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

Let \(y : (\sqrt{\frac{2}{3}}, ∞) → \R\) be the solution of
(2x − y)y' + (2y − x) = 0,
y(1) = 3.
Then

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 ?

Let \(y_c:\R \rightarrow(0,\infin)\) be the solution of the Bernoulli’s equation
\(\frac{dy}{dx}-y+y^3=0,\ \ \ \ \ \ \ y(0)=c \gt 0.\)
Then, for every 𝑐 > 0, which one of the following is true ?

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 _________.

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

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) 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

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

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

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

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

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 G be a finite group. Then G is necessarily a cyclic group if the order of G 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
\(f(x,y)=\iint\limits_{(u-x^2)+(v-y)^2 \le 1}e^{-\sqrt{(u-x)^2+(v-y)^2}}du\ dv.\)
Then \(\lim\limits_{n \rightarrow \infin}f(n,n^2)\) is

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

Consider the initial value problem
\(\frac{dy}{dx}+αy=0, \\ y(0)=1,\)
where α ∈ \(\R\). Then

Let S and T be non-empty subsets of \(\R^2\), and W be a non-zero proper subspace of \(\R^2\). Consider the following statements :
I. If span(S) = \(\R^2\) , then span(S ∩ W) = W.
II. span(S ∪ T) = span(S) ∪ span(T).
Then

Let R₁ and R₂ be the radii of convergence of the power series \(\sum\limits_{n=1}^{\infin}(-1)^nx^{n-1}\) and \(\sum\limits_{n=1}^{\infin}(-1)^n\frac{x^{n+1}}{n(n+1)}\), respectively. Then