Chapter-wise PYQ
Class 12 Mathematics

Let A = {1, 2, 3}. Define a relation R on A that is reflexive and symmetric but not transitive. Show that your relation satisfies these properties.

Show that the function f: R → R defined by f(x) = 2x − 3 is bijective.

Let f: N → N be defined by f(n) = (n+1)/2 if n is odd, and n/2 if n is even. State whether f is bijective. Justify.

Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a − b} is an equivalence relation.

Prove that: tan⁻¹(1/2) + tan⁻¹(2/11) = tan⁻¹(3/4).

Write the principal value of: (i) cos⁻¹(−√3/2) (ii) sin⁻¹(sin 3π/5) (iii) tan⁻¹(tan 3π/4).

Solve for x: tan⁻¹(2x) + tan⁻¹(3x) = π/4.

Prove that: cos⁻¹(4/5) + cos⁻¹(12/13) = cos⁻¹(33/65).

If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], verify that (AB)ᵀ = BᵀAᵀ.

Express the matrix A = [[3, 5], [1, −1]] as the sum of a symmetric and a skew-symmetric matrix.

If A is a square matrix such that A² = A, find (I + A)³ − 7A.

Find the matrix X such that: [[1, 2], [3, 4]] · X = [[7, 8], [15, 16]].

If A = [[2, 3], [1, 4]], find A⁻¹ using the formula involving cofactors. Use A⁻¹ to solve: 2x + 3y = 8, x + 4y = 9.

Using properties of determinants, prove that: |a+b+2c, a, b; c, b+c+2a, b; c, a, c+a+2b| = 2(a+b+c)³.

If |x, 2; 18, x| = |6, 2; 18, 6|, find x.

Find the area of the triangle with vertices (2, 7), (1, 1), (10, 8) using determinants.

Find dy/dx if y = (sin x)^(cos x) + (cos x)^(sin x).

If x = a(cos θ + θ sin θ) and y = a(sin θ − θ cos θ), find d²y/dx².

Verify Rolle's theorem for f(x) = x² − 4x + 3 on [1, 3].

Find the value of k so that f(x) = kx + 1, x ≤ 5 and f(x) = 3x − 5, x > 5 is continuous at x = 5.

If y = cos⁻¹(x/a), prove that a²(1−x²)y₂ − xy₁ = 0.

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

Find the local maximum and minimum values of f(x) = 2x³ − 24x + 107 in the interval [1, 3]. Also find the absolute maximum and minimum.

Find the equation of the tangent to the curve y = x³ − 2x + 7 at the point (1, 6).

Show that the rectangle of maximum area that can be inscribed in a circle of radius r is a square.

The volume of a sphere is increasing at the rate of 8 cm³/s. Find the rate at which its surface area is increasing when the radius is 12 cm.

Evaluate: ∫ x² / (x² + 3x + 2) dx using partial fractions.

Evaluate: ∫ sin x / (sin x + cos x) dx using the property of definite integrals.

Evaluate: ∫₀^π x / (1 + sin x) dx.

Evaluate: ∫ √(4 − x²) dx using substitution.

Find ∫ e^x (sin x + cos x) dx.

Find the area of the region bounded by the curve y² = 4x, x = 1, x = 4, and the x-axis.

Find the area of the region enclosed between the parabola y² = 4ax and the line y = mx.

Using integration, find the area of the triangle whose vertices are (1, 3), (2, 5) and (3, 4).

Sketch the region bounded by y = |x| and y = 1 and find its area.

Find the general solution of the differential equation: dy/dx = (y + √(x² − y²)) / x.

Solve the differential equation: (x + 1) dy/dx = 2xy; y(2) = 3.

Find the integrating factor of: dy/dx + (2/x)y = x². Hence solve the equation.

Form a differential equation representing the family of curves y = a sin(x + b), where a and b are arbitrary constants.

If vectors a⃗ and b⃗ are such that |a⃗| = 3, |b⃗| = √(2/3) and a⃗ × b⃗ is a unit vector, find the angle between a⃗ and b⃗.

Find a unit vector perpendicular to each of the vectors a⃗ = 4î − ĵ + 3k̂ and b⃗ = 2î + ĵ − k̂.

Find the projection of the vector î − ĵ on the vector î + ĵ. Also find the angle between them.

Show that the points A, B, C with position vectors 2î − ĵ + k̂, î − 3ĵ − 5k̂ and 3î − 4ĵ − 4k̂ are the vertices of a right-angled triangle.

Find the shortest distance between the lines r⃗ = (î + ĵ) + λ(2î − ĵ + k̂) and r⃗ = (2î + ĵ − k̂) + μ(3î − 5ĵ + 2k̂).

Find the equation of the plane passing through the points (2, 1, 0), (3, −2, −2) and (3, 1, 7).

Find the image of the point (1, 6, 3) in the line x/1 = (y − 1)/2 = (z − 2)/3.

Find the angle between the line x/1 = (y − 1)/2 = (z + 1)/2 and the plane 2x + y − z = 4.

A manufacturer makes two types of toys A and B. Three machines are needed for this. The time required to manufacture a toy of type A on machines I, II, and III is 12, 18, and 6 minutes respectively. The time required to manufacture a toy of type B on machines I, II, and III is 6, 0, and 9 minutes respectively. Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is ₹7.50 and for type B ₹5, formulate the LPP and solve graphically to maximise profit.

Solve the following LPP graphically: Maximise Z = 5x + 3y subject to constraints 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.

Define feasible region and optimal solution in the context of LPP.

A bag contains 5 red and 3 black balls. Two balls are drawn at random without replacement. Find the probability that (i) both are red and (ii) one is red and one is black.

State and prove Bayes' theorem. A factory has three machines A, B, C which produce 25%, 35%, and 40% of items respectively. Of these, 5%, 4%, and 2% are defective. An item chosen at random is found defective. Find the probability that it was produced by machine B.

A random variable X has the probability distribution P(X = x) as given. Find the value of k and then find the mean and variance of X.

If P(A) = 1/3, P(B) = 1/4, and P(A ∩ B) = 1/6, find P(A|B), P(B|A), and P(A' | B').

In a binomial distribution, the mean is 4 and variance is 3. Find P(X ≥ 1).