List of top Mathematics Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

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

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

Let $f$ be a function defined on $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ as \[ f(x) = \frac{\cos\left(\dfrac{\pi}{2} + |x|\right)}{\sin\left(\dfrac{\pi}{2} - |x|\right)}. \] Then,

An upstream paper mill dumps effluents in a river. The total benefit and total cost to the mill are $TB = 120Q - Q^2$ and $TC = 20Q$, respectively, where $Q$ is the amount of output it produces. The environmental cost due to the negative externality is $EC = Q^2$. The government wants to impose a production tax of $t$ per unit of output on the mill. The value of $t$ to achieve the socially optimal level of production is

Which one of the following statements is correct?

Let $\{x_1, x_2, \ldots, x_n\$ be the realization of a randomly drawn sample of size $n$ with sample mean $\bar{x}$, and let $k$ be a real number other than $\bar{x}$. Let $S_1$ and $S_2$ be the sums of squared deviations defined as} \[ S_1 = \sum_{i=1}^{n} (x_i - \bar{x})^2 \quad \text{and} \quad S_2 = \sum_{i=1}^{n} (x_i - k)^2 \] Then,

You have a budget of Rs. 4000 and would like to purchase LPG cylinders from a local seller who charges Rs. 50 per cylinder. The seller has a subsidy scheme by which if you return the empty cylinder purchased from him, you will get a refund of Rs. 20 per cylinder. You cannot borrow money from anyone. The maximum number of cylinders you can purchase is

The differential equation \[ (3x^2y + y^3)\,dx + (x^3 + 3xy^2)\,dy = 0 \] is