List of top Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

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

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:

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?

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 \( a_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \) and \( b_n = \sum_{k=1}^{n} \frac{1}{k^2} \) for all \( n \in \mathbb{N} \). Then

For \( n \in \mathbb{N} \), let \( X_1, X_2, \ldots, X_n \) be a random sample from the \( F_{20,40} \) distribution. Then, as \( n \to \infty \), \( \frac{1}{n} \sum_{i=1}^n \frac{1}{X_i} \) converges in probability to __________ (round off to 2 decimal places).

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:

For some \( a \leq 0 \) and \( b \in \mathbb{R} \), let
\[A = \begin{pmatrix} 0 & a & b \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} \end{pmatrix}.\]
If \( A \) is an orthogonal matrix, then the value of \( a\sqrt{6} + 4b\sqrt{3} \) is equal to __________ (answer in integer).

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 \( x = 0 \):

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

A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals

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?

For \( n \geq 2 \), let \( X_1, X_2, \ldots, X_n \) be a random sample from a distribution with \( E(X_1) = 0 \), \( \text{Var}(X_1) = 1 \), and \( E(X_1^4) < \infty \). Let \[ \bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i \] and \[ S_n^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X}_n)^2. \]
Then which of the following statements is/are always correct?

Let 𝑋 be a continuous random variable having the 𝑈(−2, 3) distribution. Then which of the following statements is correct?

For \( n \in \mathbb{N} \), let \( X_1, X_2, \ldots, X_n \) be a random sample from the Cauchy distribution having probability density function
\[f(x) = \frac{1}{\pi(1 + x^2)}, \quad -\infty < x < \infty.\]
Let \( g: \mathbb{R} \to \mathbb{R} \) be defined by
\[g(x) = \begin{cases} x, & \text{if } -1000 \leq x \leq 1000 \\0, & \text{otherwise}\end{cases}.\]
Let
\[\alpha = \lim_{n \to \infty} P\left( \frac{1}{n^{3/4}} \sum_{i=1}^n g(X_i) > \frac{1}{2} \right).\]
Then \( 100\alpha \) is equal to __________ (answer in integer).

Consider a coin for which the probability of obtaining head in a single toss is \( \frac{1}{3} \). Sunita tosses the coin once. If head appears, she receives a random amount of \( X \) rupees, where \( X \) has the \( \text{Exp}\left( \frac{1}{9} \right) \) distribution. If tail appears, she loses a random amount of \( Y \) rupees, where \( Y \) has the \( \text{Exp}\left( \frac{1}{3} \right) \) distribution. Her expected gain (in rupees) is equal to __________ (answer in integer).

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 \( f: \mathbb{R} \to \mathbb{R} \) 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?

Consider four dice \( D_1, D_2, D_3, \) and \( D_4 \), each having six faces marked as follows:

Die	Marks on faces
\(D_1\)	4, 4, 4, 4, 0, 0
\(D_2\)	3, 3, 3, 3, 3, 3
\(D_3\)	6, 6, 2, 2, 2, 2
\(D_4\)	5, 5, 5, 1, 1, 1

In each roll of a die, each of its six faces is equally likely to occur. Suppose that each of these four dice is rolled once, and the marks on their upper faces are noted. Let the four rolls be independent. If \( X_i \) denotes the mark on the upper face of the die \( D_i \), \( i = 1, 2, 3, 4 \), then which of the following statements is/are correct?

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 \( f(x,y) = x^2 + 3y^2 - \frac{2}{3} xy \) at \( (x,y) \in \mathbb{R}^2 \). Then

Let \( f(x) = \int_{2-2x}^{x^2-1} e^{t^2-t} \, dt \) for all \( x \in \mathbb{R} \). If \( f \) is decreasing on \( (0, m) \) and increasing on \( (m, \infty) \), then the value of \( m \) is equal to __________ (answer in integer).

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

Let \( A = \begin{pmatrix} a & 0 \\ c & d \end{pmatrix} \) be a real matrix, where \( ad = 1 \) and \( c \ne 0 \). If \( A^{-1} + (\text{adj} \, A)^{-1} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \), then \( (\alpha, \beta, \gamma, \delta) \) is equal to