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

Let the random vector \( (X, Y) \) have the joint probability density function
\[f(x, y) = \begin{cases} \frac{1}{x}, & \text{if } 0 < y < x < 1 \\0, & \text{otherwise} \end{cases}.\]
Then \( \text{Cov}(X, Y) \) is equal to:

Let \(\theta_0\) and \(\theta_1\) be real constants such that \(\theta_1 > \theta_0\). Suppose that a random sample is taken from a \(N(\theta, 1)\) distribution, \(\theta \in \mathbb{R}\). For testing \(H_0: \theta = \theta_0\) against \(H_1: \theta = \theta_1\) at level 0.05, let \(\alpha\) and \(\beta\) denote the size and the power, respectively, of the most powerful test, \(\psi_0\). Then which of the following statements is/are correct?

Let \( f(x, y) = |xy| + x \) for all \( (x, y) \in \mathbb{R}^2 \). Determine the existence of the partial derivative of \( f \) with respect to \( x \) exists:

Let \( X \) and \( Y \) be continuous random variables having the joint probability density function
\[ f(x, y) = \begin{cases} e^{-x}, & \text{if } 0 \leq y < x < \infty \\0, & \text{otherwise}\end{cases} \]
Then which of the following statements is/are correct?

Let \( f(x) = \int_{2-2x}^{x^2-1} e^{t^2-t} \, dt \) for all \( x \in \mathbb{R} \). If \( f \) is decreasing on \( (0, m) \) and increasing on \( (m, \infty) \), then the value of \( m \) is equal to __________ (answer in integer).

Let \( f(x) = 4x^2 - \sin x + \cos 2x \) for all \( x \in \mathbb{R} \). Determine the properties of the function \( f \):

Let \( f_i: \mathbb{R} \to \mathbb{R} \) for \( i = 1, 2 \) be defined as follows:
\[f_1(x) = \begin{cases} \sin\left(\frac{1}{x}\right) + \cos\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\0, & \text{if } x = 0 \end{cases}\]
and
\[f_2(x) = \begin{cases} x\left(\sin\left(\frac{1}{x}\right) + \cos\left(\frac{1}{x}\right)\right), & \text{if } x \neq 0 \\0, & \text{if } x = 0 \end{cases}\]
Then, determine the continuity of these functions at \( x = 0 \):

Let \( V = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 \mid x_1 = x_2 \} \). Consider \( V \) as a subspace of \(\mathbb{R}^4\) over the real field. Then the dimension of \( V \) is equal to __________ (answer in integer).

Let \( f(x,y) = x^2 + 3y^2 - \frac{2}{3} xy \) at \( (x,y) \in \mathbb{R}^2 \). Then

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \( f(0) = 0 \), \( f(2) = 4 \), \( f(4) = 4 \), and \( f(8) = 12 \). Then which of the following statements is/are correct?

Let 𝑃 be a 3×3 matrix having the eigenvalues 1,1 and 2. Let (1,−1,2) ^𝑇 be the only linearly independent eigenvector corresponding to the eigenvalue 1. If the adjoint of the matrix 2𝑃 is denoted by 𝑄, then which of the following statements is/are TRUE?

If, for some 𝛼∈(0, ∞),
\(∫^∞_02^{-x^2}dx=a\sqrt\pi,\) 
then 𝛼 equals (round off to two decimal places)

Let 𝑀 be a 3×3 real matrix. If 𝑃=𝑀+𝑀^𝑇 and 𝑄=𝑀-𝑀^𝑇, then which of the following statements is/are always TRUE?

Let 𝑋 be a random variable having the probability density function
\(f(x) =\begin{cases}   \frac{x}{2}       & \quad \text{if }0<x<2,\\  0, & \quad Otherwise \end{cases}\)
Then, 𝑉𝑎𝑟 (ln ( \(\frac{2}{x}\) )) equals ________

Let 𝑋 be a random variable having the probability density function
\(f(x) =\begin{cases} \frac{5}{x^6}, & \quad \text{if }x>1,\\ 0, & \quad Otherwise. \end{cases}\)
Then, which of the following statements is/are TRUE for the distribution of 𝑋 ?

