List of practice Questions

Find: \[ \int \frac{1}{x \left[(\log x)^2 - 3 \log x - 4\right]} \, dx. \]
Find the particular solution of the differential equation given by: \[ 2xy + y^2 - 2x^2 \frac{dy}{dx} = 0; \quad y = 2, \, \text{when } x = 1. \]
(a) Find the value of \( \sin^{-1}\left( -\frac{1}{2} \right) + \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) + \cot^{-1}\left( \tan \frac{4\pi}{3} \right) \).
Let \( E \) be an event of a sample space \( S \) of an experiment, then \( P(S | E) \) is:
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 \( \vec{b} \), \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \).
Determine whether the function \( f(x) = x^2 - 6x + 3 \) is increasing or decreasing in \( [4, 6] \).
(a) Find the value of \( \sin^{-1}\left( -\frac{1}{2} \right) + \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) + \cot^{-1}\left( \tan \frac{4\pi}{3} \right) \).
The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:
The anti-derivative of \( \sqrt{1 + \sin 2x}, \, x \in \left[ 0, \frac{\pi}{4} \right] \) is:
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
The coordinates of the foot of the perpendicular drawn from the point \( (0, 1, 2) \) on the \( x \)-axis are given by:
The common region determined by all the constraints of a linear programming problem is called:
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:
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:
Find the value of \( \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left[\sin\left(-\frac{\pi}{2}\right)\right]. \)
If \( A = \begin{bmatrix} 1 & \cot x \\ -\cot x & 1 \end{bmatrix} \), show that \( A^T A^{-1} = \begin{bmatrix} -\cos 2x & -\sin 2x \\ \sin 2x & -\cos 2x \end{bmatrix}. \)
If \( M \) and \( m \) denote the local maximum and local minimum values of the function \( f(x) = x + \frac{1}{x} \) (\( x \neq 0 \)) respectively, find the value of \( M - m \).
If \( F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \) and \( [F(x)]^2 = F(kx) \), then the value of \( k \) is:
Find the value of \( \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left[\sin\left(-\frac{\pi}{2}\right)\right]. \)
Find the domain of the function \( f(x) = \sin^{-1}(x^2 - 4) \). Also, find its range.

If \( y = A \sin 2x + B \cos 2x \) and \[ \frac{d^2y}{dx^2} - k y = 0, \text{ find the value of } k. \]

Show that a function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \frac{2x}{1 + x^2} \) is neither one-one nor onto. Further, find set \( A \) so that the given function \( f : \mathbb{R} \to A \) becomes an onto function.
(a) If \[ A = \begin{bmatrix} 5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1 \end{bmatrix}, \quad B^{-1} = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}, \] find \( (AB)^{-1} \). Also, find \( |AB|^{-1} \).
(b) Given \[ A = \begin{bmatrix} 2 & 3 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2 \end{bmatrix}, \text{ find } A^{-1}. \text{ Use it to solve the following system of equations:} \] \[ x + y + z = 1 \] \[ 2x + 3y + 2z = 2 \] \[ x + y + 2z = 4. \]
A relation \( R \) is defined on \( \mathbb{N} \times \mathbb{N} \) (where \( \mathbb{N} \) is the set of natural numbers) as: \[ (a, b) \, R \, (c, d) \iff a - c = b - d. \] Show that \( R \) is an equivalence relation.