List of top Questions asked in IIT JAM MS

Let $A$ be a $3 \times 3$ real matrix. Suppose that 1 and 2 are characteristic roots of $A$, and 12 is a characteristic root of $A + A^2$. Then which of the following statements is/are correct?

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(0) = 0$, $f(2) = 4$, $f(4) = 4$, and $f(8) = 12$. Then which of the following statements is/are correct?

Let $a_1 = 1$, $a_{n+1} = a_n \left( \frac{\sqrt{n} + \sin n}{n} \right)$, and $b_n = a_n^2$ for all $n \in \mathbb{N}$. Then which of the following statements is/are correct?

Let $X_1, X_2, \ldots, X_{10}$ be a random sample from a $N(0, \sigma^2)$ distribution, where $\sigma > 0$ is unknown. For testing $H_0: \sigma^2 \leq 1$ against $H_1: \sigma^2 > 1$, a test of size $\alpha = 0.05$ rejects $H_0$ if and only if $\sum_{i=1}^{10} X_i^2 > 18.307$. Let $\beta$ be the power of this test, at $\sigma^2 = 2$. Then $\beta$ lies in the interval
(You may use $\chi^2_{10,0.05} = 18.307$, $\chi^2_{10,0.1} = 15.9872$, $\chi^2_{10,0.25} = 12.5489$, $\chi^2_{10,0.5} = 9.3418$, $\chi^2_{10,0.75} = 6.7372$, $\chi^2_{10,0.9} = 4.8652$, $\chi^2_{10,0.95} = 3.9403$, $\chi^2_{10,0.975} = 3.247$)

Let $X_1, X_2, \ldots, X_n$ be a random sample from an $\text{Exp}(\lambda)$ distribution, where $\lambda \in \{1, 2\}$. For testing $H_0: \lambda = 1$ against $H_1: \lambda = 2$, the most powerful test of size $\alpha$, $\alpha \in (0,1)$, will reject $H_0$ if and only if

A biased coin, with probability of head as $p$, is tossed $m$ times independently. It is known that $p \in \left\{ \frac{1}{4}, \frac{3}{4} \right\}$ and $m \in \{3, 5\}.$ If 3 heads are observed in these $m$ tosses, then which of the following statements is correct?

For $n \geq 2$, let $\epsilon_1, \epsilon_2, \ldots, \epsilon_n$ be i.i.d. random variables having the $N(0,1)$ distribution. Consider $n$ independent random variables $Y_1, Y_2, \ldots, Y_n$ defined by $Y_i = \beta + \epsilon_i$, $i = 1,2, \ldots, n$, where $\beta \in \mathbb{R}$. Define
$$\bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i$$
$$T_1 = \frac{2\bar{Y}}{n+1}$$
$$T_2 = \frac{1}{n} \sum_{i=1}^{n} \frac{Y_i}{i}$$
Then which of the following statements is NOT correct?

Let $X_1, X_2, \ldots, X_{10}$ be a random sample from the $N(3,4)$ distribution, and let $Y_1, Y_2, \ldots, Y_{15}$ be a random sample from the $N(-3,6)$ distribution. Assume that the two samples are drawn independently. Define:
$$\bar{X} = \frac{1}{10} \sum_{i=1}^{10} X_i$$
$$\bar{Y} = \frac{1}{15} \sum_{j=1}^{15} Y_j$$
$$S = \sqrt{\frac{1}{9} \sum_{i=1}^{10} (X_i - \bar{X})^2}$$
Then the distribution of $U = \frac{\sqrt{5}(\bar{X} + \bar{Y})}{S}$ is:

For $n \in \mathbb{N}$, let $X_n$ be a random variable having the $\text{Bin}(n, \frac{1}{4})$ distribution. Then
$$\lim_{n \to \infty} \left[ P(X_n \leq 2n - \frac{\sqrt{3n}}{8}) + P\left(\frac{n}{6} \leq X_n \leq \frac{n}{3}\right) \right]$$
is equal to (You may use $\Phi(0.5) = 0.6915$, $\Phi(1) = 0.8413$, $\Phi(1.5) = 0.9332$, $\Phi(2) = 0.9772$).

Let $(X_1, Y_1), (X_2, Y_2), \ldots, (X_{20}, Y_{20})$ be a random sample from the $N_2(0, 0, 1, 1, \frac{3}{4})$ distribution. Define
$$\bar{X} = \frac{1}{20} \sum_{i=1}^{20} X_i \quad \text{and} \quad \bar{Y} = \frac{1}{20} \sum_{i=1}^{20} Y_i.$$
Then $\text{Var}(\bar{X} - \bar{Y})$ is equal to:

Let the random vector $(X, Y)$ have the joint probability density function
$$f(x, y) = \begin{cases} \frac{1}{x}, & \text{if } 0 < y < x < 1 \\0, & \text{otherwise} \end{cases}.$$
Then $\text{Cov}(X, Y)$ is equal to:

