List of practice Questions

A consumer always spends 50% of his monthly income on food. Introduction of value added tax on food items has led to a 20% increase in food prices while his monthly income remained unchanged. The consumer’s price elasticity of demand for food is __________. (in integer)

Let $X_1 \sim N(\mu_1, \sigma_1^2)$ and $X_2 \sim N(\mu_2, \sigma_2^2)$ be two normally distributed random variables, where $\mu_1 = 2, \mu_2 = 3$ and $\sigma_1^2 = 4, \sigma_2^2 = 9$. The correlation coefficient between them is 0.5. The variance of the random variable $(X_1 + X_2)$ is ___________. (in integer)

In a small open economy, the desired domestic savings ($S^d$) and the desired domestic investment ($I^d$) are as follows, where $r^w$ is the world real interest rate. \[ S^d = 10 + 100r^w, \quad I^d = 15 - 100r^w \] If $r^w = 3%$, the current account balance in the equilibrium would be ___________. (in integer)

Consider the first order difference equation \[ x_n = \left( \frac{n + 1}{n} \right) x_{n - 1}, \quad n = 1, 2, 3, \ldots \] If $x_0 = 2$, then $x_{100} - x_{50}$ equals __________. (in integer)

The value of the integral \[ \int_{0}^{9} \frac{x - 1}{1 + \sqrt{x}} \, dx \] is _____________. (in integer)

The values of normalized indices for a country are as follows.

Dimension	Value of normalized index
Standard of living	0.4
Education	0.2
Health	0.8

Following the current UNDP methodology, the value of Human Development Index (HDI) for the country is __________. (round off to 1 decimal place)

The amount of money a gambler can win in a casino is determined by three independent rolls of a six-faced fair dice. The gambler wins Rs. 800 if he gets three sixes, Rs. 400 if he gets two sixes, and Rs. 100 in the event of getting only one six. The gambler does not win or lose any money in all other possible outcomes. The probability that a gambler will win at least Rs. 400 is ___________. (round off to 2 decimal places)

Consider an economy where the full employment output is 1 trillion Rupees and the natural rate of unemployment is 6%. If actual unemployment rate is 8%, then according to Okun's law, the absolute gap between the full employment output and actual output (in billion Rupees) will be __________. (in integer)

Let $k \in \mathbb{R}$. Which of the following statements is/are correct for the roots of the quadratic equation \[ x^2 + 2(k + 1)x + 9k - 5 = 0 \] ?

Let $x, y \in \mathbb{R}$ and the matrix \[ M = \begin{bmatrix} x + y & x - y \\ x - y & x + y \end{bmatrix}. \] Also, let $\text{adj}(M)$ be the adjoint and $\det(M)$ be the determinant of the matrix $M$. If \[ M \begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}, \] then

There are two firms in an oligopolistic industry competing in prices and selling a homogeneous product. Total cost of production for firm $i$ is \[ C_i(q_i) = 10q_i, \quad i = 1, 2; \] where $q_i$ is the quantity produced by firm $i$. Suppose firm $i$ sets price $p_i$ and firm $j$ sets price $p_j$. The market demand faced by firm $i$ is given by \[ q_i(p_i, p_j) = \begin{cases} 100 - p_i, & \text{if } p_i < p_j, \\ 0, & \text{if } p_i > p_j, \\ \dfrac{100 - p_i}{2}, & \text{if } p_i = p_j, \end{cases} \] for all $i, j = 1,2$ and $i \neq j$. Price can only take integer values in this market. Nash equilibrium/equilibria is/are given by

For any two sets $S_1, S_2 \subseteq \mathbb{R}$, define the set $S_1 - S_2 = \{x \in S_1, x \notin S_2\}$. Let \[ P = \{x \in \mathbb{R} : x^2 - 2x - 3 \le 0\} \quad \text{and} \quad Q = \{x \in \mathbb{R} : \log_5(1 + x^2) \le 1\}. \] Then,

Let $X$ and $Y$ be two random variables with the joint probability density function \[ f_{X,Y}(x,y) = \begin{cases} 6xy, & 0 < y \le \sqrt{x} \le 1, \\ 0, & \text{otherwise.} \end{cases} \] Then, the conditional probability $P(Y \ge \tfrac{1}{3} \mid X = \tfrac{2}{3})$ is

