List of practice Questions

Let $ g(x) = f(x) + f(2 - x) $ for all $ x \in [0, 2] $, where $ f : [0, 2] \to \mathbb{R} $ is continuous on $ [0, 2] $ and twice differentiable on $ (0, 2) $. If $ g' $ denotes the first derivative of $ g $ and $ f'' $ denotes the second derivative of $ f $, then which of the following statements is NOT true?

Let $ X $ be a random variable with cumulative distribution function \[ F(x) = \begin{cases} 0 & \text{if } x < -1, \\ \frac{1}{4}(x + 1) & \text{if } -1 \leq x < 0, \\ \frac{1}{4}(x + 3) & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1 \end{cases} \] Which one of the following statements is true?

Let $ (X, Y) $ have joint probability mass function \[ p(x, y) = \begin{cases} \frac{c}{2x + y + 2} & \text{if } x = 0, 1, 2, \dots; \, y = 0, 1, 2, \dots; \, x \neq y, \\ 0 & \text{otherwise}. \end{cases} \] Then which one of the following statements is true?

Let $ X_1, X_2, \dots, X_{10} $ be a random sample of size 10 from a $ N_3(\mu, \Sigma) $ distribution, where $ \mu $ and non-singular $ \Sigma $ are unknown parameters. If \[ \overline{X_1} = \frac{1}{5} \sum_{i=1}^{5} X_i, \quad \overline{X_2} = \frac{1}{5} \sum_{i=6}^{10} X_i, \] \[ S_1 = \frac{1}{4} \sum_{i=1}^{5} (X_i - \overline{X_1})(X_i - \overline{X_1})^T, \quad S_2 = \frac{1}{4} \sum_{i=6}^{10} (X_i - \overline{X_2})(X_i - \overline{X_2})^T, \] then which one of the following statements is NOT true?

Let $ A $ be a $ 3 \times 3 $ real matrix such that \[ A \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ 0 \end{bmatrix}, \quad A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ 0 \end{bmatrix}, \quad A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 4 \end{bmatrix}. \] Then which of the following statements is/are true?

Let $ X $ be a random variable with probability density function \[ f(x) = \begin{cases} e^{-x} & \text{if } x \geq 0 \\ 0 & \text{otherwise} \end{cases} \] For $ a < b $, if $ U(a, b) $ denotes the uniform distribution over the interval $ (a, b) $, then which of the following statements is/are true?

Suppose that X is a discrete random variable with the following probability mass function: \[ P(X = 0) = \frac{1}{2}(1 + e^{-1}), P(X = k) = \frac{e^{-1}}{2 k!} for k = 1, 2, 3, .... \] Which of the following statements is/are true?

Suppose that $ U $ and $ V $ are two independent and identically distributed random variables each having probability density function \[ f(x) = \begin{cases} \lambda^2 x e^{-\lambda x} & \text{if } x > 0 \\ 0 & \text{otherwise}, \end{cases} \] where $ \lambda > 0 $. Which of the following statements is/are true?

Let $ (X, Y) $ have joint probability mass function \[ p(x, y) = \begin{cases} \frac{e^{-2}}{x! (y - x)!} & \text{if } x = 0, 1, 2, \dots, y; \, y = 0, 1, 2, \dots \\ 0 & \text{otherwise}. \end{cases} \] Then which of the following statements is/are true?

Let $ \{X_n\}_{n \geq 1} $ be a sequence of independent and identically distributed random variables with mean 0 and variance 1, all of them defined on the same probability space. For $ n = 1, 2, 3, \dots $, let \[ Y_n = \frac{1}{n} \left( X_1 X_2 + X_3 X_4 + \dots + X_{2n-1} X_{2n} \right). \] Then which one of the following statements is/are true?

Consider the orthonormal set \[ v_1 = \begin{bmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}, v_2 = \begin{bmatrix} \frac{1}{\sqrt{6}} \\ \frac{2}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} \end{bmatrix}, v_3 = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ \frac{1}{\sqrt{2}} \end{bmatrix} \] with respect to the standard inner product on $ \mathbb{R}^3 $. If $ u = \begin{bmatrix} a \\ b \\ c \end{bmatrix} $ is the vector such that inner products of $ u $ with $ v_1, v_2 $ and $ v_3 $ are 1, 2 and 3, respectively, then $ a^2 + b^2 + c^2 $ (in integer) equals ................

