List of practice Questions

If \( x^y = e^{x-y} \), prove that \(\frac{dy}{dx} = \frac{\log x}{(1 + \log x)^{2}}\). 
 
If \( y = \cos^3(\sec^2 2t) \), find \( \frac{dy}{dt} \).
Show that \( f(x) = \frac{4 \sin x}{2 + \cos x} - x \) is an increasing function of \( x \) in \( \left[ 0, \frac{\pi}{2} \right] \).
Find the principal value of \( \tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(-\frac{1}{\sqrt{2}}\right) \).
Express \( \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) \), where \( -\frac{\pi}{2}<x<\frac{\pi}{2} \), in the simplest form.
The volume of a cube is increasing at the rate of \( 6 \, \text{cm}^3/\text{s} \). How fast is the surface area of the cube increasing, when the length of an edge is \( 8 \, \text{cm} \)?
Assertion (A): The relation \( R = \{(x, y) : (x + y) \text{ is a prime number and } x, y \in \mathbb{N}\} \) is not a reflexive relation.
Reason (R): The number \( 2n \) is composite for all natural numbers \( n \).
Assertion (A): The corner points of the bounded feasible region of a L.P.P. are shown below. The maximum value of \( Z = x + 2y \) occurs at infinite points. 
Reason (R): The optimal solution of a LPP having bounded feasible region must occur at corner points. 
The corner points
The function \( f(x) = kx - \sin x \) is strictly increasing for:
If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}| = 1 \), \( |\vec{b}| = 2 \), and \( \vec{a} \cdot \vec{b} = \sqrt{3} \), then the angle between \( 2\vec{a} \) and \( -\vec{b} \) is:
Let \( \vec{a} \) be any vector such that \( |\vec{a}| = a \). The value of \( |\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2 \) is:
The vectors \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \), \( \vec{b} = \hat{i} - 3\hat{j} - 5\hat{k} \), and \( \vec{c} = -3\hat{i} + 4\hat{j} + 4\hat{k} \) represent the sides of:
A vector perpendicular to the line \( \vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda (3\hat{i} - \hat{j}) \) is:
The integrating factor of the differential equation \( (x + 2y^2) \frac{dy}{dx} = y \, (y>0) \) is:} 
 
Given that \(( A)^{-1} = \frac{1}{7}\)\( \begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix}\) , matrix \( A \) is: 
 
If \(A = \begin{bmatrix} 2 & 1 \\ -4 & -2 \end{bmatrix}\), then the value of \( I - A + A^2 - A^3 + \ldots \) is: 
 
If \(\begin{bmatrix} a & c & 0 \\ b & d & 0 \\ 0 & 0 & 5 \end{bmatrix}\)is a scalar matrix, then the value of \( a + 2b + 3c + 4d \) is: 
 
The number of arbitrary constants in the particular solution of the differential equation \[ \log \left( \frac{dy}{dx} \right) = 3x + 4y; \quad y(0) = 0 \] is/are:
The value of \( \int_{-1}^1 |x| \, dx \) is:
The value of constant \( c \) that makes the function \( f \) defined by \(f(x) = \ x^2 - c^2\), \(\&\ \text{if } x<4\)\(cx + 20, \&\ \text{if } x \geq 4\)   continuous for all real numbers is:
If \( e^{x^2y} = c \), then \( \frac{dy}{dx} \) is:
If \(A = \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix}\) and \( A^2 - kA - 5I = 0 \), then the value of \( k \) is:
If \(A = [a_{ij}] = \begin{bmatrix} 2 & -1 & 5 \\ 1 & 3 & 2 \\ 5 & 0 & 4 \end{bmatrix}\) and \( c_{ij} \) is the cofactor of element \( a_{ij} \), then the value of \( a_{21}c_{11} + a_{22}c_{12} + a_{23}c_{13} \) is:
The probability distribution of a random variable \( X \) is: 
\[\begin{array}{c|c|c|c|c|c} \hline X & 0 & 1 & 2 & 3 & 4 \\ \hline P(X) & 0.1 & k & 2k & k & 0.1 \\ \hline \end{array}\]
 where \( k \) is some unknown constant. The probability that the random variable \( X \) takes the value 2 is: 
 
The maximum value of \( Z = 4x + y \) for a L.P.P. whose feasible region is given below is: 
feasible region