Let $X$ and $Y$ be two independent random variables with the cumulative distribution functions \[ F_X(x) = 1 - \left(\frac{3}{4}\right)^x, \quad x = 1,2,3,\ldots \] \[ F_Y(y) = 1 - \left(\frac{2}{3}\right)^y, \quad y = 1,2,3,\ldots \] respectively. Let $Z = \min\{X, Y\}$. Then, the probability $P(Z \ge 6)$ is

Let $(x_1^* = 1, \ x_2^* = 0, \ x_3^* = 2)$ be an optimal solution of the linear programming problem \[ \text{Minimize } \quad x_1 + 5x_2 + 2x_3 \] subject to \[ \begin{cases} x_1 - x_2 \le 1, \\ x_1 + x_2 + x_3 \ge 3, \\ x_1, x_2, x_3 \ge 0. \end{cases} \] If $(\lambda_1^*, \lambda_2^*)$ is an optimal solution of its dual, then

Let $f, g : \mathbb{R} \to \mathbb{R}$ be defined by \[ f(x) = x e^{-x} \quad \text{and} \quad g(x) = x|x|. \] Then, on $\mathbb{R}$,

Let $W$ be a subspace of the vector space $\mathbb{R}^3$ over the field $\mathbb{R}$ spanned by

Which one of the following vectors lies in $W$?

Let $X$ be a uniformly distributed random variable in $[a, b]$. The values of an independently drawn sample of size five from $X$ are given by 1.3, 0.8, 9.5, 20.2, 8.2. Let $\hat{a}$ and $\hat{b}$ denote the Maximum Likelihood Estimates for the parameters $a$ and $b$, respectively. Then,

Let $X$ be a uniformly distributed random variable in $[0, b]$. If the critical region for testing the null hypothesis $H_0: b = 2$ against the alternative $H_A: b \ne 2$ is $\{x \le 0.1 \text{ or } x \ge 1.9\}$, where $x$ is the value of a single draw of $X$, then the probability of Type-I error is

Let $\bar{\alpha}_1$ and $\bar{\alpha}_2$ be two independent unbiased estimators of the parameter $\alpha$ with standard errors $\sigma_1$ and $\sigma_2$, respectively, with $\sigma_1 \ne \sigma_2$. The linear combination of $\bar{\alpha}_1$ and $\bar{\alpha}_2$ that yields an unbiased estimator of $\alpha$ with the minimum variance is

Consider a Solow growth model without technological progress. The production function is \[ Y_t = K_t^{\alpha} N_t^{1-\alpha}, \] where $Y_t$, $K_t$, and $N_t$ are aggregate output, capital, and population at time $t$, respectively. The population grows at a constant rate $g_N>0$, savings rate is constant at $s \in (0,1)$, and capital depreciates at a constant rate $\delta \ge 0$. Denote per capita capital as \[ k_t = \frac{K_t}{N_t}, \] and define the steady state as a situation where $k_{t+1} = k_t = k^*$, where $k^*$ is a positive constant. Suppose the population growth rate exogenously increases to $g'_N$. At the new steady state, the aggregate output will grow at a rate

Let $\|.\|$ and $\langle .,.\rangle$ denote the standard norm and inner product in $\mathbb{R}^n$, respectively. If $u, v \in \mathbb{R}^3$ such that $\|u\| = \|v\| = 2$ and the angle between $u$ and $v$ is $\pi/3$, then

There are two sellers, $H$ and $L$, in a second-hand goods market where product quality varies. The sellers know the quality of their own product but the buyers cannot distinguish the product quality without further information. Sellers’ valuation of their own product is based on the quality. $H$ is willing to sell his product with quality $Q_H$ at a price $P_H$ per unit and $L$ is willing to sell the product with quality $Q_L$ at a price $P_L$ per unit such that \[ Q_H>Q_L \quad \text{and} \quad P_H>P_L. \] This market will suffer from

Which of the following statements is/are correct about the Indian economy during the colonial period?

In the context of Expectations Augmented Phillips Curve (EAPC), which of the following statements is/are correct?