Let 𝑋 be a random variable such that the moment generating function of 𝑋 exists in a neighborhood of zero and
\(E(X^n)=(-1)^n\frac{2}{5}+\frac{2^{n+1}}{5}+\frac{1}{5},\) n=1,2,3,…..
Then, 𝑃 (|𝑋-\(\frac{1}{2}\)|>1) equals

Let 𝑋 be a random variable having a probability mass function 𝑝(𝑥) which is positive only for non-negative integers. If
𝑝(𝑥+1)=(\(\frac{ ln\,\, 3}{𝑥+1}\)) 𝑝(𝑥), 𝑥=0, 1, 2, … , 
then 𝑉𝑎𝑟(𝑋) equals

Let 𝑋 be a random variable having a probability density function
\(f(x; θ) =\begin{cases}  (3-θ){x^2-θ}      & \quad \text{if }0<x<1,\\  0, & \quad Otherwise \end{cases}\)
where 𝜃 ∈ {0, 1}. For testing the null hypothesis 𝐻₀ : 𝜃=0 against 𝐻₁ : 𝜃=1, the power of the most powerful test, at the level of significance 𝛼=0.125, equals

Let 𝑋₁~𝑁(2, 1), 𝑋₂~𝑁(−1,4) and 𝑋₃~𝑁(0, 1) be mutually independent random variables. Then, the probability that exactly two of these three random variables are less than 1, equals____(round off to two decimal places)

Let (𝑋, 𝑌) be a random vector having the joint probability density function

where 𝛼 is a positive constant. Then, 𝑃(𝑋 > 𝑌) equal

Among all the points on the sphere 𝑥²+𝑦²+𝑧²=24, the point (𝛼, 𝛽, 𝛾) is closest to the point (1, 2,−1). Then, the value of 𝛼+𝛽+𝛾 equals

Let 𝑥₁=2.1, 𝑥₂=4.2, 𝑥₃=5.8 and 𝑥₄=3.9 be the observed values of a random sample 𝑋₁, 𝑋₂, 𝑋₃ and 𝑋₄ from a population having a probability density function
\(f(x) =\begin{cases}   \frac{x}{θ^2}e^-\frac{x}{θ}     & \quad \text{if }x >0,\\  0, & \quad Otherwise \end{cases}\)
where 𝜃 ∈ (0, ∞) is an unknown parameter. Then, the maximum likelihood estimate of 𝑉𝑎𝑟(𝑋₁) equals ________

A point (𝑎, 𝑏) is chosen at random from the rectangular region [0, 2]×[0, 4]. Then, the probability that the area of the region
𝑅={(𝑥, 𝑦) ∈ ℝ × ℝ ∶ 𝑏𝑥 + 𝑎𝑦 ≤ 𝑎𝑏, 𝑥, 𝑦 ≥ 0} 
will be less than 2, equals

Let 𝑓: ℝ→ ℝ be a function defined by 𝑓(𝑥)=𝑥²−𝑥, 𝑥 ∈ ℝ. Let 𝑔: ℝ→ℝ be a twice differentiable function such that 𝑔(𝑥)=0 has exactly three distinct roots in the open interval (0, 1). Let ℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥), 𝑥 ∈ ℝ, and ℎ′′ be the second order derivative of the function ℎ. If 𝑛 is the number of roots of ℎ′′(𝑥)=0 in (0, 1), then the minimum possible value of 𝑛 equals ________

Let 𝑋₁, 𝑋₂,𝑋₁₀ be a random sample from an 𝑁(0, 𝜎² ) distribution, where 𝜎>0 is an unknown parameter. For some real constant 𝑐, let 𝑌=\(\frac{ 𝑐}{10} ∑^{10}_i=1 |𝑋_𝑖 |\) be an unbiased estimator of 𝜎 . Then, the value of 𝑐 equals ________ (round off to two decimal places)