List of top Questions asked in IIT JAM MA

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 G be a finite group. Then G is necessarily a cyclic group if the order of G is

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

Let p(x) = x⁵⁷ + 3x¹⁰ - 21x³ + x² + 21 and
\(q(x)=p(x)+\sum\limits_{j=1}^{57}p^{(j)}(x) \ \text{for all }x \in \R,\)
where p^(j)(x) denotes the j^th derivative of p(x). Then the function q admits

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

Which one of the following groups has elements of order 1, 2, 3, 4, 5 but does not have an element of order greater than or equal to 6 ?

Consider the following statements.
P : If a system of linear equations Ax = b has a unique solution, where A is an m × n matrix and b is an m × 1 matrix, then m = n.
Q : For a subspace W of a nonzero vector space V, whenever u ∈ V ∖ W and v ∈ V ∖ W, then u + v ∈ V ∖ W.
Which one of the following holds ?

Let G be a finite group containing a non-identity element which is conjugate to its inverse. Then, which one of the following is TRUE ?

Consider the 4 × 4 matrix \(M=\begin{pmatrix} 11 & 10 & 10 & 10 \\ 10 & 11 & 10 & 10 \\ 10 & 10 & 11 & 10 \\ 10 & 10 & 10 & 11 \end{pmatrix}\). Then the value of the determinant of M is equal to _________.

Let a, b be positive real numbers such that \(a\lt b\). Given that
\(\lim\limits_{N\rightarrow\infin}\displaystyle\int^{N}_{0}e^{-t^2}dt=\frac{\sqrt \pi}{2},\) the value of
\(\lim\limits_{N\rightarrow\infin}\displaystyle\int^{N}_{0}\frac{1}{t^2}(e^{-at^2}-e^{-bt^2})dt\) is equal to

Let V be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 5, together with the zero polynomial. Let T: V→ \(\R\) be the linear map defined by 7(1) = 1 and T(x(x-1)... (x-k+1)) = 1 for 1 ≤ k ≤ 5.
Then which of the following is/are true?

Let \( P \in \mathbb{M}_4 (\mathbb{R}) \) be such that \( P^4 \) is the zero matrix, but \( P^3 \) is a nonzero matrix. Then which one of the following is FALSE?

Define f : \(\R^2 → \R\) by
f(x, y) = x² + 2y² - x for (x, y) ∈ \(\R^2\).
Let D = {(x, y) ∈ \(\R^2\) ∶ x² + y² ≤ 1} and \(E=\left\{(x,y)\in \R^2:\frac{x^2}{4}+\frac{y^2}{9} \le 1\right\}.\)
Consider the sets
D_max = {(a, b) ∈ D ∶ f has absolute maximum on D at (a, b)}, 
D_min = {(a, b) ∈ D ∶ f has absolute minimum on D at (a, b)}, 
E_max = {(c, d) ∈ E ∶ f has absolute maximum on E at (c, d)}, 
E_min = {(c, d) ∈ E ∶ f has absolute minimum on E at (c, d)}.
Then the total number of elements in the set
D_max ∪ D_min ∪ E_max ∪ E_min
is equal to _________.

Let \(A=\begin{pmatrix} 1 & 1 \\ 0 & 1 \\ -1 & 1 \end{pmatrix}\)and let A^T denote the transpose of A. Let \(u=\begin{pmatrix} u_1 \\ u_2 \end{pmatrix}\) and \(v=\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\)be column vectors with entries in \(\R\) such that \(u^2_1+u_2^2=1\) and \(v_1^2+v_2^2+v^2_3=1\). Suppose
\(Au=\sqrt2v\) and \(A^Tv=\sqrt2u.\)
Then |\(𝑢1 + 2 \sqrt2 𝑣_1\)| is equal to _________. (Rounded off to two decimal places)

Let \( V \) be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 6, together with the zero polynomial. Then which one of the following is true?

Let M be a positive real number and let u, v: \(\R^2\rightarrow\R\) be continuous functions satisfying \(\sqrt{u(x,y)^2+v(x,y)^2}\ge M\sqrt{x^2+y^2}\ for\ all\ (x,y)\isin\R^2.\)
Let F: \(\R^2\rightarrow\R^2\) be given by
F(x, y) = (u(x, y), v(x, y)) for (x, y)\(\isin\R^2\).
Then which of the following is/are true?

Let \( (x_n) \) and \( (y_n) \) be sequences of real numbers defined by
\[ x_1 = 1, \quad y_1 = \frac{1}{2}, \quad x_{n+1} = \frac{x_n + y_n}{2}, \quad \text{and} \quad y_{n+1} = \sqrt{x_n y_n} \quad \text{for all} \ n \in \mathbb{N}. \]
Then which one of the following is true?

consider the differential equation
y''+ay'+y=sin x for x for xεR. (**) 
then which one of the following is true?

On the open interval \((-c, c)\), where \( c \) is a positive real number, \( y(x) \) is an infinitely differentiable solution of the differential equation\[ \frac{dy}{dx} = y^2 - 1 + \cos x, \] with the initial condition \( y(0) = 0 \). Then which one of the following is correct?

Let P be a fixed 3×3 matrix with entries in \(\R\). Which of the following maps from \(M_3(\R)\) to \(M_3(\R)\) is/are linear?

Let \(\theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)\). Consider the functions
\[ u : \mathbb{R}^2 - \{ (0, 0) \} \to \mathbb{R} \quad \text{and} \quad v : \mathbb{R}^2 - \{ (0, 0) \} \to \mathbb{R} \]
given by
\[ u(x,y) = x - \frac{x}{x^2 + y^2} \quad \text{and} \quad v(x,y) = y + \frac{y}{x^2 + y^2}. \]
The value of the determinant \(\begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}\) at the point \((\cos \theta, \sin \theta)\) is equal to

Consider the function u:R³→R given by
\(u(x_1,x_2,x_3)=x_1x^4_2x^2_3-x^3_1x^4_3-26x^2_1x^2_2x^3_3\)
Let CεN be such that 
\(x_1\frac{ ∂u}{∂x_2}+2x_2\frac{ ∂u}{∂x_3} \)
evaluate at the point (t, t²,t³), eqals ct^k for every tεR, The n the value of K is equal to____.

If G is the region in \(\R^2\) given by
\(G=\left\{(x,y)\in \R^2:x^2+y^2<1,\frac{x}{\sqrt3}< y < \sqrt3x,x>0,y>0\right\}\)
then the value of
\(\frac{200}{\pi}\iint_Gx^2dx\ dy\)
is equal to _________. (Rounded off to two decimal places)

Consider the family F1 of curve lying in the region
{(x,y) εR²: y>0 and 0<x<π }
and given by 
\(y=\frac{c(1-cosx)}{sinx},\) Where c is a positive real number.
Let F₂ be the family of orthogonal trajectories to f₁. consider the curve C belonging to the family F2 passing through the point (\(\frac{π}{3}\),1). if a is a real number such that (\(\frac{π}{4}\), a) lies on C, then the value of a⁴ is equal _____to (Rounded off to two decimal places).

For \(-1\le x\le1\), if f (x) is the sum of the convergent power series \(x+\frac{x^2}{2^2}+\frac{x^3}{3^2}+...+\frac{x^n}{n^2}+...\) then \(f(\frac{1}{2})\) is equal to