CBSE Class 12 Mathematics Important Questions 2027 (Repeated in Board Exams)

A relation R is defined on Z, the set of integers, as R = {(x, y) : |x - y| is divisible by a prime number 'p', x, y in Z} check whether R is an equivalence relation or not.

A function f: R - {3/5} -> R - {3/5} is defined as f(x) = (3x + 2)/(5x - 3). Show that f is one-one and onto.

[On the set A = {1, 2, 3}, R₂ = {(1, 2), (1, 3), (3, 2)}.] What pairs should be added to the relation R₂ to make it an equivalence relation?

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

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

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

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

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

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

Value of the determinant |[cos 67°, sin 67°], [sin 23°, cos 23°]| is (A) 0 (B) 1/2 (C) sqrt(3)/2 (D) 1

A school wants to allocate students into three clubs: Sports, Music and Drama, under following conditions: The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. The number of students in Music club should be 20 more than half the number of students in Sports club. The total number of students to be allocated in all three clubs are 180. Find the number of students allocated to different clubs, using matrix method.

If A and B are non-singular matrices of same order with det(A) = 5, then det(B^-1AB)² is equal to (A) 5 (B) 5² (C) 5⁴ (D) 5⁵

Check whether function f(x) defined as f(x) = { |x - 3|/(2(x - 3)), x < 3 ; (x - 6)/6, x >= 3 } is continuous at x = 3 or not?

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

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

If x = a(theta - sin theta), y = a(1 - cos theta) find d²y/dx².

If x = cos t, y = cos mt, prove that (1 - x²) d²y/dx² - x dy/dx + m² y = 0.

If f(x) = x tan^-1 x, then f'(1) is equal to (A) pi/4 - 1/2 (B) pi/4 + 1/2 (C) -pi/4 - 1/2 (D) -pi/4 + 1/2

Differentiate 2^{cos2 x} w.r.t cos² x.

Assertion-Reason based question. Select the correct answer from the options: (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. Assertion (A): Consider the function defined as f(x) = |x| + |x - 1|, x in R. Then f(x) is not differentiable at x = 0 and x = 1. Reason (R): Suppose f be defined and continuous on (a,b) and c in (a,b), then f(x) is not differentiable at x = c if lim_h->0^- (f(c+h)-f(c))/h != lim_h->0^+ (f(c+h)-f(c))/h.

Differentiate the following function with respect to x: (cos x)^x; (where x in (0, pi/2)).

[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?

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.

[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.] Find the sub-intervals of (0, 5000) in which R(x) is increasing and decreasing.

[case context] The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is theta. (i) Express theta in terms of height of the camera installed on the pole and x.

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

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)

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

Evaluate: integral from 0 to 1 of log(1+x)/(1+x²) dx

Find: integral (x-3)/(x-1)³ e^x dx

Sketch the graph y = |x + 1|. Evaluate integral from -4 to 2 of |x + 1| dx. What does the value of this integral represent on the graph?

The value of integral_-1¹ x³/(x² + 2|x| + 1) dx is (A) 0 (B) log 2 (C) 2 log 2 (D) (1/2) log 2

If f(a+b-x) = f(x), then integral a to b of x f(x) dx is equal to (A) (a+b)/2 integral a to b f(b-x) dx (B) (a+b)/2 integral a to b f(a-x) dx (C) (b-a)/2 integral a to b f(x) dx (D) (a+b)/2 integral a to b f(x) dx

Evaluate: integral₀^pi/4 dx/(cos³ x sqrt(2 sin 2x))

Evaluate: integral₀¹ x tan^-1 x dx.

Find out the area of shaded region in the enclosed figure [region bounded by the parabola x² = y and the line y = 4, shaded between the y-axis and the parabola].

[Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such round-about is made such that equation representing its boundary is given by C₁ : x² + y² = 64. There is a circular pond with a fountain in the middle of the roundabout whose equation is given by C₂ : x² + y² = 4.] Represent the given equations C₁ and C₂ with the help of a diagram.

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

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

Find the particular solution of the differential equation xy dy/dx = (x + 2)(y + 2), given that y(1) = -1.

