List of top Questions asked in IIT JAM MS

Let \( f : [0, \frac{\pi}{2}] \to \mathbb{R} \) be defined as \[ f(x) = ax + \beta \sin x, \] where \( a, \beta \in \mathbb{R} \). Let \( f \) have a local minimum at \( x = \frac{\pi}{4} \) with \[ f' \left( \frac{\pi}{4} \right) = -\frac{4}{\sqrt{2}}. \] Then \( 8\sqrt{2a} + 4 \beta \) equals

Let \( f(x) \) be a nonnegative differentiable function on \( [a, b] \subset \mathbb{R} \) such that \( f(a) = 0 = f(b) \) and \( |f'(x)| \leq 4 \). Let \( L_1 \) and \( L_2 \) be the straight lines given by the equations \[ y = 4(x - a) \quad \text{and} \quad y = -4(x - b), \text{ respectively. Then which of the following statements is (are) TRUE?} \]

Let \( E \) and \( F \) be two events with \( 0<P(E)<1 \), \( 0<P(F)<1 \) and \( P(E) + P(F) \geq 1 \). Which of the following statements is (are) TRUE?

The cumulative distribution function of a random variable \( X \) is given by
\[ F(x) = \begin{cases} 0, & \text{if } x < 0 \\ \frac{4}{9}, & \text{if } 0 \leq x < 1 \\ \frac{8}{9}, & \text{if } 1 \leq x < 2 \\ 1, & \text{if } x \geq 2 \end{cases} \] Which of the following statements is (are) TRUE?

Let \( X_1, X_2 \) be a random sample from a distribution with the probability mass function
\[ f(x|\theta) = \begin{cases} 1 - \theta, & \text{if } x = 0 \\ \theta, & \text{if } x = 1 \\ 0, & \text{otherwise}, \quad 0 < \theta < 1. \end{cases} \] Which of the following is (are) unbiased estimator(s) of \( \theta \)?

Let \( X_1, X_2, X_3 \) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} \frac{1}{\theta} e^{-x/\theta}, & \text{if } x > 0, \, \theta > 0 \\ 0, & \text{otherwise} \end{cases} \] If \( \hat{X}(X_1, X_2, X_3) \) is an unbiased estimator of \( \theta \), which of the following CANNOT be attained as a value of the variance of \( \hat{X} \) at \( \theta = 1 \)?

Let \( X_1, X_2, \dots, X_n \) (where \( n \geq 2 \)) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} \frac{x}{\theta^2} e^{-x/\theta}, & x > 0, \, \theta > 0 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \). Which of the following statistics is (are) sufficient but NOT complete?

Let \( \mathbf{v} \in \mathbb{R}^k \) with \( \mathbf{v}^T \mathbf{v} \neq 0 \). Let \[ P = I - 2 \frac{ \mathbf{v} \mathbf{v}^T }{ \mathbf{v}^T \mathbf{v} }, \] where \( I \) is the \( k \times k \) identity matrix. Then which of the following statements is (are) TRUE?

Let \( \{a_n\} \) and \( \{b_n\} \) be sequences of real numbers such that \( \{a_n\} \) is increasing and \( \{b_n\} \) is decreasing. Under which of the following conditions, the sequence \( \{a_n + b_n\} \) is always convergent?

Let \( f: [0,1] \to [0,1] \) be defined as follows:
\[ f(x) = \begin{cases} x, & \text{if } x \in \mathbb{Q} \cap [0,1] \\ x + \frac{2}{3}, & \text{if } x \in \mathbb{Q}^c \cap (0, \frac{1}{3}) \\ x - \frac{1}{3}, & \text{if } x \in \mathbb{Q}^c \cap (\frac{1}{3}, 1] \end{cases} \] Which of the following statements is (are) TRUE?

Let \( x(t), y(t), 1 \leq t \leq \pi \), be the curve defined by \[ x(t) = \int_1^t \frac{\cos z}{z^2} \, dz \quad \text{and} \quad y(t) = \int_1^t \frac{\sin z}{z^2} \, dz. \] Let \( L \) be the length of the arc of this curve from the origin to the point \( P \) on the curve at which the tangent is perpendicular to the x-axis. Then \( L \) equals

Let \( Q, A, B \) be matrices of order \( n \times n \) with real entries such that \( Q \) is orthogonal and \( A \) is invertible. Then the eigenvalues of \( Q^T A^{-1} B Q \) are always the same as those of

