List of top Questions asked in IIT JAM MA

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 family of curves x² + y² = 2x + 4y + k with a real parameter k > −5. Then the orthogonal trajectory to this family of curves passing through (2, 3) also passes through

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

Suppose f : (−1, 1) → \(\R\) is an infinitely differentiable function such that the series \(\sum\limits_{j=0}^{\infin}a_j\frac{x^j}{j^!}\) converges to f(x) for each x ∈ (−1, 1), where,
\(a_j=\int\limits_{0}^{\pi/2}\theta^j\cos^j(\tan\theta)d\theta+\int\limits^{\pi}_{\pi/2}(\theta-\pi)^2\cos^j(\tan\theta)d\theta\)
for j ≥ 0. Then

Let f(x) = cos(x) and g(x) =\(1-\frac{x^2}{2}\) for  \(x \in (-\frac{\pi}{2},\frac{\pi}{2})\). 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

How many group homomorphisms are there from \(\Z_2\) to S₅ ?

Let y : \(\R → \R\) be a twice differentiable function such that y" is continuous on [0, 1] and y(0) = y(1) = 0. Suppose y"(x) + x² < 0 for all x ∈ [0, 1]. Then

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

Let f : \(\R → \R\) be an infinitely differentiable function such that f" has exactly two distinct zeroes. 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 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 \(y : (\sqrt{\frac{2}{3}}, ∞) → \R\) be the solution of
(2x − y)y' + (2y − x) = 0,
y(1) = 3.
Then

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 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 \(y_c:\R \rightarrow(0,\infin)\) 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 ?

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(𝑥) be the solution of the differential equation
\(\frac{dy}{dx}=1+y\sec x\ \ \text{for}\ x \in(-\frac{\pi}{2},\frac{\pi}{2})\)
that satisfies 𝑦(0) = 0. Then, the value of \(y(\frac{\pi}{6})\) equals

Let F be the family of curves given by
x² + 2hxy + y² = 1, -1 < h < 1.
Then, the differential equation for the family of orthogonal trajectories to F is

Let G be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE ?

For a positive integer n, let U(n) = {\(\bar{r}\) ∈ ℤ_n ∶ gcd(r, n) = 1} be the group under multiplication modulo n. Then, which one of the following statements is TRUE ?

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

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

Which one of the following is TRUE for the symmetric group S₁₃ ?

Which of the following is a subspace of the real vector space \(\R^3\) ?