List of top Questions asked in IIT JAM MA

Let y : ℝ → ℝ be the solution to the differential equation
\(\frac{d^2y}{dx^2}+2\frac{dy}{dx}+5y=1\)
satisfying y(0) = 0 and y'(0) = 1.
Then, \(\lim\limits_{x \rightarrow \infin}y(x)\) equals __________ (rounded off to two decimal places).

Consider the function f: ℝ → ℝ given by f(x) = x³ - 4x² + 4x - 6.
For c ∈ ℝ, let
\(S(c)=\left\{x \in \R : f(x)=c\right\}\)
and |S(c)| denote the number of elements in S(c). Then, the value of
|S(-7)| + |S(-5)| + |S(3)|
equals _________

For a > b > 0, consider
\(D=\left\{(x,y,z) \in \R^3 :x^2+y^2+z^2 \le a^2\ \text{and } x^2+y^2 \ge b^2\right\}.\)
Then, the surface area of the boundary of the solid D is

Let P₇(x) be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 7, together with the zero polynomial.Let T : P₇(x) → P₇(𝑥) be the linear transformation defined by
\(T(f(x))=f(x)+\frac{df(x)}{dx}\).
Then, which one of the following is TRUE ?

For a positive integer \( n \), let \( U(n) = \{ r \in \mathbb{Z}_n : \gcd(r, n) = 1 \} \) be the group under multiplication modulo \( n \). Then, which one of the following statements is TRUE?

Consider the group \( G = \{ A \in M_2(\mathbb{R}) : AA^T = I_2 \ \) with respect to matrix multiplication. Let \[ Z(G) = \{ A \in G : AB = BA, \, for all \, B \in G \}. \] Then, the cardinality of \( Z(G) \) is:}

Let \( V \) be a nonzero subspace of the complex vector space \( M_7(\mathbb{C}) \) such that every nonzero matrix in \( V \) is invertible. Then, the dimension of \( V \) over \( \mathbb{C} \) is:

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?

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?

Let \( g: \mathbb{R} \to \mathbb{R} \) be a continuous function. Which one of the following is the solution of the differential equation \[ \frac{d^2 y}{dx^2} + y = g(x) \quad \text{for} \quad x \in \mathbb{R}, \] satisfying the conditions \( y(0) = 0 \), \( y'(0) = 1 \)?

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

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:

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

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 twice continuously differentiable function \(g: \mathbb{R} \to \mathbb{R}\), define \[ u_g(x, y) = \frac{1}{y} \int_{-y}^{y} g(x + t) \, dt \quad \text{for} \quad (x, y) \in \mathbb{R}^2, \, y>0. \] Which one of the following holds for all such \(g\)?

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

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 A be a 6 × 5 matrix with entries in ℝ and B be a 5 × 4 matrix with entries in ℝ. Consider the following two statements.
P : For all such nonzero matrices A and B, there is a nonzero matrix Z such that AZB is the 6 × 4 zero matrix.
Q : For all such nonzero matrices A and B, there is a nonzero matrix Y such that BYA is the 5 × 5 zero matrix.
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 ?