For \( c \in \mathbb{R} \), let the sequence \( \{ u_n \} \) be defined by \[ u_n = \frac{(1 + \frac{c}{n})^{n^2}}{(3 - \frac{1}{n})^n} \] Then the values of \( c \) for which the series \[ \sum_{n=1}^{\infty} u_n \] converges are

Let \( X_1, X_2, X_3, X_4 \) be a random sample from \( N(\theta_1, \sigma^2) \) distribution and \( Y_1, Y_2, Y_3, Y_4 \) be a random sample from \( N(\theta_2, \sigma^2) \) distribution, where \( \theta_1, \theta_2 \in (-\infty, \infty) \) and \( \sigma>0 \). Further suppose that the two random samples are independent. For testing the null hypothesis \( H_0 : \theta_1 = \theta_2 \) against the alternative hypothesis \( H_1 : \theta_1 \neq \theta_2 \), suppose that a test \( \psi \) rejects \( H_0 \) if and only if \( \sum_{i=1}^{4} X_i>\sum_{i=1}^{4} Y_i \). The power of the test \( \psi \) at \( \theta_1 = 1 + \sqrt{2}, \theta_2 = 1 \) and \( \sigma^2 = 4 \) is

Let \( X_1, X_2 \) be a random sample from an \( N(0, \theta) \) distribution, where \( \theta>0 \). Then the value of \( k \), for which the interval \[ \left( 0, \frac{X_1^2 + X_2^2}{k} \right) \] is a 95% confidence interval for \( \theta \), equals

Let \( x_1 = 2, x_2 = 1, x_3 = \sqrt{5}, x_4 = \sqrt{2} \) be the observed values of a random sample of size four from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} \frac{1}{2\theta}, & \text{if } -\theta \leq x \leq \theta \\ 0, & \text{otherwise}, \quad \theta > 0 \end{cases} \] Then the method of moments estimate of \( \theta \) is

Let \( \{X_n\}_{n \geq 1} \) be a sequence of i.i.d. random variables with the probability mass function
\[ f(x) = \begin{cases} \frac{1}{4}, & \text{if } x = 4 \\ \frac{3}{4}, & \text{if } x = 8 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i, n = 1, 2, \dots \). If \( \lim_{n \to \infty} P(m \leq \bar{X}_n \leq M) = 1 \), then possible values of \( m \) and \( M \) are

Let \( X_1, X_2, \dots, X_m, Y_1, Y_2, \dots, Y_n \) be i.i.d. \( N(0,1) \) random variables. Then \[ W = \frac{n \sum_{i=1}^m X_i^2}{m \sum_{j=1}^n Y_j^2} \] has

Let \( X \) and \( Y \) be continuous random variables with the joint probability density function
\[ f(x, y) = \begin{cases} c x (1 - x), & 0 < x < y < 1 \\ 0, & \text{otherwise} \end{cases} \] where \( c \) is a positive real constant. Then \( E(X) \) equals

Let \( X_1, X_2, X_3 \) be i.i.d. discrete random variables with the probability mass function
\[ p(k) = \left( \frac{2}{3} \right)^{k-1} \left( \frac{1}{3} \right), \quad k = 1, 2, 3, \dots \] Let \( Y = X_1 + X_2 + X_3 \). Then \( P(Y \geq 5) \) equals

Let the random variable \( X \) have uniform distribution on the interval \( \left( \frac{\pi}{6}, \frac{\pi}{2} \right) \). Then
\[ P(\cos X>\sin X) \] is

Let \( X \) be a continuous random variable with the probability density function
\[ f(x) = \begin{cases} x^3, & 0 < x \leq 1 \\ 3x^5, & x > 1 \\ 0, & \text{otherwise} \end{cases} \] Then \( P\left( \frac{1}{2} \leq X \leq 2 \right) \) equals

Let \( u_n = \frac{(4 - n)}{n} \), \( n \in \mathbb{N} \), and let \( l = \lim_{n \to \infty} u_n \).
Which of the following statements is TRUE?

The imaginary parts of the eigenvalues of the matrix
\[ P = \begin{pmatrix} 3 & 2 & 5 \\ 2 & -3 & 6 \\ 0 & 0 & -3 \end{pmatrix} \] are

If for a suitable \( \alpha>0 \), \[ \lim_{x \to 0} \left( \frac{1}{e^{2x} - 1} - \frac{1}{\alpha x} \right) \] exists and is equal to \( l \) (\( |l|<\infty \)), then \( \alpha = 2, l = -\frac{1}{2} \) is given by