List of top Questions asked in IIT JAM MA

Let \( f(x) = 10x^2 + e^x - \sin(2x) - \cos x \), \( x \in \mathbb{R} \). The number of points at which the function \( f \) has a local minimum is:

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = (y - 1)(y - 3), \] satisfying \( \varphi(0) = 2 \). Then, which one of the following is TRUE?

Let \( C \) denote the family of curves described by \( yx^2 = \lambda \), for \( \lambda \in (0, \infty) \) and lying in the first quadrant of the \( xy \)-plane. Let \( O \) denote the family of orthogonal trajectories of \( C \). Which one of the following curves is a member of \( O \), and passes through the point \( (2, 1) \)?

Let \( \varphi : (0, \infty) \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = \left( \ln y - \ln x \right) y, \] satisfying \( \varphi(1) = e^2 \). Then, the value of \( \varphi(2) \) is equal to:

Let \( T : P_2(\mathbb{R}) \to P_2(\mathbb{R}) \) be the linear transformation defined by \[ T(p(x)) = p(x + 1), \quad \text{for all } p(x) \in P_2(\mathbb{R}) \] If \( M \) is the matrix representation of \( T \) with respect to the ordered basis \( \{1, x, x^2\} \) of \( P_2(\mathbb{R}) \), then which one of the following is TRUE?

Let \( G \) be a finite abelian group of order 10. Let \( x_0 \) be an element of order 2 in \( G \). If \( X = \{ x \in G : x^3 = x_0 \} \), then which one of the following is TRUE?

Which one of the following is the general solution of the differential equation \[ \frac{d^2 y}{dx^2} - 8 \frac{dy}{dx} + 16y = 2e^{4x} ? \]

Define \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) by \[ T(x, y, z) = (x + z, 2x + 3y + 5z, 2y + 2z), \quad \text{for all } (x, y, z) \in \mathbb{R}^3 \] Then, which one of the following is TRUE?

For which one of the following choices of \( N(x, y) \), is the equation \[ (e^x \sin y - 2y \sin x) \, dx + N(x, y) \, dy = 0 \] an exact differential equation?

The sum of the infinite series \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \pi^{2n+1}}{2^{2n+1} (2n)!} \] is equal to

For n ≥ 3, let a regular n-sided polygon P_n be circumscribed by a circle of radius R_n and let r_n be the radius of the circle inscribed in P_n. Then
\(\lim\limits_{n \rightarrow \infin}(\frac{R_n}{r_n})^{n^2}\)
equals

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).

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 ?

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?

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?

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 \( g: \mathbb{R} \to \mathbb{R} \) 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 \( y(0) = 0 \), \( y'(0) = 1 \)?

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?

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:

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: