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

If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.

A relation R is defined on Z as aRb if and only if a² - 7ab + 6b² = 0. Then R is: (a) reflexive and symmetric (b) transitive but not reflexive (c) symmetric but not reflexive (d) reflexive but not symmetric

Let f(x) = 4 - (x - 7)³ be an invertible function, then find f^-1(x).

Write the smallest equivalence relation from the set A to A, where A = {1, 2, 3}.

If f: R -> R is defined by f(x) = (2x-7)/4, show that f(x) is one-one and onto.

Solve: sin^-1(x) + sin^-1(1 - x) = cos^-1 x.

The value of cosec(sin^-1(-1/2)) - sec(cos^-1(-1/2)) is equal to: (a) -4 (b) 0 (c) -1 (d) 4

For what value of x, is A = [[0, 1, -2], [-1, 0, 3], [x, -3, 0]] a skew symmetric matrix?

A ball is thrown vertically downwards from the top of a cliff, tracked from when first thrown until it hits the ground. The height above the ground after t seconds is S(t) = at² + bt + c, where a, b, c are real constants and S is in meters. After 1 second the ball is 285 m above the ground; after 2 seconds its height is 260 m; after 4 seconds it is 180 m. Set up and solve a matrix equation to find the height of the cliff.

The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of C²?

Find the value of the determinant given below, without expanding it at any stage. | beta gamma, 1, alpha(beta + gamma) ; gamma alpha, 1, beta(gamma + alpha) ; alpha beta, 1, gamma(alpha + beta) |

If y = x³ log(1/x), then prove that x d²y/dx² - 2 dy/dx + 3x² = 0.

Determine the value of k for which the following function is continuous at x = 3. f(x) = { ((x+3)² - 36)/(x-3), x != 3 ; k, x = 3 }

The surface of a spherical balloon is increasing at the rate of 4 cm²/sec. Find the rate of change of volume when its radius is 12 cm.

A person manufactured a water tank as a closed right circular cylinder of volume 539/2 cubic units with height h and radius r. (a) Express h in terms of radius r and given volume. (b) Express the total surface area S of the closed cylinder tank in terms of radius r. (c) If the total surface area of the tank is minimum, prove that radius r = 7/2 units. (d) Find the height of the tank.

Find the point on the curve y = (x - 2)² at which the tangent is parallel to the chord joining the end points (2, 0) and (4, 4).

A paper weight manufacturer designs a paper weight as a cylinder inscribed in a sphere and wants the volume of the cylinder to be maximum. The radius of the sphere is 3 sqrt(3) cm. Find the maximum volume of the cylinder.

Find the point on the curve y = 2x² - 6x - 4 at which the tangent is parallel to the x-axis.

Find the interval in which the function f(x) = x² e^-x is strictly increasing or decreasing.

Sumit bought a closed cylindrical dustbin of radius r cm and height h cm with volume 20pi cm³. (a) Express h in terms of r using the volume. (b) Prove the total surface area is 2pi r² + 40pi/r. (c) Cost of painting base and top is rupees 2 per cm² and curved side is rupees 25 per cm²; find the total cost in terms of r. (d) Calculate the minimum cost for painting the dustbin.

Using properties of definite integral, calculate the value of: \int₀^pi/2 sin² x / (1 + sin x cos x) dx

The value of \int₀^3/2 |x| dx is: (a) 1/8 (b) 9/8 (c) 9/4 (d) 3/4

Evaluate: \int 5/sqrt(2x+7) dx

Evaluate: \int (x² + x + 1)/((x + 2)(x² + 1)) dx

Evaluate: \int (log x)² dx

Evaluate: \int sqrt(sec(x/2) - 1) dx

If \int x⁵ cos(x⁶) dx = k sin(x⁶) + C, find the value of k.

Evaluate: \int e^x cosec x (1 - cot x) dx

Evaluate: \int (2x³ - 1)/(x⁴ + x) dx

Evaluate: \int sin x dx / (cos x (1 - sin x))

Solve the following differential equation: dx/dy = x/y + f(x/y)/f'(x/y)

Solve the following differential equation: (1 + x²) dy/dx + 2xy = 4x²

Show that the general solution of the differential equation dy/dx = y cot 2x is log y = (1/2) log|sin 2x| + C.

Find the particular solution for the following differential equation: sqrt(1 - y²) dx = (sin^-1 y - x) dy, given that y(0) = 0

Statement 1: If a and b represent two adjacent sides of a parallelogram then the diagonals are represented by a + b and a - b. Statement 2: If a and b represent two diagonals of a parallelogram then the adjacent sides are represented by 2(a + b) and 2(a - b). Which is correct? (a) Statement 1 is true and Statement 2 is false. (b) Statement 2 is true and Statement 1 is false. (c) Both the statements are true. (d) Both the statements are false.

Shown is a cuboid (with sides 2 cm, 3 cm, 4 cm). Find BA . BC.

What are the values of x for which the angle between the vectors 2x² i + 3x j + k and i - 2j + x² k is obtuse?

The projection of i + 2j - 3k on 2i - j + k is: (a) sqrt(3)/sqrt(2) (b) -sqrt(3)/sqrt(2) (c) -3/2 (d) -sqrt(3)/2

Find a vector of magnitude 20 units parallel to the vector 2i + 5j + 4k.

If a and b are two non-zero vectors such that |a x b| = a . b, find the angle between a and b.

If the direction cosines of a line are <1/c, 1/c, 1/c> then: (a) c > 0 (b) 0 < c < 1 (c) c = +/-sqrt(3) (d) c > 2

Find the equation of the plane containing the line x/(-2) = (y-1)/3 = (1-z)/1 and the point (-1, 0, 2).

Raunak sells sewing machines in a rural market with total capital of rupees 5,760 and a godown storing a maximum of 20 machines. Electronic sewing machine costs him rupees 360; manually operated costs him rupees 240. Profit is rupees 22 on an electronic machine and rupees 18 on a manually operated one. Using Linear Programming, find the number of sewing machines of each type Raunak should sell to maximise his profit.

A linear programming problem is given by Z = px + qy where p, q > 0 subject to the constraints x + y <= 60, 5x + y <= 100, x >= 0 and y >= 0. (a) Solve graphically to find the corner points of the feasible region. (b) If Z = px + qy is maximum at (0, 60) and (10, 50), find the relation of p and q. Also mention the number of optimal solution(s) in this case.

Maximise Z = 4x - 3y, subject to the constraints 2x + y <= 20, 3x - 2y >= 16, x <= 9, y >= 1. Also, find Z's maximum value.

In a large organisation, only 1 out of every 1,500 emails contains a harmful attachment. An automated filter correctly marks a harmful email as dangerous 97% of the time and correctly marks a safe email as safe 97% of the time. (a) Find the probability that the email is safe. (b) Find the probability that given the email is harmful the filter marks it safe. (c) One email has been flagged as dangerous by the system. Find the probability that this email is actually safe.

Kiran plays a game of throwing a fair die 3 times but quits as and when she gets a six. Kiran gets +1 point for a six and -1 for any other number. (i) If X denotes the random variable 'points earned', what are the possible values X can take? (ii) Find the probability distribution of X. (iii) Find the expected value of the points she gets.

Five numbers x1, x2, x3, x4, x5 are randomly selected from 1, 2, 3, ..., 18 and arranged in increasing order such that x1 < x2 < x3 < x4 < x5. What is the probability that x2 = 7 and x4 = 11? (a) 26/51 (b) 3/104 (c) 1/68 (d) 1/34

Given two events A and B such that P(A/B) = 0.25 and P(A ∩ B) = 0.12. The value P(A' ∩ B) is: (a) 0.36 (b) 0.48 (c) 0.88 (d) 0.036

There are three machines and 2 of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. What is the probability that only two tests are needed to identify the faulty machines?

A gardener plants saplings on a day when rain is not predicted. Forecast: probability of rain today is 0.4; if it rains today, probability of raining tomorrow is 0.8; if it does not rain today, probability it will rain tomorrow is 0.7. (i) What is the probability that he will not plant the saplings tomorrow? (ii) Find the probability that he will plant them tomorrow. (iii) Given that he does not plant them tomorrow, what is the probability that he did not plant them today? (iv) What is the probability that he can plant saplings on both the days?