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

Consider the matrix $ A = \begin{bmatrix} 4 & -1 \\ 12 & -3 \end{bmatrix} $.
The value of the determinant of $ A^5 $ is .......... (rounded off to two decimal places).

You are tossing a fair coin repeatedly until a ‘Head’ appears. The expected number of tosses required for a ‘Head’ to appear is ...........

Anubhab faces a multiple-choice question (MCQ) with four alternative choices, where only one is the right choice. There is no negative mark for making a wrong choice. Also, assume that the probability that he knows the answer is 0.50. The probability of making the correct choice is 0.25, if he does not know the answer. He got full marks in this question.
The probability that he knew the answer is ......... (rounded off to one decimal place).

The value of the derivative of the function $ y = x e^x $ at $ x = 1 $ is ........... (rounded off to two decimal places).

A fair coin is tossed 3 times in succession. The probability of the event that ‘both first and second toss result in head’ is (rounded off to two decimal places).

Two cards are drawn from a full pack of 52 cards at random. The probability of getting a heart and a diamond is (rounded off to two decimal places).

The correlation coefficient between $x$ and $y$ using the following information is (rounded off to two decimal places). \[ \sum_{i=1}^{100} x_i = 280, \quad \sum_{i=1}^{100} y_i = 60, \quad \sum_{i=1}^{100} x_i^2 = 2384, \quad \sum_{i=1}^{100} y_i^2 = 117, \quad \sum_{i=1}^{100} x_i y_i = 438 \]

Which of the following is/are NOT TRUE?

In the context where sample mean of the dependent variable $ Y $ lies closer to the observed $ Y_1 $ than its least square predictor $ \hat{Y}_1 $, ............

Power of a statistical test is defined as

In statistical hypothesis testing, the area of non-rejection is defined as

Let $X_1, X_2, X_3, X_4$ be independent random variables following the standard normal distribution. Let $Y$ be defined as \[ Y = (X_1 + X_2)^2 + (X_3 + X_4)^2. \] Then the variance of $Y$ equals __________. (in integer)

For $\beta>0$, let the variables $x_1$ and $x_3$ be the optimal basic feasible solution of the linear programming problem \[ \text{maximize } \; z = x_1 + 2x_2 + 3x_3 \]

If the optimal value is 7, then $\beta$ equals ____________. (in integer)

The optimal value of the constrained optimization problem \[ \text{minimize } 2xy \quad \text{subject to } 9x^2 + 4y^2 \leq 36 \] is _____________. (in integer)

Let $y(x) > 0$ be a solution of the differential equation \[ \frac{dy}{dx} + y = y^2. \] If $y(\ln 2) = \dfrac{1}{3}$, where $\ln$ denotes the natural logarithmic function, then $y(\ln 3)$ equals ___________. (round off to 2 decimal places)

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)

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 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)

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

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