Suppose that the random variable $X$ has an $\text{Exp}\left(\frac{1}{5}\right)$ distribution, and for any $x > 0$, the conditional distribution of the random variable $Y$, given $X = x$, is $N(x, 2)$. Then $\text{Var}(X + Y)$ is equal to:

Let $X$ be a random variable such that $X$ and $-X$ have the same distribution. Let $Y = X^2$ be a continuous random variable with the probability density function
$$g(y) = \begin{cases} \frac{e^{-y/2}}{\sqrt{2\pi y}}, & \text{if } y > 0 \\0, & \text{if } y \leq 0 \end{cases}.$$
Then $E((X - 1)^4)$ is equal to:

Let $f_1(x)$ be the probability density function of the $N(0,1)$ distribution, and $f_2(x)$ be the probability density function of the $N(0, 6)$ distribution. Let $Y$ be a random variable with probability density function
$$f(x) = 0.6 f_1(x) + 0.4 f_2(x), \quad -\infty < x < \infty.$$
Then $\text{Var}(Y)$ is equal to:

Two fair coins $S_1$ and $S_2$ are tossed independently once. Let the events $E, F,$ and $G$ be defined as follows:
$E$: Head appears on $S_1$
$F$: Head appears on $S_2$  
$G$: The same outcome (head or tail) appears on both $S_1$ and $S_2$  
Then which of the following statements is NOT correct?

Let $\Omega = \{1,2,3,4,5,6\}$. Then which of the following classes of sets is an algebra?

Let $A$ be a $3 \times 5$ matrix defined by
$$A = \begin{pmatrix}0 & 1 & 3 & 1 & 2 \\1 & 6 & 2 & 3 & 4 \\1 & 8 & 8 & 5 & 8\end{pmatrix}.$$
Consider the system of linear equations given by
$$A \begin{pmatrix}x_1 \\x_2 \\x_3 \\x_4 \\x_5\end{pmatrix} = \begin{pmatrix}3 \\1 \\10\end{pmatrix},$$
where $x_1, x_2, x_3, x_4, x_5$ are real variables. Then

Consider the improper integrals
$$I_1 = \int_{1}^{\infty} \frac{t \sin t}{e^t} \, dt$$
and 
$$I_2 = \int_{1}^{\infty} \frac{1}{\sqrt{t}} \ln\left(1 + \frac{1}{t}\right) \, dt.$$
Then:

Let $f(x) = 4x^2 - \sin x + \cos 2x$ for all $x \in \mathbb{R}$. Determine the properties of the function $f$:

Let $f(x, y) = |xy| + x$ for all $(x, y) \in \mathbb{R}^2$. Determine the existence of the partial derivative of $f$ with respect to $x$ exists:

Let $f_i: \mathbb{R} \to \mathbb{R}$ for $i = 1, 2$ be defined as follows:
$$f_1(x) = \begin{cases} \sin\left(\frac{1}{x}\right) + \cos\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\0, & \text{if } x = 0 \end{cases}$$
and
$$f_2(x) = \begin{cases} x\left(\sin\left(\frac{1}{x}\right) + \cos\left(\frac{1}{x}\right)\right), & \text{if } x \neq 0 \\0, & \text{if } x = 0 \end{cases}$$
Then, determine the continuity of these functions at $x = 0$:

For $n \in \mathbb{N}$, let $$a_n = \sqrt{n} \sin^2\left(\frac{1}{n}\right) \cos n,$$ and $$b_n = \sqrt{n} \sin\left(\frac{1}{n^2}\right) \cos n.$$ Then

Let $x_1, x_2, x_3, x_4$ be the observed values from a random sample drawn from a $N(\mu, \sigma^2)$ distribution, where $\mu \in \mathbb{R}$ and $\sigma \in (0, \infty)$ are unknown parameters. Let $\bar{x}$ and $s = \sqrt{\frac{1}{3} \sum_{i=1}^{4} (x_i - \bar{x})^2}$ be the observed be the observed sample mean sample standard deviation,repectively. For testing the hypotheses $H_0: \mu = 0$ against $H_1: \mu \neq 0$, the likelihood ratio test of size $\alpha = 0.05$ rejects $H_0$ if and only if $$\frac{|\bar{x}|}{s} > k.$$ Then the value of $k$ is given by:

Let $X_1, X_2, \ldots, X_{20}$ be a random sample from the $N(5, 2)$ distribution, and let $Y_i = X_{2i} - X_{2i-1}$ for $i = 1, 2, \ldots, 10$. Then $W = \frac{1}{4} \sum_{i=1}^{10} Y_i^2$ has the distribution:

For $n \in \mathbb{N}$, let $Z_n$ be the smallest order statistic based on a random sample of size $n$ from the $U(0,1)$ distribution. Let $nZ_n \xrightarrow{d} Z$ as $n \to \infty$ for some random variable $Z$. Then $P(Z \leq \ln 3)$ is equal to: