Let $f : [-1,3] \to \mathbb{R}$ be a continuous function such that $f$ is differentiable on $(-1,3)$, $|f'(x)| \le \dfrac{3}{2}$ for all $x \in (-1,3)$, $f(-1) = 1$ and $f(3) = 7$. Then $f(1)$ equals .................
Find the rank of the matrix: \[ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 2 \\ 2 & 5 & 6 & 4 \\ 2 & 6 & 8 & 5 \end{bmatrix} \] Rank = ?
Consider a sequence of independent Bernoulli trials with probability of success in each trial being $\dfrac{1}{5}$. Then which of the following statements is/are TRUE?
For real constants $a$ and $b$, let \[ M = \begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} \\ a & b \end{bmatrix} \] be an orthogonal matrix. Then which of the following statements is/are always TRUE?
Let the sequence $\{x_n\}_{n \ge 1}$ be given by $x_n = \sin \dfrac{n\pi}{6}$, $n = 1, 2, \ldots$. Then which of the following statements is/are TRUE?
Let $X_1, X_2, \ldots, X_n$ be a random sample from a distribution with probability density function \[ f_{\theta}(x) = \begin{cases} \theta (1 - x)^{\theta - 1}, & 0 < x < 1, \\ 0, & \text{otherwise}, \end{cases} \theta > 0. \] To test $H_0: \theta = 1$ against $H_1: \theta > 1$, the uniformly most powerful (UMP) test of size $\alpha$ would reject $H_0$ if
Let $X_1, X_2, \ldots, X_n$ be a random sample from $U(\theta - 0.5, \theta + 0.5)$ distribution, where $\theta \in \mathbb{R}$. If $X_{(1)} = \min(X_1, X_2, \ldots, X_n)$ and $X_{(n)} = \max(X_1, X_2, \ldots, X_n)$, then which one of the following estimators is NOT a maximum likelihood estimator (MLE) of $\theta$?
Consider a sequence of independent Bernoulli trials with probability of success in each trial being $\dfrac{1}{3}$. Let $X$ denote the number of trials required to get the second success. Then $P(X \ge 5)$ equals
Let $M$ be an $n \times n$ non-zero skew symmetric matrix. Then the matrix $(I_n - M)(I_n + M)^{-1}$ is always
Let $T: \mathbb{R}^3 \to \mathbb{R}^4$ be a linear transformation. If $T(1,1,0) = (2,0,0,0)$, $T(1,0,1) = (2,4,0,0)$, and $T(0,1,1) = (0,0,2,0)$, then $T(1,1,1)$ equals
Let $\{a_n\}_{n \ge 1}$ be a sequence of real numbers such that $a_1 = 1, a_2 = 7$, and $a_{n+1} = \dfrac{a_n + a_{n-1}}{2}$, $n \ge 2$. Assuming that $\lim_{n \to \infty} a_n$ exists, the value of $\lim_{n \to \infty} a_n$ is
Let $X_1, X_2, \dots, X_n$ be i.i.d. random variables having $N(\mu, \sigma^2)$ distribution, where $\mu \in \mathbb{R}$ and $\sigma > 0$. Define \[ W = \frac{1}{2n^2} \sum_{i=1}^{n} \sum_{j=1}^{n} (X_i - X_j)^2. \] Then $W$, as an estimator of $\sigma^2$, is
Let $\{X_n\}_{n \ge 1}$ be a sequence of i.i.d. random variables such that $E(X_i) = 1$ and $\text{Var}(X_i) = 1$. Then the approximate distribution of $\dfrac{1}{\sqrt{n}} \sum_{i=1}^n (X_{2i} - X_{2i-1})$, for large $n$, is
Consider the following system of linear equations: \[ \begin{cases} ax + 2y + z = 0 \\ y + 5z = 1 \\ by - 5z = -1 \end{cases} \]
Which one of the following statements is TRUE?
For real constants $a$ and $b$, let \[ f(x) = \begin{cases} \frac{a \sin x - 2x}{x}, & x < 0 \\ bx, & x \ge 0 \end{cases} \]
If $f$ is a differentiable function, then the value of $a + b$ is
If $\{x_n\}_{n \ge 1}$ is a sequence of real numbers such that $\lim_{n \to \infty} \frac{x_n}{n} = 0.001$, then