Class 12 MathematicsImportant Questions for Board Exams

Show that the relation R in the set A = {1,2,3,4,5} defined as R = {(a,b): |a−b| is even} is an equivalence relation.

Let f: R→R defined by f(x) = 3x+5. Show f is bijective and find f⁻¹.

Find the principal value of sin⁻¹(−√3/2).

Prove: tan⁻¹(1/2) + tan⁻¹(1/3) = π/4.

Express the matrix A = [[2,−2,−4],[−1,3,4],[1,−2,−3]] as the sum of a symmetric and skew-symmetric matrix.

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

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

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

Find the intervals in which f(x) = 2x³ − 9x² + 12x − 5 is increasing or decreasing.

Find the dimensions of a rectangle with maximum area inscribed in a circle of radius r.

Evaluate: ∫(x² + 1)/(x⁴ + x² + 1) dx

Evaluate: ∫₀^π x/(1 + sin x) dx

Find the area enclosed by the ellipse x²/4 + y²/9 = 1.

Find the area of the region bounded by y² = 4x and the line x = 3.

Solve: (x + y)dy + (x − y)dx = 0

Solve: dy/dx + 2y = e^(2x) sin x

If a⃗ × b⃗ = c⃗ × d⃗ and a⃗ × c⃗ = b⃗ × d⃗, show that (a⃗ − d⃗) is parallel to (b⃗ − c⃗).

Find the shortest distance between the lines r⃗ = (1+λ)î + (2−λ)ĵ + (1+λ)k̂ and r⃗ = 2î − ĵ − k̂ + μ(2î + ĵ + 2k̂).

Maximise Z = 3x + 4y subject to x + y ≤ 4, x ≥ 0, y ≥ 0.

A factory makes tennis rackets and cricket bats. Formulate as LPP and solve graphically.

A card from a pack of 52 cards is lost. From the remaining 51 cards, two are drawn and both are found to be spades. Find the probability that the missing card is a spade.

Five cards are drawn from a pack of 52. Find the probability of getting 3 diamonds and 2 spades.