List of top Questions asked in IIT JAM MA

Let \( F \) be the family of curves given by \[ x^2 + 2hxy + y^2 = 1, \quad -1<h<1. \] 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?

Which one of the following is TRUE for the symmetric group \( S_{13} \)?

Let \( y_c : \mathbb{R} \to (0, \infty) \) be the solution of the Bernoulli’s equation \[ \frac{dy}{dx} - y + y^3 = 0, \quad y(0) = c>0. \] Then, for every \( c>0 \), which one of the following is true?

For \(n\ \in \N\), let
\(a_n=\frac{1}{(3n+2)(3n+4)}\) and \(b_n=\frac{n^3+\cos(3^n)}{3n+n^3}\).
Then, which one of the following is TRUE ?

Let \(a=\begin{bmatrix} \frac{1}{\sqrt3} \\ \frac{-1}{\sqrt2} \\ \frac{1}{\sqrt6} \\ 0\end{bmatrix}\). Consider the following two statements.
P : The matrix I₄ - aa^T is invertible.
Q: The matrix I₄ - 2aa^T is invertible.
Then, which one of the following holds ?

Let P₁₁(x) be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 11, together with the zero polynomial. Let
E = {s₀(x), s₁(𝑥), … , s₁₁(x)}, F = {r0 (x), r₁ (x), … , r₁₁(x)}
be subsets of P₁₁(x) having 12 elements each and satisfying
s₀ (3) = s₁ (3) = ⋯ = s₁₁(3) = 0 , r₀(4) = r₁(4) = ⋯ = r₁₁(4) = 1.
Then, which one of the following is TRUE ?

For the differential equation
y(8x − 9y)dx + 2x(x − 3y)dy = 0 ,
which one of the following statements is TRUE ?

For x ∈ ℝ, let ⌊x⌋ denote the greatest integer less than or equal to x.
For x, y ∈ ℝ, define
\(\min\left\{x,y\right\} = \begin{cases}     x & \text{if } x \le y, \\     y & \text{otherwise.} \end{cases}\)
Let f:[−2𝜋, 2𝜋] → ℝ be defined by
f(x) = sin(min{x, x − ⌊x⌋}) for x ∈ [−2𝜋, 2𝜋].
Consider the set S = {x ∈ [−2𝜋, 2𝜋]: f is discontinuous at x}.
Which one of the following statements is TRUE ?

Define the sequences \(\left\{a_n\right\}^{\infin}_{n=3}\) and \(\left\{b_n\right\}^{\infin}_{n=3}\) as
\(a_n=(\log n+\log\ \log n)^{\log n}\) and \(b_n=n^{(1+\frac{1}{\log n})}\).
Which one of the following is TRUE ?

For p, q, r ∈ ℝ, r ≠ 0 and n ∈ \(\N\), let
\(a_n=p^n n^q(\frac{n}{n+2})^{n^2}\) and \(b_n=\frac{n^n}{n!r^n}(\sqrt{\frac{n+2}{n}})\).
Then, which one of the following statements is TRUE ?

For a ∈ ℝ, let ya(x) be the solution of the differential equation
\(\frac{dy}{dx}+2y=\frac{1}{1+x^2}\) for x ∈ \(\R\)
satisfying y(0) = a. Then, which one of the following is TRUE ?

Consider the following two statements.
P : There exist functions f: ℝ → ℝ, g: ℝ → ℝ such that f is continuous at x = 1 and g is discontinuous at x = 1 but g ∘ f is continuous at x = 1.
Q : There exist functions f: ℝ → ℝ, g: ℝ → ℝ such that both f and g are discontinuous at x = 1 but g ∘ f is continuous at x = 1.
Which one of the following holds ?

Let f: ℝ → ℝ be defined by
\(f(x)=\frac{(x^2+1)^2}{x^4+x^2+1}\) for x ∈ \(\R\).
Then, which one of the following is TRUE ?

Let f: ℝ → ℝ be a solution of the differential equation
\(\frac{d^2y}{dx^2}-2\frac{dy}{dx}+y=2e^x\) for x ∈ \(\R\).
Consider the following statements.
P : If f(x) > 0 for all x ∈ ℝ, then f'(x) > 0 for all x ∈ ℝ.
Q : If f'(x) > 0 for all x ∈ ℝ, then f(x) > 0 for all x ∈ ℝ.
Then, which one of the following holds ?

Let L₁ denote the line y = 3x + 2 and L₂ denote the line y = 4x + 3. Suppose that f: ℝ → ℝ is a four times continuously differentiable function such that the line L₁ intersects the curve y = f(x) at exactly three distinct points and the line L₂ intersects the curve y = f(x) at exactly four distinct points. Then, which one of the following is TRUE ?

Define the function \(f: \R^2 → \R\) by
f(x, y) = 12xy e^-(2x+3y-2).
If (a, b) is the point of local maximum of f, then f(a, b) equals

Let \(\left\{a_n\right\}^{\infin}_{n=1}\) be a sequence of real numbers.
Then, which of the following statements is/are always TRUE ?

Which of the following statements is/are TRUE ?

Which of the following statements is/are TRUE ?

Let
\(𝑆 = x\left\{(𝑥, 𝑦) ∈ ℝ^2 : 𝑥 > 0, 𝑦 > 0\right\} ,\)
and f: S → ℝ be given by
\(f(x,y)=2x^2+3y^2-\log x-\frac{1}{6}\log y.\)
Then, which of the following statements is/are TRUE ?

The center Z(G) of a group G is defined as
z(G) = {x ∈ G ∶ xg = gx for all g ∈ G}.
Let |G| denote the order of G. Then, which of the following statements is/are TRUE for any group G ?

For a matrix M, let Rowspace(M) denote the linear span of the rows of M and Colspace(M) denote the linear span of the columns of M. Which of the following hold(s) for all A, B, C ∈ M10(\(\R\)) satisfying A = BC ?

Define f: ℝ → ℝ and g: ℝ → ℝ as follows
\(f(x)=\sum\limits^{\infin}_{m=0}\frac{(-1)^mx^{2m}}{2^{2m}(m!)^2}\) and \(g(x)=\frac{x}{2}\sum\limits^{\infin}_{m=0}\frac{(-1)^mx^{2m}}{2^{2m}(m+1)!m!}\) for x ∈ \(\R\).
Let x₁, x₂, x₃, x₄ ∈ ℝ be such that 0 < x₁ < x₂ , 0 < x₃ < x₄,
f(x₁) = f(x₂) = 0,    f(x) ≠ 0 when x₁ < x < x₂,
g(x₃) = g(x₄) = 0 and g(x) ≠ 0 when x₃ < x < x₄.
Then, which of the following statements is/are TRUE ?

For 0 < a < 4, define the sequence \(\left\{x_n\right\}^{\infin}_{n=1}\) of real numbers as follows :
x₁ = a and x_n+1 + 2 = −x_n(x_n - 4) for n ∈ \(\N\).
Which of the following is/are TRUE ?