Let X be a random sample of size 1 from a population with cumulative distribution function \[ F(x) = \begin{cases} 0 & \text{if } x < 0, \\ 1 - (1 - x)^\theta & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1 \end{cases} \] where $ \theta > 0 $ is an unknown parameter. To test $ H_0 : \theta = 1 $ against $ H_1 : \theta = 2 $, consider using the critical region $ \{ x \in \mathbb{R} : x < 0.5 \} $. If $ \alpha $ and $ \beta $ denote the level and power of the test, respectively, then $ \alpha + \beta $ (rounded off to two decimal places) equals:

Let $ \{0.13, 0.12, 0.78, 0.51\} $ be a realization of a random sample of size 4 from a population with cumulative distribution function $ F(.) $. Consider testing $ H_0: F = F_0 $ against $ H_1: F \neq F_0 $, where \[ F_0(x) = \begin{cases} 0 & \text{if } x < 0, \\ x & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1. \end{cases} \] Let $ D $ denote the Kolmogorov-Smirnov test statistic. If $ P(D > 0.669) = 0.01 $ under $ H_0 $ and $ \psi = \begin{cases} 1 & \text{if } H_0 \text{ is accepted at level 0.01}, \\ 0 & \text{otherwise} \end{cases} $
then based on the given data, the observed value of } $ D + \psi $
(rounded off to two decimal places) equals ...............

Consider the probability space $(\Omega, \mathcal{G}, P)$, where $\Omega = [0,2]$ and $\mathcal{G} = \{\emptyset, \Omega, [0,1], (1,2]\}$. Let $X$ and $Y$ be two functions on $\Omega$ defined as} \[ X(\omega) = \begin{cases} 1 & \text{if } \omega \in [0,1] \\ 2 & \text{if } \omega \in (1,2] \end{cases} \] and \[ Y(\omega) = \begin{cases} 2 & \text{if } \omega \in [0,1.5] \\ 3 & \text{if } \omega \in (1.5,2] \end{cases} \] Then which one of the following statements is true?

Rafi told Mary, "I am thinking of watching a film this weekend."
The following reports the above statement in indirect speech:
Rafi told Mary that he _________ of watching a film that weekend.

Consider the following statements. (I) Let $ A $ and $ B $ be two $ n \times n $ real matrices. If $ B $ is invertible, then $ \text{rank}(BA) = \text{rank}(A) $. (II) Let $ A $ be an $ n \times n $ real matrix. If $ A^2x = b $ has a solution for every $ b \in \mathbb{R}^n $, then $ Ax = b $ also has a solution for every $ b \in \mathbb{R}^n $. Which of the above statements is/are true?

Which of the following show(s) dissociation between cognitive disorder and language abilities?

A square of side length 4 cm is given. The boundary of the shaded region is defined by one semi-circle on the top and two circular arcs at the bottom, each of radius 2 cm, as shown. The area of the shaded region is ............... cm$^2$.

Which one of the options can be inferred about the mean, median, and mode for the given probability distribution (i.e. probability mass function), $P(x)$, of a variable $x$?

A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line. The figure below consists of 20 unit squares arranged as shown. In addition to the given black squares, up to 5 more may be coloured black. Which one among the following options depicts the minimum number of boxes that must be coloured black to achieve two lines of symmetry? (The figure is representative)

In the given figure, PQRSTV is a regular hexagon with each side of length 5 cm. A circle is drawn with its centre at V such that it passes through P. What is the area (in cm$^2$) of the shaded region? (The diagram is representative)

Match the following historical sound changes in Column X with the processes in Column Y

Consider the following morphological break-up of pfeifing produced by a simultaneous bilingual child. What phenomenon does this example indicate?

What sociolinguistic phenomenon does the following sentence exemplify?