List of practice Questions

Let \( X \) be a continuous random variable with the probability density function \[ f(x) = \frac{1}{3} x^2 e^{-x^2}, \quad x>0 \] Then the distribution of the random variable \[ W = 2X^2 \quad \text{is} \]
Let \( X \) be a continuous random variable with the probability density function \[ f(x) = \frac{e^x}{(1 + e^x)^2}, \quad -\infty<x<\infty \] Then \( E(X) \) and \( P(X>1) \), respectively, are
Let \( E \) and \( F \) be any two independent events with \( 0<P(E)<1 \) and \( 0<P(F)<1 \). Which one of the following statements is NOT TRUE?
Let \( P \) be a \( 3 \times 3 \) non-null real matrix. If there exists a \( 3 \times 2 \) real matrix \( Q \) and a \( 2 \times 3 \) real matrix \( R \) such that \( P = QR \), then
Let \( E, F \), and \( G \) be any three events with \( P(E) = 0.3 \), \( P(F|E) = 0.2 \), \( P(G|E) = 0.1 \). Then \( P(E - (F \cup G)) \) equals
The length of the curve \[ y = \frac{3}{4} x^{4/3} - \frac{3}{8} x^{2/3} + 7 \] from \( x = 1 \) to \( x = 8 \) equals
The volume of the solid generated by revolving the region bounded by the parabola \[ x = 2y^2 + 4 \quad \text{and the line} \quad x = 6 \quad \text{about the line} \quad x = 6 \] is
Evaluate the limit \[ \lim_{n \to \infty} \frac{1 + \frac{1}{2} + \dots + \frac{1}{n}}{(n + e^n)^{1/n} \log_e n} \]
A possible value of \( b \in \mathbb{R} \) for which the equation \[ x^4 + bx^3 + 1 = 0 \] has no real root is
Let \( X_1, X_2, \dots, X_n \) be a random sample from a \( N(\theta, 1) \) distribution. Instead of observing \( X_1, X_2, \dots, X_n \), we observe \( Y_i = e^{X_i}, i = 1, 2, \dots, n \). To test the hypothesis \[ H_0: \theta = 1 \quad \text{against} \quad H_1: \theta \neq 1 \] based on the random sample \( Y_1, Y_2, \dots, Y_n \), the rejection region of the likelihood ratio test is of the form, for some \( c_1<c_2 \),
The sum \[ \sum_{n=4}^{\infty} \frac{6}{n^2 - 4n + 3} \quad \text{equals} \]
Let \( X \) be a Poisson random variable and \( P(X = 1) + 2P(X = 0) = 12P(X = 2) \). Which one of the following statements is TRUE?
Let \( X_1, X_2, \dots \) be a sequence of i.i.d. discrete random variables with the probability mass function \[ P(X_1 = m) = \frac{(\log 2)^m}{2(m!)} \quad \text{for} \quad m = 0, 1, 2, \dots \] If \( S_n = X_1 + X_2 + \dots + X_n \), then which one of the following sequences of random variables converges to 0 in probability?
Let \( X_1, X_2, \dots, X_n \) be a random sample from a continuous distribution with the probability density function \[ f(x) = \frac{1}{2\sqrt{2\pi}} \left[ e^{-\frac{1}{2}(x - 2)^2} + e^{-\frac{1}{2}(x - 4)^2} \right], \quad -\infty<x<\infty \] If \( T_n = X_1 + X_2 + \dots + X_n \), then which one of the following is an unbiased estimator of \( \mu \)?
If \( F(x) = \int_{x}^{4} \sqrt{4 + t^2} \, dt \), for \( x \in \mathbb{R} \), then \( F'(1) \) equals
Two biased coins \( C_1 \) and \( C_2 \) have probabilities of getting heads \( \frac{2}{3} \) and \( \frac{3}{4} \), respectively. When tossed. If both coins are tossed independently two times each, then the probability of getting exactly two heads out of these four tosses is
Let \( \{x_n\}_{n \geq 1} \) be a sequence of positive real numbers. Which one of the following statements is always TRUE?
Consider the function \( f(x, y) = x^3 - 3xy^2, x, y \in \mathbb{R} \). Which one of the following statements is TRUE?
If \( y(x) = 2e^{2x} + e^{\beta x}, \beta \neq 2 \), is a solution of the differential equation \[ \frac{d^2 y}{dx^2} + \frac{dy}{dx} - 6y = 0, \] satisfying \( \frac{dy}{dx} (0) = 5 \), then \( y(0) \) is equal to
Let \( M \) and \( N \) be any two \( 4 \times 4 \) matrices with integer entries satisfying

\[ MN = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} = 2 \]

Then the maximum value of \( \det(M) + \det(N) \) is ...............
Let \( M \) be a \( 3 \times 3 \) matrix with real entries such that

\[ M^2 = M + 2I, \quad \text{where} \quad I \text{ denotes the } 3 \times 3 \text{ identity matrix.} \]

If \( \alpha, \beta, \gamma \) are eigenvalues of \( M \) such that \( \alpha\beta\gamma = -4, \) then \( \alpha + \beta + \gamma \) is equal to .............

Let \( y(x) = x v(x) \) be a solution of the differential equation \[ x^2 \frac{d^2y}{dx^2} - 3x \frac{dy}{dx} + 3y = 0. \] If \( v(0) = 0 \) and \( v(1) = 1, \) then \( v(-2) \) is equal to .................

If \( y(x) \) is the solution of the initial value problem \[ \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + 4y = 0, \quad y(0) = 2, \quad \frac{dy}{dx}(0) = 0, \] then \( y(\ln 2) \) is (round off to 2 decimal places) equal to ...............

If

\[ \begin{pmatrix} z \\ y \end{pmatrix} \]

is an eigenvector corresponding to a real eigenvalue of the matrix

\[ \begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & -4 \\ 0 & 1 & 3 \end{pmatrix} \]

then \( z - y \) is equal to .................
Let

\[ f(x, y) = \begin{cases} \frac{x^3 + y^3}{x^2 - y^2}, & x^2 - y^2 \neq 0 \\ 0, & x^2 - y^2 = 0 \end{cases} \]

Then the directional derivative of \( f \) at \( (0, 0) \) in the direction of \( \frac{4}{5} \hat{i} + \frac{3}{5} \hat{j} \) is ............