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

Let A be a 3 × 3 real matrix such that det(A) = 6 and
\(adj\ A=\begin{pmatrix} 1 & -1 & 2 \\ 5 & 7 & 1 \\ -1 & 1 & 1 \end{pmatrix}\)
where adj A denotes the adjoint of A.
Then the trace of A equals __________ (round off to 2 decimal places)

For n = 1, 2, 3, …, let the joint moment generating function of (X, Y_n) be
\(M_{X,Y_n}(t_1,t_2)=e^{\frac{t^2_1}{2}(1-2t_2)^{-\frac{n}{2}}}, t_1 \in \R,t_2 \lt \frac{1}{2}.\)
If \(T_n=\frac{\sqrt{n}X}{\sqrt{Y_n}},n \ge1,\) then which one of the following statements is true ?

Let 0.2, 1.2, 1.4, 0.3, 0.9, 0.7 be the observed values of a random sample of size 6 from a continuous distribution with the probability density function
\(f(x) = \begin{cases}    1, & 0< x \le \frac{1}{2} \\     \frac{1}{2\theta-1}, & \frac{1}{2} \lt x \le \theta \\ 0, & \text{otherwise,}\end{cases}\)
where θ > \(\frac{1}{2}\) is unknown. Then the maximum likelihood estimate and the method of moments estimate of θ, respectively, are

Let X and Y be two independent and identically distributed random variables having U(0, 1) distribution. Then P(X² < Y < X) equals __________ (round off to 2 decimal places)

Let X and Y be two independent random variables having N(0, \(σ^2_1\)) and N(0, \(σ^2_2\)) distributions, respectively, where 0 < σ₁ < σ₂. Then which of the following statements is/are true ?

Let X be a random variable denoting the amount of loss in a business. The moment generating function of X is
\(M(t)=(\frac{2}{2-t})^2,\ \ \ \ t< 2.\)
If an insurance policy pays 60% of the loss, then the variance of the amount paid by the insurance policy equals __________ (round off to 2 decimal places)

Let X be a random variable having binomial distribution with parameters n(> 1) and p(0 < p < 1). Then \(E(\frac{1}{1+X})\) equals

Let \(A=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{pmatrix}\). Then the sum of all the elements of A¹⁰⁰ equals

Let X be a random variable having the probability density function
\(f(x)=\begin{cases} ax^2+b, & 0 \le x \le 3\\ 0, & \text{otherwise,} \end{cases}\)
where a and b are real constants, and \(P(X \ge 2)=\frac{2}{3}.\)
Then E(X) equals __________ (round off to 2 decimal places)

Let X be a random variable with the moment generating function
\(M(t)=\frac{1}{(1-4t)^5},t \lt\frac{1}{4}.\)
Then the lower bounds for P(X < 40), using Chebyshev’s inequality and Markov’s inequality, respectively, are

Let {an}_n≥1 be a sequence of non-zero real numbers. Then which one of the following statements is true ?

Let (X, Y) be a random vector having bivariate normal distribution with parameters E(X) = 0, Var(X) = 1, E(Y) = −1, Var(Y) = 4 and ρ(X, Y) = \(-\frac{1}{2}\) , where ρ(X, Y) denotes the correlation coefficient between X and Y. Then P(X + Y > 1 | 2X − Y = 1) equals

Let \(A=\begin{pmatrix} 1 & -1 & 2 \\ -1 & 0 & 1 \\ 2 & 1 & 1 \end{pmatrix}\) and let \(\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\)be an eigenvector corresponding to the smallest eigenvalue of A, satisfying \(x^2_1+x^2_2+x^2_3=1\). Then the value of |x₁| + |x₂| + |x₃| equals __________ (round off to 2 decimal places)

Let (X, Y) be a discrete random vector. Then which of the following statements is/are true ?

Let {a_n}_n≥1 be a sequence of real numbers such that \(a_n=\frac{1}{3^n}\) for all n ≥ 1. Then which of the following statements is/are true ?

Let (X, Y) be a random vector having the joint probability density function
\(f(x,y)=\begin{cases} \frac{\sqrt2}{\sqrt{\pi}}e^{-2x}e^{-\frac{(y-x)^2}{2}}, & 0 <x< ∞,-∞<y<∞\\ 0,& \text{otherwise} \end{cases}\)
Then E(Y) equals

Let X₁ and X₂ be two independent and identically distributed discrete random variables having the probability mass function
\(f(x) = \begin{cases} (\frac{1}{2})^x, & x=1,2,3,... \\   0, & \text{otherwise. }  \end{cases}\)
Then P(min{X₁ , X₂} ≥ 5) equals

Let {a_n}_n≥1 be a sequence of real numbers such that a_1+5m = 2, a_2+5m = 3, a_3+5m = 4, a_4+5m = 5, a_5+5m = 6, m = 0, 1, 2, … . Then limsup_n→∞ a_n + liminf_n→∞ a_n equals __________

Let X₁ and X₂ be two independent and identically distributed random variables having \(X^2_2\) distribution and W = X₁ + X₂ . Then P(W > E(W)) equals __________ (round off to 2 decimal places)

Let (X, Y) be a random vector having the joint moment generating function
\(M(t_1,t_2)=(\frac{1}{2}e^{-t_1}+\frac{1}{2}e^{t_1})^2(\frac{1}{2}+\frac{1}{2}et_2)^2,\ \ (t_1,t_2)\in \R^2.\)
Then P( |X + Y| = 2) equals __________ (round off to 2 decimal places)

Let Ω = {1, 2, 3, … } be the sample space of a random experiment and suppose that all subsets of Ω are events. Further, let P be a probability function such that P({i}) > 0 for all i ∈ Ω. Then which of the following statements is/are true ?

A university bears the yearly medical expenses of each of its employees up to a maximum of Rs. 1000. If the yearly medical expenses of an employee exceed Rs. 1000, then the employee gets the excess amount from an insurance policy up to a maximum of Rs. 500. If the yearly medical expenses of a randomly selected employee has U(250, 1750) distribution and 𝑌 denotes the amount the employee gets from the insurance policy, then which of the following statements is/are true ?

Consider a sequence of independent Bernoulli trials, where \(\frac{3}{4}\) is the probability of success in each trial. Let X be a random variable defined as follows : If the first trial is a success, then X counts the number of failures before the next success. If the first trial is a failure, then X counts the number of successes before the next failure. Then 2E(X) equals __________

Let F : [0, 2] → \(\R\) be the function defined by
\(F(x)=\int_{x^2}^{x+2}e^{x[t]}dt,\)
where [t] denotes the greatest integer less than or equal to t. Then the value of the derivative of F at x = 1 equals

Let f : [1, 2] → \(\R\) be the function defined by
\(f(t)=\int^t_1\sqrt{x^2e^{x^2}-1}\ dx.\)
Then the arc length of the graph of f over the interval [1, 2] equals