List of top Questions asked in IIT JAM MS

Let {𝑎_𝑛 }_𝑛≥1 be a sequence such that 𝑎₁=1 and 4𝑎_𝑛+1=\(\sqrt{45 + 16𝑎_𝑛} ,𝑛\)=1, 2, 3, … . Then, which one of the following statements is TRUE?

Let the series 𝑆 and 𝑇 be defined by
\(∑^∞_{n-0}\frac{ 2 ⋅ 5 ⋅ 8 ⋯ (3𝑛 + 2) }{1 ⋅ 5 ⋅ 9 ⋯ (4𝑛 + 1)}\) and \(∑^∞ _{n=1}(1 + \frac{1 }{𝑛} ) ^{−𝑛^2}\)
respectively. Then, which one of the following statements is TRUE?

The volume of the region 𝑅={(𝑥, 𝑦, 𝑧) ∈ ℝ × ℝ × ℝ ∶ 𝑥 ² + 𝑦 ² ≤ 4, 0 ≤ 𝑧 ≤ 4 − 𝑦} is

For real constants 𝛼 and 𝛽, suppose that the system of linear equations
𝑥+2𝑦+3𝑧=6; x+ 𝑦 +𝛼𝑧 = 3; 2𝑦+𝑧=𝛽, 
has infinitely many solutions. Then, the value of 4𝛼 + 3𝛽 equals

Let M= \(\begin{pmatrix} 1 & -1 & 0\\ -a & 2 & -1 \\0&-1&1\end{pmatrix}\)If a non-zero vector 𝑋=(𝑥, 𝑦, 𝑧)^T∈ℝ³ satisfies 𝑀⁶𝑋=𝑋, then a subspace of ℝ3 that contains the vector 𝑋 is

Let 𝑥₁, 𝑥₂, 𝑥₃ and 𝑥₄ be observed values of a random sample from an 𝑁(𝜃, 𝜎 ² ) distribution, where 𝜃∈ℝ and 𝜎>0 are unknown parameters. Suppose that 𝑥̅=\(\frac{1}{4} ∑^4_{i=1} 𝑥_𝑖 = 3.6 \) and \(\frac{1}{3} ∑^4_{i=1} (𝑥_𝑖-\overline{x} )^2= 20.25\) . For testing the null hypothesis 𝐻₀ ∶ 𝜃=0 against 𝐻₁ ∶𝜃≠0, the 𝑝-value of the likelihood ratio test equals

Let 𝑀=𝑀₁𝑀₂, where 𝑀₁ and 𝑀₂ are two 3×3 distinct matrices. Consider the following two statements:
(I) The rows of 𝑀 are linear combinations of rows of 𝑀₂. 
(II) The columns of 𝑀 are linear combinations of columns of 𝑀₁. 
Then,

Let 𝑋~𝐹_6,2 and 𝑌~𝐹_2,6 . If 𝑃(𝑋 ≤ 2) =\( \frac{216}{343}\) and 𝑃 (𝑌≤\(\frac{1}{2}\) )=𝛼, then 686𝛼 equals

Let 𝑋 and 𝑌 be jointly distributed random variables such that, for every fixed 𝜆>0, the conditional distribution of 𝑋 given 𝑌=𝜆 is the Poisson distribution with mean 𝜆. If the distribution of 𝑌 is 𝐺𝑎𝑚𝑚𝑎 (2, \(\frac{1}{2}\) ), then the value of 𝑃(𝑋= 0)+𝑃(𝑋=1) equals

Let 𝑌~𝐹_4,2 . Then, 𝑃(𝑌≤2) equals

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, 𝑋₂, be a sequence of 𝑖.𝑖.𝑑. random variables each having 𝑈(0,1) distribution. Let 𝑌 be a random variable having the distribution function 𝐺. Suppose that
\(lim𝑛_{n→∞} \frac{( 𝑋-1+𝑋_2+𝑋_𝑛}{4} ≤ 𝑥)=𝐺(𝑥)\), for all 𝑥 ∈ ℝ . 
Then, 𝑉𝑎𝑟(𝑌) equals

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 𝑋₁, 𝑋₂, 𝑋₃ be a random sample from an 𝑁(𝜃, 1) distribution, where 𝜃 ∈ ℝ is an unknown parameter. Then, which one of the following conditional expectations does NOT depend on 𝜃 ?

Let 𝑋₁, 𝑋₂, 𝑋₃ be 𝑖. 𝑖. 𝑑. random variables, each having the 𝑁(0, 1) distribution. Then, which of the following statements is/are TRUE?

For the function 𝑓∶ℝ×ℝ→ℝ defined by 𝑓(𝑥, 𝑦)=2𝑥²-𝑥𝑦-3𝑦 ²-3𝑥+7𝑦 , the point (1,1) is

Let 𝑥₁, 𝑥₂, … , 𝑥₁₀ be the observed values of a random sample of size 10 from an 𝑁(𝜃, 𝜎 2 ) distribution, where 𝜃 ∈ ℝ and 𝜎 > 0 are unknown parameters. If
𝑥̅=\(\frac{1}{10}\) \(∑^{10}_{i=1}x_i=0\) and s=\(\sqrt{\frac{1}{9}∑^{10}_{i=1}(x_i-\overline{x})^2=2,}\)
then based on the values of 𝑥̅ and 𝑠 and using Student’s 𝑡-distribution with 9 degrees of freedom, 90% confidence interval(s) for 𝜃 is/are

Let 𝐸₁, 𝐸₂ and 𝐸₃ be three events such that 𝑃(𝐸₁∩𝐸₂ )=\(\frac{1}{4}\) , 𝑃(𝐸₁ ∩ 𝐸₃ )=𝑃(𝐸₂ ∩ 𝐸₃ )=\(\frac{1}{5}\) and 𝑃(𝐸₁ ∩ 𝐸₂ ∩ 𝐸₃ ) =\(\frac{1}{6}\) . Then, among the events 𝐸₁ 𝐸₂ and 𝐸₃, the probability that at least two events occur, equals

Let 𝑋 be a continuous random variable such that 𝑃(𝑋 ≥ 0)=1 and 𝑉𝑎𝑟(𝑋)<∞ . Then, 𝐸(𝑋 ²) is

Let (𝑋₁, 𝑋 ) be a random vector having the probability mass function

where 𝑐 is a real constant. Then, which of the following statements is/are TRUE?

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 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?

Let 𝑓:ℝ×ℝ→ℝ be a function defined by
\(f(x,y)=\begin{cases} \frac{xy(x+y}{x^2+y^2} & \quad \text{if }(x,y)≠(0,0),\\  0, & \quad if\,(x,y)=(0,0). \end{cases}\)
Then, which of the following statements is/are TRUE?

Let 𝑋1 and 𝑋2 be 𝑖. 𝑖. 𝑑. random variables having the common probability density function
\(f(x) =\begin{cases}   e^{-x}     & \quad x ≥0,\\  0, & \quad Otherwise \end{cases}\)
Define 𝑋₍₁₎=min{𝑋₁, 𝑋₂} and 𝑋₍₂₎ = max{𝑋₁, 𝑋₂}. Then, which one of the following statements is FALSE?

Let 𝑋 and 𝑌 be i.i.d. random variables each having the 𝑁(0, 1) distribution. Let 𝑈=\(\frac{𝑋}{𝑌}\) and 𝑍=|𝑈|. Then, which of the following statements is/are TRUE?