List of top Questions asked in IIT JAM MA

Let
S = {(x, y, x) ∈ \(\R^3\) ∶ x² + y² + z² = 4, (x - 1)² + y² ≤ 1, 𝑧 ≥ 0}.
Then, the surface area of S equals ___________ (rounded off to two decimal places).

Let P₇(x) be the real vector space of polynomials in x with degree at most 7, together with the zero polynomial. For r = 1, 2, … , 7, define
s_r(x) = x(x - 1) ⋯ (x − (r - 1)) and s₀(x) = 1.
Consider the fact that B = {s₀(x), s₁(x), … , s₇(x)} is a basis of P₇(𝑥).
If
\(x^5=\sum\limits_{k=0}^7a_{5,k}s_k(x),\)
where a_5,k ∈ \(\R\), then a_5,2 equals ___________ (rounded off to two decimal places)

Let
\(M=\begin{pmatrix} 0 & 0 & 0 & 0 & -1 \\ 2 & 0 & 0 & 0 & -4 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 3 \\ 0 & 0 & 0 & 2 & 2 \end{pmatrix}\)
If p(x) is the characteristic polynomial of M, then p(2) - 1 equals _________

For a ∈ (−2π, 0), consider the differential equation
\(x^2\frac{d^2y}{dx^2}+ax\frac{dy}{dx}+y=0\) for x > 0.
Let D be the set of all a ∈ (−2π, 0) for which all corresponding real solutions to the above differential equation approach zero as x → 0⁺. Then, the number of elements in D ∩ \(\Z\) equals ___________

Let T be the planar region enclosed by the square with vertices at the points (0,1), (1,0), (0, −1) and (−1,0). Then, the value of
\(\iint\limits_T\left(\cos(\pi(x-y))-\cos(\pi(x+y))\right)^2dx\ dy\)
equals ____________ (rounded off to two decimal places).

Let
S = {(x, y, z) ∈ \(\R^3\) : x² + y² + z² < 1}.
Then, the value of
\(\frac{1}{\pi}\iiint_s\left((x-2y+z)^2+(2x-y-z)+(x-y+2z)^2\right)dxdydz\)
equals ________ (rounded off to two decimal places).

For n ∈ \(\N\), if
\(a_n=\frac{1}{n^3+1}+\frac{2^2}{n^3+2}+...+\frac{n^2}{n^3+n}\)
then the sequence \(\left\{a_n\right\}_{n=1}^{\infin}\) converges to ____________ (rounded off to two decimal places)

Let c > 0 be such that
\(\int^c_0e^{s^2}ds=3\)
Then, the value of
\(\)\(\int\limits_{0}^c\left(\int\limits^c_0 e^{x^2+y^2}dy\right)dx\)
equals __________(rounded off to one decimal place).

For k ∈ \(\N\), let 0 = t₀ < t₁ < ⋯ < t_k < t_k+1 = 1. A function f : [0,1] → ℝ is said to be piecewise linear with nodes t₁, … ,t_k, if for each j = 1, 2, … , k + 1, there exist a_j ∈ ℝ, b_j ∈ ℝ such that
f(t) = a_j + b_jt for t_j-1 < t < t_j.
Let V be the real vector space of all real valued continuous piecewise linear functions on [0,1] with nodes \(\frac{1}{4},\frac{1}{2},\frac{3}{4}\) . Then, the dimension of V equals ______________

For n ∈ \(\N\), let
\(a_n=\frac{1}{n^{n-1}}\sum\limits_{k=0}^n\frac{n!}{k!(n-k)!}\frac{n^k}{k+1}\)
and \(\beta=\lim\limits_{n\rightarrow \infin}a_n\). Then, the value of log 𝛽 equals ___________ (rounded off to two decimal places).

Define the function \(f:(-1,1)\rightarrow (-\frac{\pi}{2},\frac{\pi}{2})\) by
\(f(x)=\sin^{-1}x\)
Let a₆ denote the coefficient of x⁶ in the Taylor series of (f(x))² about x = 0. Then, the value of 9a₆ equals ____________ (rounded off to two decimal places).

Let f: (1, ∞) → (0, ∞) be a continuous function such that for every n ∈ \(\N\), f(n) is the smallest prime factor of n. Then, which of the following options is/are CORRECT ?

Let f : \(\R^2 → \R\) be the function defined as follows :
\(f(x,y)=\begin{cases}     (x^2-1)^2\cos^2(\frac{y^2}{(x^2-1)^2}) & \text{if }x \ne ±1 \\     0 & \text{if } x=±1\end{cases}\)
The number of points of discontinuity of f(x, y) is equal to _________.

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

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

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
\(\begin{pmatrix} 1 & -1 & 0 \\ 0 & 0 & 0 \\ -2 & 2 & 2 \end{pmatrix}\)
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 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 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 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

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