List of practice Questions

Evaluate \( \int_{0}^{\pi/4} \frac{\cos^2 x}{\cos^2 x + 4 \sin^2 x} \, dx \):
The inverse of the matrix \[ \begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 3 \\ 5 & 2 & -1 \end{bmatrix} \] is:
The tangent at the point \( (x_1, y_1) \) on the curve \( y = x^3 + 3x^2 + 5 \) passes through the origin. Then \( (x_1, y_1) \) does NOT lie on the curve:
Let the following system of equations: $$ kx + y + z = 1, \quad x + ky + z = k, \quad x + y + kz = k^2 $$ have no solution. Find $|k|$:
If \( x \neq 0 \), then \[ \frac{\sin(\pi + x)\cos\left(\frac{\pi}{2} + x\right)\tan\left(\frac{3\pi}{2} - x\right)\cot(2\pi - x)}{\sin(2\pi - x)\cos(2\pi + x)\csc(-x)\sin\left(\frac{3\pi}{2} + x\right)} = \]
If \( x = \frac{1 - t^2}{1 + t^2} \) and \( y = \frac{2t}{1 + t^2} \), then \( \frac{dy}{dx} \) is equal to:
Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \, \vec{b} = \hat{i} + 3\hat{j} + 5\hat{k} \) and \( \vec{c} = 7\hat{i} + 9\hat{j} + 11\hat{k} \). Then the area of a parallelogram having diagonals \( \vec{a} + \vec{b} \) and \( \vec{b} + \vec{c} \) is:
The integral \( \int e^x \frac{2 + \sin 2x}{1 + \cos 2x} \, dx \) is equal to
If the vector equation of the line \[ \frac{x - 2}{2} = \frac{2y - 5}{-3} = z + 1, \] is given by: \[ \vec{r} = \left(2\hat{i} + \frac{5}{2}\hat{j} - \hat{k}\right) + \lambda\left(2\hat{i} - \frac{3}{2}\hat{j} + p\hat{k}\right), \] then \( p \) is equal to:
If \( y = \log_e \left[ e^{3x} \left( \frac{x - 4}{x + 3} \right)^{3/2} \right] \), then find \( \frac{dy}{dx} \):
Find the differential equation of the family of all circles, whose center lies on the x-axis and touches the y-axis at the origin.
The number of four-letter words that can be formed using the letters of the word "BARRACK" is:
The maximum value of \(\frac{\log(x)}{x}\) is:

If two numbers \( p \) and \( q \) are chosen randomly from the set \( \{1, 2, 3, 4\} \) with replacement, what is the probability that \( p^2 \geq 4q \)?

Let \( f(x) = \frac{2 - \sqrt{x + 4}}{\sin 2x}, \, x \neq 0 \). In order that \( f(x) \) is continuous at \( x = 0 \), \( f(0) \) is to be defined as:

If \( y = 5\cos x - 3\sin x \), then \( \frac{d^2y}{dx^2} + y \) equals:
If \( A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{bmatrix} \), then \( A^{-1} \) is:
If the statement \( p \leftrightarrow (q \rightarrow p) \) is false, then the true statement is:

Given, the function \( f(x) = \frac{a^x + a^{-x}}{2} \) (\( a > 2 \)), then \( f(x+y) + f(x-y) \) is equal to

The length of the perpendicular drawn from the point \((1, 2, 3)\) to the line \((X - 6)/3 = (y - 7)/2 = (Z - 7)/-2\) is:
If \( f(x) = |x| - |1| \), then points where \( f(x) \) is not differentiable, is/are:
The absolute maximum value of the function \( f(x) = 2x^3 - 3x^2 - 36x + 9 \) defined on \( [-3, 3] \) is
The integral of \( \sec^{2/3}x \csc^{4/3}x \, dx \) from \( \pi/6 \) to \( \pi/3 \) is equal to:
A student scores the following marks in five tests: 45, 54, 41, 57, 43. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is:
The distribution function \( F(X) \) of a discrete random variable \( X \) is given. Then \( P[X = 4] + P[X = 5] \):
\[ \begin{array}{|c|c|c|c|c|c|c|} \hline X & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline F(X = x) & 0.2 & 0.37 & 0.48 & 0.62 & 0.85 & 1 \\ \hline \end{array} \]