List of practice Questions

Let \( X \) and \( Y \) be random variables having chi-square distributions with 6 and 3 degrees of freedom respectively. Then, which of the following statements is TRUE?

If the series \(\sum_{n=1}^{\infty} a_n\) converges absolutely, then which of the following series diverges?

There are three urns labeled 1, 2, 3.
Urn 1: 2 white, 2 black; Urn 2: 1 white, 3 black; Urn 3: 3 white, 1 black.
Two coins are tossed independently, each with \(P(\text{head}) = 0.2\).
Urn 1 is selected if 2 heads occur, Urn 3 if 2 tails occur, otherwise Urn 2 is selected. A ball is drawn at random from the chosen urn. Find \[ P(\text{Urn 1 is selected} \mid \text{Ball drawn is white}) \]

Let \(X\) be a random variable with \[ f(x) = \frac{1}{2} e^{-|x|}, \quad -\infty<x<\infty \] Then, which of the following statements is FALSE?

Let \( \{X_n\}_{n \ge 1} \) be i.i.d. random variables distributed as \( N(0,1) \). Then find \[ \lim_{n \to \infty} P\left( \frac{\sum_{i=1}^{n} X_i^2 - 3n}{\sqrt{32n}} \le \sqrt{6} \right) \] is equal to:

Consider independent Bernoulli trials with success probability \( p = \frac{1}{3} \). The probability that three successes occur before four failures is:

Let \(X\) and \(Y\) be independent \(N(0,1)\) random variables and \(Z = \left|\frac{X}{Y}\right|\). Then, which of the following expectations is finite?

Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = x^7 + 5x^3 + 11x + 15 \). Then, which of the following statements is TRUE?

The value of the limit \[ \lim_{x \to 0} \frac{e^{-3x} - e^{x} + 4x}{5(1 - \cos x)} \] is equal to:

The value of the limit \[ \lim_{n \to \infty} \sum_{k=0}^{n} \binom{2n}{k} \frac{1}{4^n} \] is equal to:

Let \(X\) be a \(U(0,1)\) random variable and \(Y = X^2\). If \(\rho\) is the correlation coefficient between \(X\) and \(Y\), then \(48\rho^2\) is equal to:

The value of the limit \[ \lim_{n \to \infty} \left( \left(1 + \frac{1}{n}\right) \left(1 + \frac{2}{n}\right) \cdots \left(1 + \frac{n}{n}\right) \right)^{\frac{1}{n}} \] is equal to:

Define the sequence \[ s_n = \begin{cases} \dfrac{1}{2^n}\displaystyle\sum_{j=0}^{n-2} 2^{2j}, & \text{if } n \text{ is even and } n \gt 0, \\[8pt] \dfrac{1}{2^n}\displaystyle\sum_{j=0}^{n-1} 2^{2j}, & \text{if } n \text{ is odd and } n \gt 0. \end{cases} \] Define \[ \sigma_m = \frac{1}{m}\sum_{n=1}^{m} s_n. \] The number of limit points of the sequence \(\{\sigma_m\}\) is _________.

The determinant of the matrix \[ \begin{pmatrix} 2021 & 2020 & 2020 & 2020 \\ 2021 & 2021 & 2020 & 2020 \\ 2021 & 2021 & 2021 & 2020 \\ 2021 & 2021 & 2021 & 2021 \end{pmatrix} \] is _________.

Let \[ A = \begin{pmatrix} 2 & -1 & 3 \\ 2 & -1 & 3 \\ 3 & 2 & -1 \end{pmatrix}. \] Then the largest eigenvalue of \(A\) is _________.

Let \[ A = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}. \] Consider the linear map \(T_A : M_4(\mathbb{R}) \to M_4(\mathbb{R})\) defined by \[ T_A(X) = AX - XA. \] Then the dimension of the range of \(T_A\) is _________.

Consider the four functions from \(\mathbb{R}\) to \(\mathbb{R}\): \[ f_1(x) = x^4 + 3x^3 + 7x + 1, \quad f_2(x) = x^3 + 3x^2 + 4x, \quad f_3(x) = \arctan(x), \] and \[ f_4(x) = \begin{cases} x, & \text{if } x \notin \mathbb{Z}, \\ 0, & \text{if } x \in \mathbb{Z}. \end{cases} \] Which of the following subsets of \(\mathbb{R}\) are open?

Let \( A \) be an \( n \times n \) invertible matrix and \( C \) be an \( n \times n \) nilpotent matrix. If \[ X = \begin{pmatrix} X_{11} & X_{12} \\ X_{21} & X_{22} \end{pmatrix} \] is a \( 2n \times 2n \) matrix (each \( X_{ij} \) is \( n \times n \)) that commutes with the \( 2n \times 2n \) matrix \[ B = \begin{pmatrix} A & 0 \\ 0 & C \end{pmatrix}, \] then

Consider the function \[ f(x) = \begin{cases} 1, & \text{if } x \in (\mathbb{R} \setminus \mathbb{Q}) \cup \{0\}, \\ 1 - \frac{1}{p}, & \text{if } x = \frac{n}{p},\ n \in \mathbb{Z}\setminus\{0\},\ p \in \mathbb{N},\ \gcd(n,p)=1 . \end{cases} \] Then

The largest positive number \(a\) such that \[ \int_0^5 f(x)\,dx + \int_0^3 f^{-1}(x)\,dx \ge a \] for every strictly increasing surjective continuous function \(f : [0, \infty) \to [0, \infty)\) is _________.

The value of \[ \lim_{n \to \infty} \int_0^1 e^{x^2} \sin(nx)\,dx \] is _________.

Let \(S\) be the surface defined by \[ \{(x, y, z) \in \mathbb{R}^3 : z = 1 - x^2 - y^2,\, z \ge 0\}. \] Let \[ \vec{F} = -y\hat{i} + (x - 1)\hat{j} + z^2\hat{k}, \] and let \(\hat{n}\) be the continuous unit normal field to the surface \(S\) with positive \(z\)-component. Then the value of \[ \frac{1}{\pi} \iint_S (\nabla \times \vec{F}) \cdot \hat{n}\, dS \] is _________.

Let \(V\) be the real vector space of all continuous functions \( f : [0,2] \to \mathbb{R} \) such that the restriction of \(f\) to the interval \([0,1]\) is a polynomial of degree \(\le 2,\) the restriction of \(f\) to \([1,2]\) is a polynomial of degree \(\le 3,\) and \(f(0) = 0.\) Then the dimension of \(V\) is _________.

The number of group homomorphisms from the group \(\mathbb{Z}_4\) to the group \(S_3\) is _________.

Let \(y : \left(\frac{9}{10}, 3\right) \to \mathbb{R}\) be a differentiable function satisfying \[ (x - 2y)\frac{dy}{dx} + (2x + y) = 0, \quad x \in \left(\frac{9}{10}, 3\right), \] and \(y(1) = 1.\) Then \(y(2)\) equals _________.