List of top Questions asked in IIT JAM MA

Let y(x) be the solution of the deferential equation \(\frac{dy}{dx}+3x^2y=x^2\), for xεR, 
satisfying the initial condition y(0)=4. 
Then \(lim \,\,y_{n→∞}\) y(x) is equal to ____ (Rounded off to decimal places)

Let y(x) be the solution of the differential equation
\(x,y^2y'+y^3=\frac{sin x}{x} for x>0\)
satisfying (\(\frac{π}{2}\))=0 
The the value of y (\(\frac{5π}{2}\)) is equal to ____. (Rounded off to two decimal places)

Suppose \[ a_n = \frac{3^n + 3}{5^n - 5} \quad \text{and} \quad b_n = \frac{1}{(1 + n^2)^{1/4}} \quad \text{for} \ n = 2, 3, 4, \ldots \] Then which one of the following is true?

The number of distinct subgroup of Z₉₉₉ is_____.

The number of elements of order 12 in the symmetric group S₇ is equal to____

The radius of convergence of the power series \[\sum_{n=1}^{\infty} \left(\frac{n^3}{4^n}\right) x^{5n}\] is

The rank of the \( 4 \times 6 \) matrix \( \begin{pmatrix} 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \end{pmatrix} \) with entries in \( \mathbb{R} \), is

The sum of the series
\(∑^∞_{ n=1} \frac{1}{(4n-3)(4n+1) }\) 
is equal to_____.(Round off to two decimal places)

The value of
\[ \lim_{n \to \infty} \sum_{k=2}^n \frac{\sqrt{n} + 1 - \sqrt{n}}{k(\ln k)^2} \]
is equal to

The value of the limit
\(lim _{n→∞}(\frac{1^4+2^4+…+n^4}{n^5}+\frac{1}{√n}(\frac{1}{\sqrt{n+1}}+\frac{1}{\sqrt{n+2}}+...+\frac{1}{\sqrt{4n}}))\)
is equal to___(Rounded off two decimal places)

A real-valued function y(x) defined on \(\R\) is said to be periodic if there exists a real number \(T\gt0\) such that y(x + T) = y(x) for all \(x\isin\R\). Consider the differential equation
\(\frac{d^2y}{dx^2}+4y=\sin(ax), x\isin\R, \quad (*)\)
where \(a\isin\R\) is a constant. 
Then which of the following is/are true?

Define f: [0,1] → [0,1] by
\(f(x) = \begin{cases} 1 & \text{if } x =0 \\ \frac{1}{n} & \text{if } x=\frac{m}{n}\ \text{for some}\ m, n \isin \N\ \text{with}\ m\le n\ \text{and gcd(m, n)} = 1, \\ 0 & \text{if} x\isin[0,1]\ \text{is irrational}\end{cases}\)
and define g: [0,1]→ [0,1] by
\(g(n) = \begin{cases} 0 & \text{if } x=0 \\ 1 & \text{if } x\isin(0,1]. \end{cases}\)
Then which of the following is/are true?

Let be a finite group of order at least two and let e denote the identity element of G. Let σ: G →g be a bijective group homomorphism that satisfies the following two conditions:
(i): if σ(g)=g for some gεG, then g=e, 
(ii) (σ o σ) (g)=g for all g σ G. 
The n which of the following is/are correct ?

Let (-c,c) be the largest open interval in \(\R\) (where c is either a positive real number or c = ∞) on which the solution y(x) of the differential equation \(\frac{dy}{dx}=x^2+y^2+1\) with initial condition y(0) = 0 exists and is unique. Then which of the following is/are true?

Let S be the set of all continuous functions f: [-1,1]→\(\R\) satisfying the following three conditions:
(i) f is infinitely differentiable on the open interval (-1,1),
(ii) the Taylor series
\(f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+...\) of f at 0 converges to f(x) for each x ∈ (-1,1),
(iii) \(f(\frac{1}{n})=0\ \text{for all}\ n\isin\N\)
Then which of the following is/are true?

Let S be the set of all functions f: \(\R\rightarrow\R\) satisfying \(|f(x)-f(y) |^2 \le |x - y|^3\ for\ all\ x, y \isin \R.\) Then which of the following is/are true?

Let (x_n) be a sequence of real numbers. Consider the set P = {n\(\isin\N:x_n\gt x_m\) for all \(m\isin\N\) with \(m\gt n\)}. Then which of the following is/are true?