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

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

The value of the integral
\[\int_{0}^{\sqrt{8}} \int_{x^2}^{\sqrt{8 - x^2}} \frac{3\sqrt{x^2 + y^2}}{\sqrt{8}} \, dy \, dx \]
is equal to __________ (answer in integer).

For \( n \in \mathbb{N} \), let \[a_n = \sqrt{n} \sin^2\left(\frac{1}{n}\right) \cos n,\] and \[b_n = \sqrt{n} \sin\left(\frac{1}{n^2}\right) \cos n.\] Then

Let \( A = \begin{pmatrix} a & 0 \\ c & d \end{pmatrix} \) be a real matrix, where \( ad = 1 \) and \( c \ne 0 \). If \( A^{-1} + (\text{adj} \, A)^{-1} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \), then \( (\alpha, \beta, \gamma, \delta) \) is equal to

Let \( A \) be a \( 3 \times 5 \) matrix defined by
\[A = \begin{pmatrix}0 & 1 & 3 & 1 & 2 \\1 & 6 & 2 & 3 & 4 \\1 & 8 & 8 & 5 & 8\end{pmatrix}.\]
Consider the system of linear equations given by
\[A \begin{pmatrix}x_1 \\x_2 \\x_3 \\x_4 \\x_5\end{pmatrix} = \begin{pmatrix}3 \\1 \\10\end{pmatrix},\]
where \( x_1, x_2, x_3, x_4, x_5 \) are real variables. Then

Let \( \Omega = \{1,2,3,4,5,6\} \). Then which of the following classes of sets is an algebra?

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?

For some \( a \leq 0 \) and \( b \in \mathbb{R} \), let
\[A = \begin{pmatrix} 0 & a & b \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} \end{pmatrix}.\]
If \( A \) is an orthogonal matrix, then the value of \( a\sqrt{6} + 4b\sqrt{3} \) is equal to __________ (answer in integer).

Let \( A \) be a \( 3 \times 3 \) real matrix. Suppose that 1 and 2 are characteristic roots of \( A \), and 12 is a characteristic root of \( A + A^2 \). Then which of the following statements is/are correct?

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

Let \( a_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \) and \( b_n = \sum_{k=1}^{n} \frac{1}{k^2} \) for all \( n \in \mathbb{N} \). Then

Consider the improper integrals
\[I_1 = \int_{1}^{\infty} \frac{t \sin t}{e^t} \, dt \]
and 
\[I_2 = \int_{1}^{\infty} \frac{1}{\sqrt{t}} \ln\left(1 + \frac{1}{t}\right) \, dt.\]
Then:

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 \( a_1 = 1 \), \( a_{n+1} = a_n \left( \frac{\sqrt{n} + \sin n}{n} \right) \), and \( b_n = a_n^2 \) for all \( n \in \mathbb{N} \). Then which of the following statements is/are correct?

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

Which of the following functions represents a cumulative distribution function?

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 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 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?

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,be a sequence of 𝑖. 𝑖. 𝑑. random variables, each having the probability density function
\(f(x) =\begin{cases}   \frac{x^2r^{-x}}{2}       & \quad \text{if }x ≥0,\\  0, & \quad Otherwise \end{cases}\)
For some real constants 𝛽, 𝛾 and 𝑘, suppose that 
\(lim_{→∞} ( \frac{1}{ 𝑛} ∑^n_{i=1}𝑋_𝑖≤ 𝑥) \)=\(\begin{cases}   0     & \quad if\,x< β.\\  kx, & \quad if\,β≤x≤y.\\  ky, & \quad if\,x>y.\end{cases}\)
Then, the value of 2𝛽 + 3𝛾 + 6𝑘 equals _______

For 𝑥∈ℝ, the curve 𝑦=𝑥² intersects the curve 𝑦=𝑥 sin 𝑥+cos 𝑥 at exactly 𝑛 points. Then, 𝑛 equals