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

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

The integrating factor of differential equation R dx/dy + Px = Q where P, Q, R are functions of y is (A) e^{integral (P/Q) dy} (B) e^{integral P dy} (C) e^{integral (P/R) dy} (D) e^{integral (P/R) dx}

Which of the following is not a homogeneous function of x and y? (A) y² - xy (B) x - 3y (C) sin²(y/x) + y/x (D) tan x - sec y

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 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

The diagonals of a parallelogram are given by vector a = 2i^ - j^ + k^ and vector b = i^ + 3j^ - k^. Find the area of the parallelogram.

Find the vector of magnitude 14 in the direction of QP, where P and Q are the points (1, 3, 2) and (-1, 0, 8) respectively.

Position vectors of the points A, B and C as shown in the figure below are a->, b-> and c-> respectively. (A, B, C are collinear points shown in that order on a line.) If AC-> = (5/4) AB->, express c-> in terms of a-> and b->.

In the given figure, ABCD is a parallelogram. If AB-> = 2i^ - 4j^ + 5k^ and DB-> = 3i^ - 6j^ + 2k^, then find AD-> and hence find the area of parallelogram ABCD.

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'.

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

Assertion (A): Lines given by x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular to each other when pp' + rr' = 1. Reason (R): Two lines r = a₁ + lambda b₁ and r = a₂ + mu b₂ are perpendicular to each other if b₁ . b₂ = 0. (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 and Reason (R) is false. (D) Assertion (A) is false and Reason (R) is true.

Find a point P on the line (x + 5)/1 = (y + 3)/4 = (z - 6)/(-9) such that its distance from point Q(2, 4, -1) is 7 units. Also, find the equation of line joining P and Q.

Assertion (A) : Equation of a line passing through the points (1, 2, 3) and (3, -1, 3) is (x-3)/2 = (y+1)/3 = (z-3)/0. Reason (R) : Equation of a line passing through points (x₁, y₁, z₁), (x₂, y₂, z₂) is given by (x-x₁)/(x_2-x₁) = (y-y₁)/(y_2-y₁) = (z-z₁)/(z_2-z₁). (a) Both Assertion (A) and Reason (R) are true and 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 and Reason (R) is false. (d) Assertion (A) is false and Reason (R) is true.

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

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

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)

A linear programming problem (LPP) along with the graph of its constraints is shown below. The corresponding objective function is: Z = 18x + 10y, which has to be minimized. The smallest value of the objective function Z is 134 and is obtained at the corner point (3,8). (Corner points B1(0,20), B2(0,10), P(3,8), A1(5,0), A2(15,0); constraints 18x+10y=134, 4x+y=20, 2x+3y=30.) (Note: The figure is not to scale.) The optimal solution of the above linear programming problem ___________. (A) does not exist as the feasible region is unbounded. (B) does not exist as the inequality 18x + 10y < 134 does not have any point in common with the feasible region. (C) exists as the inequality 18x + 10y > 134 has infinitely many points in common with the feasible region. (D) exists as the inequality 18x + 10y < 134 does not have any point in common with the feasible region.

The degree of an objective function of a linear programming problem is (A) 0 (B) 1 (C) 2 (D) Any natural number

The number of corner points of the feasible region determined by constraints x >= 0, y >= 0, x + y >= 4 is : (A) 0 (B) 1 (C) 2 (D) 3

[case context] According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55%, 37% and 17% due to severe, moderate and light turbulence respectively. (i) Find the probability that an airplane reached its destination late.

[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?

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 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.

If E and F are two independent events such that P(E) = 2/3, P(F) = 3/7, then P(E/F̄) is equal to: (A) 1/6 (B) 1/2 (C) 2/3 (D) 7/9

For any two events A and B, if P(A_bar) = 1/2, P(B_bar) = 2/3 and P(A intersect B) = 1/4, then P(A_bar / B_bar) equals: (A) 3/8 (B) 8/9 (C) 5/8 (D) 1/4

Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number given that number appearing on the die being odd is 2/3. Reason (R): For any two events A and B, P(A|B) = P(A union B)/P(B). (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 and Reason (R) is false. (D) Assertion (A) is false and Reason (R) is true.