List of top Questions asked in IIT JAM MA

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

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

Let y : (1, ∞) → \(\R\) be the solution of the differential equation
\(y"-\frac{2y}{(1-x)^2}=0\)
satisfying y(2) = 1 and\(\lim\limits_{x→∞}y(x) = 0\). Then y(3) is equal to __________. (rounded off to two decimal places)

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

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

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

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 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

Let f : \(\R^2 → \R\) be defined as follows :
\(f(x,y)=\begin{cases}    \frac{x^4y^3}{x^6+y^6} & \text{if }(x,y) \ne (0,0)\\     0  & \text{if } (x,y)=(0,0) \end{cases}\)
Then

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 V be the volume of the region S ⊆ \(\R^3\) defined by
S = {(x, y, z) ∈ \(\R^3\) : xy ≤ z ≤ 4, 0 ≤ x² + y² ≤ 1}.
Then \(\frac{V}{π}\) is equal to ________ . (rounded off to two decimal places)

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

Let f : \(\R → \R\) be an infinitely differentiable function such that f" has exactly two distinct zeroes. Then

Which of the following functions is/are Riemann integrable on [0, 1] ?

Let f(x) = \(\sqrt[3]{x}\) for x ∈ (0, ∞), and θ(h) be a function such that
f(3 + h) − f(3) = hf′ (3 + θ(h)h)
for all h ∈ (−1, 1). Then \(\lim\limits_{h→0} θ(h)\) is equal to _________. (rounded off to two decimal places)

Let v₁, . . . , v₉ be the column vectors of a non-zero 9 × 9 real matrix A. Let a₁, . . . , a₉ ∈ \(\R\), not all zero, be such that \(\sum^9_{i=1}a_iv_i=0\). Then the system \(Ax=\sum^9_{i=1}v_i\) has

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

The value of \(\lim\limits_{n \rightarrow \infin}\left(1+\frac{1}{2^n}+\frac{1}{3^n}+...+\frac{1}{(2023)^n}\right)^\frac{1}{n}\) is equal to _____________ . (rounded off to two decimal places)

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 W be the subspace of \(M_3(\R)\) consisting of all matrices with the property that the sum of the entries in each row is zero and the sum of the entries in each column is zero. Then the dimension of W is equal to ___________.

Let S be the set of all real numbers α such that the solution y of the initial value problem
\(\frac{dy}{dx}=y(2-y),\\y(0)=\alpha,\)
exists on [0, ∞). Then the minimum of the set S is equal to __________. (rounded off to two decimal places)

Let S be the triangular region whose vertices are (0, 0), \((0,\frac{\pi}{2})\) and \((\frac{\pi}{2},0)\). The value of \(\iint\limits_S\sin(x)\cos(y)dx\ dy\) is equal to ________. (rounded off to two decimal places)

Which of the following is/are true ?