CBSE Class 12 · Mathematics

Algebra and Calculus

19 practice questions, each with the answer and an explanation.

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Let $R$ be a binary relation defined on a non-empty set $A$ . Under which condition is $R$ classified as an equivalence relation?
What is the principal value branch of the inverse trigonometric function $f (x) = sin^{- 1} x$ ?
What is the exact principal value of $cos^{- 1} (- \frac{1}{2}) + 2 sin^{- 1} (\frac{1}{2})$ ?
For two conformable square matrices $A$ and $B$ , which of the following expressions is always identical to $(A B)^{T}$ ?
What is the value of all main diagonal elements in any real skew-symmetric matrix $A$ ?
If $A$ is an invertible square matrix of order $3 \times 3$ with determinant $∣ A ∣ = 4$ , what is the value of $∣2 A ∣$ ?
If $A$ is a non-singular square matrix of order $3 \times 3$ with determinant $∣ A ∣ = 5$ , what is the value of $∣ adj (A) ∣$ ?
What is the derivative with respect to $x$ of the function $y = x^{x}$ for $x > 0$ ?
If $x = a cos^{3} θ$ and $y = a sin^{3} θ$ , what is $\frac{d y}{d x}$ at $θ = \frac{π}{4}$ ?
The radius of a spherical bubble is increasing at the rate of $0.5 cm/s$ . At what rate is its volume increasing when the radius is $10 cm$ ?
On which interval is the function $f (x) = 2 x^{3} - 9 x^{2} + 12 x + 5$ strictly increasing?
What is the value of the definite integral $I = \int_{0}^{π /2} \frac{sin ^{3} x}{sin ^{3} x + cos ^{3} x} d x$ ?
What is the value of the definite integral $\int_{- π /2}^{π /2} x^{3} cos x d x$ ?
What is the area of the region bounded by the parabola $y^{2} = 4 x$ and the line $y = 2 x$ ?
What are the order and degree of the differential equation $(\frac{d ^{2} y}{d x ^{2}})^{3} + (\frac{d y}{d x})^{4} + sin (\frac{d y}{d x}) = 0$ ?
What is the integrating factor (I.F.) for the linear first-order differential equation $\frac{d y}{d x} + \frac{2}{x} y = x^{2}$ for $x > 0$ ?
If the two vectors $a = 2 \hat{i} + 3 \hat{j} + λ \hat{k}$ and $b = 4 \hat{i} - 2 \hat{j} + 2 \hat{k}$ are mutually perpendicula…
What is the scalar projection of the vector $a = 2 \hat{i} + 3 \hat{j} + 2 \hat{k}$ on the vector…
If vectors $a$ and $b$ represent the adjacent sides of a parallelogram such that $a = \hat{i} - \hat{j} + 3 \hat{k}$ and…

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Algebra and Calculus

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