List of practice Questions

Let $ x_1 = 1.1, x_2 = 2.2, x_3 = 3.3 $ be the observed values of a random sample of size three from a distribution with the probability density function \[ f(x; \theta) = \begin{cases} \dfrac{1}{\theta} e^{-x/\theta}, & x > 0, \\ 0, & \text{otherwise}, \end{cases} \] where $ \theta \in \Theta = \{ 1, 2, \dots \} $ is the unknown parameter. Then the maximum likelihood estimate of $ \theta $ equals .........

Let $ M = \sum_{i=1}^{4} X_i X_i^T $, where \[ X_1^T = [1 \ -1 \ 1 \ 0], X_2^T = [1 \ 1 \ 0 \ 1], X_3^T = [1 \ 3 \ 1 \ 0] \, \text{and} \, X_4^T = [1 \ 1 \ 1 \ 0]. \] Then the rank of $ M $ equals ...............

Let $ f : \mathbb{R} \to \mathbb{R} $ be a differentiable function such that $ f^\prime $ is continuous on $ \mathbb{R} $ with $ f^\prime(3) = 18 $. Define \[ g_n(x) = n \left( f \left( x + \frac{5}{n} \right) - f \left( x - \frac{2}{n} \right) \right). \] Then $ \lim_{n \to \infty} g_n(3) $ equals ...............

Let $ f : \mathbb{R} \to \mathbb{R} $ be a differentiable function with $ \lim_{x \to \infty} f(x) = \infty $ and $\lim_{x \to \infty} f'(x) = 2. $ Then \[ \lim_{x \to \infty} \left( 1 + \frac{f(x)}{x^2} \right)^x \] equals ..............

The value of \[ \int_0^\frac{\pi}{2} \left( \int_0^x e^{\sin y} \sin x \, dy \right) dx \] equals .............

Let $ X $ be a random variable with the probability density function \[ f(x) = \begin{cases} 4x^k, & 0 < x < 1, \\ x - \dfrac{x^2}{2}, & 1 \leq x < 2, \\ 0, & \text{otherwise}, \end{cases} \] where $ k $ is a positive integer. Then $ P\left( \frac{1}{2} \leq X \leq \frac{3}{2} \right) \textbf{ equals ............} $

Let $ X $ and $ Y $ be two discrete random variables with the joint moment generating function \[ M_{X,Y}(t_1, t_2) = \left( \frac{1}{3} e^{t_1} + \frac{2}{3} \right)^2 \left( \frac{2}{3} e^{t_2} + \frac{1}{3} \right)^3, t_1, t_2 \in \mathbb{R}. \] Then $ P(2X + 3Y > 1) $ equals .........

Let $ X_1, X_2, X_3 $ and $ X_4 $ be i.i.d. discrete random variables with the probability mass function

\[ P(X_1 = n) = \begin{cases} \dfrac{3^{n-1}}{4^n}, & n = 1, 2, \dots, \\ 0, & \text{otherwise}. \end{cases} \]

Then $ P(X_1 + X_2 + X_3 + X_4 = 6) $ equals ...............

What is the voltage gain in a common emitter amplifier, where input resistance is $3 \,\Omega$ and load resistance $24\,\Omega \, \beta = 0.6 $?

Infinite number of masses, each $1 \,kg$ are placed along the $x-$ axis at $x=\pm 1 m, \pm 2 m, \pm 4 m, \pm 8 m, \pm 16 \,m, \ldots$ the magnitude of the resultant gravitational potential in terms of gravitational constant $G$ at the origin $(x=0)$ is

If a stone of mass $0.05 \,kg$ is thrown out a window of a train moving at a constant speed of $100\, km/h$ then magnitude of the net force acting on the stone is

In the circuit shown, the symbols have their usual meanings. The cell has emf $E$. $X$ is initially joined to $Y$ for a long time. Then, $X$ is joined to $Z$. The maximum charge on $C$ at any later time will be

How many $\sigma$ bonds and $\pi$ bonds are present in

Which of the following element do not form complex with EDTA?

At a place, if the earth's horizontal and vertical components of magnetic fields are equal, then the angle of dip will be

Which of the following will have a meso-isomer also?

Which of the following is the correct and increasing order of lone pair of electrons on the central atom?

Let $ X $ be a random variable with the probability mass function

\[P(X = n) = \begin{cases} \dfrac{1}{10}, & n = 1,2,\dots,10, \\ 0, & \text{otherwise}.\end{cases}\]

Then $ E(\max\{X, 5\}) $ equals ............

Let $ X $ be a sample observation from $ U(\theta, \theta^2) $ distribution, where $ \theta \in \Theta = \{2, 3\} $ is the unknown parameter. For testing \[ H_0 : \theta = 2 \,\, \text{against} \,\, H_1 : \theta = 3, \] let $ \alpha $ and $ \beta $ be the size and power, respectively, of the test that rejects $ H_0 $ if and only if $ X \geq 3.5 $. Then $ \alpha + \beta $ equals ...............

A fair die is rolled four times independently. For $ i = 1, 2, 3, 4 $, define \[ Y_i = \begin{cases} 1, & \text{if 6 appears in the $ i $-th throw}, \\ 0, & \text{otherwise}. \end{cases} \] Then $ P(\max\{Y_1, Y_2, Y_3, Y_4\} = 1) $ equals ............

Let $ \mathbb{C}^* = \mathbb{C} \setminus \{0\} $ denote the group of non-zero complex numbers under multiplication. Suppose \[ Y_n = \{ z \in \mathbb{C} \mid z^n = 1 \}, n \in \mathbb{N}. \] Which of the following is (are) subgroup(s) of $ \mathbb{C}^* $?

Let $ P $ and $ Q $ be two non-empty disjoint subsets of $ \mathbb{R} $. Which of the following is (are) FALSE?

Suppose $ f, g, h $ are permutations of the set $ \{ \alpha, \beta, \gamma, \delta \} $, where
f interchanges $ \alpha $ and $ \beta $ but fixes $ \gamma $ and $ \delta $,
g interchanges $ \beta $ and $ \gamma $ but fixes $ \alpha $ and $ \delta $,
h interchanges $ \gamma $ and $ \delta $ but fixes $ \alpha $ and $ \beta $.
Which of the following permutations interchange(s) $ \alpha $ and $ \delta $ but fix(es) $ \beta $ and $ \gamma $?

The solution(s) of the differential equation $ \frac{dy}{dx} = (\sin 2x) y^{1/3}$ satisfying $y(0) = 0$ is(are)

Let $ f : \mathbb{R} \setminus \{0\} \to \mathbb{R} $ be defined by $ f(x) = x + \dfrac{1}{x^3} $. On which of the following interval(s) is $ f $ one-to-one?