List of practice Questions

If \[ x \cos y = \sin(x + y), \] find \(\frac{dy}{dx}\).
Differentiate \[ \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \] with respect to \[ \sin^{-1} \left( \frac{2x}{1 + x^2} \right). \]
If \[ x = \sqrt{1 + t^2}, \quad y = \sqrt{1 - t^2}, \] then find \(\frac{dy}{dx}\).
If \[ y = x^{\sin x}, \] find \(\frac{dy}{dx}\).
If \( y = \sin(xy) \), then find \( \frac{dy}{dx} \):
Evaluate the integral: \[ \int (x+2)^2 \, dx = ? \]
Evaluate \( P(A \cup B) \) if \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A|B) = \frac{2}{5} \):
Prove that \[ 4\left(\cot^{-1} 3 + \cot^{-1} 2 \right) = \pi. \]
Prove that \[ \tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x+y}{1 - xy} \right), \] provided \( xy<1 \).
Find \( \frac{dy}{dx} \) if \[ y = \sqrt{\sin x^2}. \]
Find \( f \circ g \) and \( g \circ f \) if \[ f(x) = 8x^3 \quad \text{and} \quad g(x) = x^{1/3}. \]
Find the angle between the vectors \[ \mathbf{A} = 5\vec{i} + 3\vec{j} + 4\vec{k} \quad \text{and} \quad \mathbf{B} = 6\vec{i} - 8\vec{j} - \vec{k}. \]
If \( y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \ldots \text{ to } \infty}}} \), then \( \frac{dy}{dx} = \):
If \( y = x^{20} \), then \( \frac{d^2y}{dx^2} = \):
Evaluate the integral: \[ \int \sqrt{1 + \cos 2x} \, dx = ? \]
Evaluate the integral: \[ \int \frac{\log x}{x} \, dx = ? \]
Evaluate the integral: \[ \int \frac{\cos \sqrt{x}}{\sqrt{x}} \, dx = ? \]
Evaluate the integral: \[ \int \sqrt{\cos x} \cdot \sin x \, dx = ? \]
Evaluate the following matrix multiplication: \[ \begin{pmatrix} -3 & 5 & 2 \end{pmatrix} \times \begin{pmatrix} 1 \\ 6 \\ -4 \end{pmatrix} \]
Let \( A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \). Then, \( A^5 = \):
If \( A = \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix} \), then \( \text{adjoint } A = \):
Find \( \frac{d}{dx} \log (\sec x + \tan x) \):
Find \( \frac{d}{dx} \left( \sec^{-1} x + \csc^{-1} x \right) \):
If \( y = \tan^{-1} \left( \frac{1 - \cos x}{\sin x} \right) \), then \( \frac{dy}{dx} = \):
If \( x = a \sec \theta, \, y = b \tan \theta \), then \( \frac{dy}{dx} = \):