List of practice Questions

Which one of the following prismatic crystal forms belongs to the hexagonal crystal system?
Which one of the following minerals exhibits luminescence when exposed to ultraviolet light?
In which one of the following mass extinction periods trilobites became extinct?
Let \[ f(x, y) = \frac{xz y}{x^2 + y^2 + z^2}, \quad (x, y) \neq (0, 0). \] Then \[ \frac{\partial f}{\partial x} \text{ and } f \text{ are bounded and unbounded.} \]
Let \( 0<a_1<b_1 \), For \( n \geq 1 \), define \[ a_{n+1} = \sqrt{a_n b_n} \quad \text{and} \quad b_{n+1} = \frac{a_n + b_n}{2}. \] Then which one of the following is NOT TRUE?
The area of the surface \[ z = \frac{xy}{3} \] intercepted by the cylinder \[ x^2 + y^2 \leq 16 \] lies in the interval \[ \left( 20\pi, 22\pi \right) \]
For \( a>0, b>0 \), let \[ \mathbf{F} = \frac{xj - yk}{b x^2 + a y^2}. \] Let \[ C = \{(x, y) \in \mathbb{R}^2 | x^2 + y^2 = a^2 + b^2\}. \] Then the line integral \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \]
If \( \lim_{t \to \infty} \int_0^t e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \), then \[ \lim_{t \to \infty} \int_0^t x^2 e^{-x^2} dx = \]
The flux of the vector field \[ \mathbf{F} = \left( \frac{2\pi x + 2x^2 y^2}{\pi} \right) \hat{i} + \left( \frac{2\pi x y - 4y}{\pi} \right) \hat{j} \] along the outward normal, across the ellipse \( x^2 + 16y^2 = 4 \) is equal to
The number of generators of the additive group \( \mathbb{Z}_{36} \) is equal to
Find the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{n} \sin \left( \frac{\pi}{2} + \frac{5\pi}{2} \cdot \frac{k}{n} \right) = \]
Evaluate the integral: \[ \int_0^1 \int_x^1 \sin(y^2) \, dy \, dx \]
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
Let \[ P = \int_0^1 \frac{dx}{\sqrt{8 - x^2 - x^3}}. \] Which of the following statements is TRUE?
Let \( X \) be a random variable having a probability density function \( f \in \{ f_0, f_1 \} \), where
\[ f_0(x) = \begin{cases} 1, & 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases} \quad \text{and} \quad f_1(x) = \begin{cases} \frac{1}{2}, & 0 \leq x \leq 2 \\ 0, & \text{otherwise} \end{cases} \] For testing the null hypothesis \( H_0 : f = f_0 \) against \( H_1 : f = f_1 \), based on a single observation on \( X \), the power of the most powerful test of size \( \alpha = 0.05 \) equals
If \[ \int_0^1 \int_0^{\sqrt{1 - (y - 1)^2}} f(x, y) \, dx \, dy \] equals \[ \int_0^1 \int_0^x f(x, y) \, dy \, dx, \] then \( \alpha(x) \) and \( \beta(x) \) are
Let \( f: [0, 1] \to \mathbb{R} \) be a function defined as
\[ f(t) = \begin{cases} t^3 \left( 1 + \frac{1}{5} \cos(\log(e^t)) \right), & \text{if } t \in (0,1] \\ 0, & \text{if } t = 0 \end{cases} \] Let \( F: [0, 1] \to \mathbb{R} \) be defined as
\[ F(x) = \int_0^x f(t) \, dt \] Then \( F''(0) \) equals
Consider the function \[ f(x, y) = x^3 - y^3 - 3x^2 + 3y^2 + 7, \, x, y \in \mathbb{R}. \] Then the local minimum (\( m \)) and the local maximum (\( M \)) of \( f \) are given by
Let \( X \) and \( Y \) be continuous random variables with the joint probability density function
\[ f(x, y) = \begin{cases} x + y, & \text{if } 0 < x < 1, 0 < y < 1 \\ 0, & \text{otherwise} \end{cases} \] Then \( P(X + Y > \frac{1}{2}) \) equals
Let \( x_1 = 1.1, x_2 = 0.5, x_3 = 1.4, x_4 = 1.2 \) be the observed values of a random sample of size four from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} e^{-\theta x}, & \text{if } x \geq \theta \\ 0, & \text{otherwise}, \quad \theta \in (-\infty, \infty) \end{cases} \] Then the maximum likelihood estimate of \( \theta^2 \) is
Let \( X \) be a discrete random variable with the probability mass function
\[ p(x) = k(1 + |x|)^2, \quad x = -2, -1, 0, 1, 2, \] where \( k \) is a real constant. Then \( P(X = 0) \) equals
Let \( \{X_n\}_{n \geq 1} \) be a sequence of i.i.d. random variables having common probability density function
\[ f(x) = \begin{cases} x e^{-x}, & x \geq 0 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i \), \( n = 1, 2, \dots \). Then \[ \lim_{n \to \infty} P(\bar{X}_n = 2) \] equals
Let \( X_1, X_2, X_3 \) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \frac{1}{\theta} e^{-x/\theta}, \quad x>0, \ \theta>0 \] Which of the following estimators of \( \theta \) has the smallest variance for all \( \theta>0 \)?
Player \( P_1 \) tosses 4 fair coins and player \( P_2 \) tosses a fair die independently of \( P_1 \). The probability that the number of heads observed is more than the number on the upper face of the die, equals
Let \( X_1 \) and \( X_2 \) be i.i.d. continuous random variables with the probability density function
\[ f(x) = \begin{cases} 6x(1 - x), & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \] Using Chebyshev's inequality, the lower bound of \( P \left( |X_1 + X_2 - 1| \leq \frac{1}{2} \right) \) is