List of practice Questions

Show that: \[ \frac{d}{dx} \left(|x|\right) = \frac{x}{|x|}, \quad x \neq 0. \]
Evaluate: \[ \int_{-2}^{2} \sqrt{\frac{2 - x}{2 + x}} \, dx. \]
If \( y = \csc(\cot^{-1} x) \), then prove that \( \sqrt{1 + x^2} \frac{dy}{dx} - x = 0 \).
If \( x = e^{\cos 3t} \) and \( y = e^{\sin 3t} \), prove that \( \frac{dy}{dx} = -\frac{y \log x}{x \log y} \).
Find the domain of the function \( f(x) = \sin^{-1(x^2 - 4) \). Also, find its range.}
If \( f(x) = |\tan 2x| \), then find the value of \( f'(x) \) at \( x = \frac{\pi}{3} \).
Assertion (A): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \).
Reason (R): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \).
If a line makes an angle of \( 30^\circ \) with the positive direction of \( x \)-axis, \( 120^\circ \) with the positive direction of \( y \)-axis, then the angle which it makes with the positive direction of \( z \)-axis is:
The unit vector perpendicular to both vectors \( \hat{i} + \hat{k} \) and \( \hat{i} - \hat{k} \) is:
Direction ratios of a vector parallel to the line \( \frac{x - 1}{2} = -y = \frac{2z + 1}{6} \) are:
Let \( E \) be an event of a sample space \( S \) of an experiment, then \( P(S | E) \) is:
The derivative of \( \tan^{-1}(x^2) \) w.r.t. \( x \) is:
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
The common region determined by all the constraints of a linear programming problem is called:
The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:
\( \int_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} \, dx \) is equal to:
Let \( f : \mathbb{R}_+ \to [-5, \infty) \) be defined as \( f(x) = 9x^2 + 6x - 5 \), where \( \mathbb{R}_+ \) is the set of all non-negative real numbers. Then, \( f \) is:
If \( \begin{vmatrix} -a & b & c
a & -b & c
a & b & -c \end{vmatrix} = kabc \), then the value of \( k \) is:
If \( A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} \) and \( A^{-1} = \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix} \), find the value of \((a + x) - (b + y)\).
Evaluate: \[ \int_{-2}^{2} \frac{x^3 + |x| + 1}{x^2 + 4|x| + 4} \, dx. \]
Using integration, find the area of the ellipse: \[ \frac{x^2}{16} + \frac{y^2}{4} = 1, \] included between the lines \(x = -2\) and \(x = 2\).
Solve the following differential equation: \[ (\tan^{-1}y - x) \, dy = (1 + y^2) \, dx. \]
Find the equation of a line \( l_2 \) which is the mirror image of the line \( l_1 \) with respect to line \( l : \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \), given that line \( l_1 \) passes through the point \( P(1, 6, 3) \) and is parallel to line \( l \).
A relation \( R \) on set \( A = \{1, 2, 3, 4, 5\} \) is defined as \[ R = \{(x, y) : |x^2 - y^2| <8\}. \] Check whether the relation \( R \) is reflexive, symmetric, and transitive.
If \( \sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y) \), prove that \( \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} \).