general solution of the following differential equation: x² dy/dx = x² + xy + y²

If A = [[0,2,1],[-2,-1,-2],[1,-1,0]], find A^-1 and use it to solve the following system of equations: -2y + z = 7, 2x - y - z = 8, x - 2y = 10

Check whether the lines given by (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 4)/5 = (y - 1)/2 = z are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.

Find the image A' of the point A(1, 6, 3) in the line x/1 = (y - 1)/2 = (z - 2)/3. Also, find the equation of the line joining A and A'.

Solve the differential equation: y + d/dx(xy) = x(sin x + x)

Using integration, find the area of the smaller region enclosed by the curve 4x² + 4y² = 9 and the line 2x + 2y = 3.

[An online delivery company in a city has 5000 subscribers and collects annual subscription fees of Rs 300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that, for every increase of Rs 1, ten subscribers will discontinue. Assume that the company increased the annual fee by Rs x.] How many subscribers will discontinue after an increase of Rs x in annual fee?

[Case Study 3: three groups by screen time - high (>4 hrs) 60%, moderate (2-4 hrs) 30%, low (<2 hrs) 10%; anxiety/low-retention rates 80%, 70%, 30% respectively.] 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?