List of top Questions asked in Indian Institute Of Technology Joint Admission Test for MSc- 2023

Which one of the following is TRUE for the symmetric group S₁₃ ?

Let
\(A=\begin{pmatrix} 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 3 \\ 1 & 1 & 4 & 4 &4 \\ \end{pmatrix}\)
and B be a 5 × 5 real matrix such that AB is the zero matrix. Then the maximum possible rank of B is equal to ____________.

Let f : \(\R → \R\) be a bijective function such that for all x ∈ R, f(x) = \(\sum\limits^{\infin}_{n=1}a_nx^n\) and \(f^{-1}(x)=\sum\limits_{n=1}^{\infin}b_nx^n\), where f^-1 is the inverse function of f. If a₁ = 2 and a₂ = 4, then b₁ is equal to _________.

For a twice continuously differentiable function 𝑔: ℝ → ℝ, define
\(u_g(x,y)=\frac{1}{y}\int^y_{-y}g(x+t)dt\ \ \ \text{for}(x,y)\in \R^2, \ \ \ y \gt0.\)
Which one of the following holds for all such 𝑔 ?

From the additive group Q to which one of the following groups does there exist a non-trivial group homomorphism ?

Let \(a_n=\frac{1+2^{-2}+...+n^{-2}}{n}\) for n ∈ \(\N\). Then

Let \(a_n=\sin(\frac{1}{n^3})\) and \(b_n=\sin(\frac{1}{n})\) for n ∈ \(\N\). Then

Consider the following statements :
I. There exists a linear transformation from \(\R^3\) to itself such that its range space and null space are the same.
II. There exists a linear transformation from \(\R^2\) to itself such that its range space and null space are the same.
Then

Let (a_n) be a sequence of real numbers such that the series \(\sum\limits_{n=0}^{\infin}a_n(x-2)^n\) converges at x = −5. Then this series also converges at

The area of the curved surface
\(S = \left\{ (x, y, z) ∈ \R^3 : z^2 = (x − 1)^2 + (y − 2)^2 \right\}\)
lying between the planes z = 2 and z = 3 is

Let
\(\begin{pmatrix} 1 & -1 & 0 \\ 0 & 0 & 0 \\ -2 & 2 & 2 \end{pmatrix}\)
and B = A⁵ + A⁴ + I₃. Which of the following is NOT an eigenvalue of B ?

The system of linear equations in x₁, x₂, x₃
\(\begin{pmatrix} 1 & 1 & 1 \\ 0 & -1 & 1 \\ 2 & 3 & \alpha \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix} 3 \\ 1 \\ \beta \end{pmatrix}\)
where α, β ∈ R, has

Let f(x, y) = ln(1 + x² + y² ) for (x, y) ∈ \(\R^2\). Define
\(\begin{matrix} P=\frac{∂^2f}{∂x^2}|_{(0,0)} &  Q=\frac{∂^2f}{∂x∂y}|_{(0,0)} \\  R=\frac{∂^2f}{∂y∂x}|_{(0,0)} &  S=\frac{∂^2f}{∂y^2}|_{(0,0)} \end{matrix}\)
Then

Let f(x, y) = \(e^{x^2}+y^2\) for (x, y) ∈ \(\R^2\) , and a_n be the determinant of the matrix
\(\begin{pmatrix} \frac{∂^2f}{∂x^2} &  \frac{∂^2f}{∂x∂y} \\  \frac{∂^2f}{∂y∂x} &  \frac{∂^2f}{∂y^2} \end{pmatrix}\)
evaluated at the point (cos(n),sin(n)). Then the limit \(\lim\limits_{n \rightarrow \infin}\) a_n is

Let \(a_n=\left(1+\frac{1}{n}\right)^n\) and \(b_n=n\cos(\frac{n!\pi}{2^{10}})\) for n ∈ \(\N\). Then

For each t ∈ (0, 1), the surface P_t in \(\R^3\) is defined by
\(P_t = \left\{(x, y, z) : (x^2 + y^2 )z = 1, t^2 ≤ x^2 + y^2 ≤ 1\right\}.\)
Let a_t ∈ R be the surface area of P_t. Then

Let A ⊆ \(\Z\) with 0 ∈ A. For r, s ∈ \(\Z\), define
rA = {ra : a ∈ A},   rA + sA = {ra + sb : a, b ∈ A}.
Which of the following conditions imply that A is a subgroup of the additive group \(\Z\) ?

Let (a_n) and (b_n) be sequences of real numbers such that
\(|a_n-a_{n+1}|=\frac{1}{2^n}\) and \(|b_n-b_{n+1}|=\frac{1}{\sqrt{n}}\) for n ∈ \(\N\).
Then

Let f : \(\R^2 → \R^2\) be defined by f(x, y) = (e^x cos(y), e^x sin(y)). Then the number of points in \(\R^2\) that do NOT lie in the range of f is

Consider the family of curves x² + y² = 2x + 4y + k with a real parameter k > −5. Then the orthogonal trajectory to this family of curves passing through (2, 3) also passes through

For g ∈ \(\Z\), let \(\bar{g}\) ∈ \(\Z_8\) denote the residue class of g modulo 8. Consider the group \(\Z^×_8\) = {\(\bar{x}\) ∈ \(\Z_8\) : 1 ≤ x ≤ 7, gcd(x, 8) = 1} with respect to multiplication modulo 8. The number of group isomorphisms from \(\Z^×_8\) onto itself is equal to ________

Let (an) be a sequence of real numbers defined by
\(a_n=\begin{cases} 1 & \text{if } n \text{ is prime}\\    -1 & \text{if } n \text{ is not prime} \end{cases}\)
Let \(b_n=\frac{a_n}{n}\) for n ∈ \(\N\). Then

For σ ∈ S₈, let o(σ) denote the order of σ. Then max{o(σ) : σ ∈ S₈} is equal to __________.

A subset S ⊆ \(\R^2\) is said to be bounded if there is an M > 0 such that |x| ≤ M and |y| ≤ M for all (x, y) ∈ S. Which of the following subsets of \(\R^2\) is/are bounded ?

Consider the following statements :
I. Every infinite group has infinitely many subgroups.
II. There are only finitely many non-isomorphic groups of a given finite order.
Then