List of top Mathematics Questions on Calculus asked in IIT JAM MA

Let \( T \) denote the triangle in the \( xy \)-plane bounded by the \( x \)-axis and the lines \( y = x \) and \( x = 1 \). The value of the double integral (over \( T \)) \[ \iint_T (5 - y) \, dx \, dy \] is equal to ............. (rounded off to two decimal places).

The value of \[ \int_0^1 \left( \int_0^{\sqrt{y}} 3e^{x^3} \, dx \right) dy \] is equal to

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = \begin{cases} \frac{x^2 + y^5}{x^2 + y^4} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Then, which of the following is/are TRUE?

Consider the following subspaces of the real vector space \( \mathbb{R}^3 \): \[ V_1 = \text{span} \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} \right\}, \quad V_2 = \text{span} \left\{ \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} \right\}, \quad V_3 = \text{span} \left\{ \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \right\}, \quad V_4 = \text{span} \left\{ \begin{pmatrix} 1 \\ 3 \\ 6 \end{pmatrix} \right\}, \quad V_5 = \text{span} \left\{ \begin{pmatrix} 1 \\ 0 \\ -3 \end{pmatrix} \right\}. \] Then, which of the following is/are TRUE?

Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \[ \int_0^{\frac{\pi}{4}} \left( \sin(x) f(x) + \cos(x) \int_0^x f(t) \, dt \right) \, dx = \sqrt{2}. \] Then, the value of \[ \int_0^{\frac{\pi}{4}} f(x) \, dx \] is equal to ............... (rounded off to two decimal places).

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ f(0) = 4, \, f(1) = -2, \, f(2) = 8, \, f(3) = 2. \] Then, which of the following is/are TRUE?

For \( n \in \mathbb{N} \), consider the set \( U(n) = \{ x \in \mathbb{Z}_n : \gcd(x, n) = 1 \} \) as a group under multiplication modulo \( n \). Then, which of the following is/are TRUE?

Let \( u_1 = (1, 0, 0, -1) \), \( u_2 = (2, 0, 0, -1) \), \( u_3 = (0, 0, 1, -1) \), \( u_4 = (0, 0, 0, 1) \) be elements in the real vector space \( \mathbb{R}^4 \). Then, which of the following is/are TRUE?

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ f(0) = 0, \, f'(0) = 2, \, f(1) = -3. \] Then, which of the following is/are TRUE?

Let \( \Omega \) be the bounded region in \( \mathbb{R}^3 \) lying in the first octant \( (x \geq 0, y \geq 0, z \geq 0) \), and bounded by the surfaces \( z = x^2 + y^2 \), \( z = 4 \), \( x = 0 \) and \( y = 0 \). Then, the volume of \( \Omega \) is equal to:

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = e^{y}(x^2 + y^2) \quad \text{for all } (x, y) \in \mathbb{R}^2. \] Then, which one of the following is TRUE?

The largest positive number \(a\) such that \[ \int_0^5 f(x)\,dx + \int_0^3 f^{-1}(x)\,dx \ge a \] for every strictly increasing surjective continuous function \(f : [0, \infty) \to [0, \infty)\) is _________.

The value of \[ \frac{\pi}{2} \lim_{n \to \infty} \cos\!\left(\frac{\pi}{4}\right) \cos\!\left(\frac{\pi}{8}\right) \cos\!\left(\frac{\pi}{16}\right) \cdots \cos\!\left(\frac{\pi}{2^{n+1}}\right) \] is _________.

The value of \[ \lim_{n \to \infty} \left(3^n + 5^n + 7^n \right)^{\tfrac{1}{n}} \] is _________.

Consider the two functions \( f(x,y) = x + y \) and \( g(x,y) = xy - 16 \) defined on \( \mathbb{R}^2 \). Then

Let \( D = \mathbb{R}^2 \setminus \{(0,0)\} \). Consider the two functions \( u,v : D \to \mathbb{R} \) defined by \[ u(x,y) = x^2 - y^2 \quad \text{and} \quad v(x,y) = xy. \] Consider the gradients \(\nabla u\) and \(\nabla v\) of the functions \(u\) and \(v\), respectively. Then

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function with the property that for every \( y \in \mathbb{R} \), the value of the expression \[ \sup_{x \in \mathbb{R}} [xy - f(x)] \] is finite. Define \( g(y) = \sup_{x \in \mathbb{R}} [xy - f(x)] \) for \(y \in \mathbb{R}\). Then

Consider the equation \[ x^{2021} + x^{2020} + \cdots + x + 1 = 0. \] Then

Let \( D \subseteq \mathbb{R}^2 \) be defined by \[ D = \mathbb{R}^2 \setminus \{(x,0) : x \in \mathbb{R}\}. \] Consider the function \( f : D \to \mathbb{R} \) defined by \[ f(x,y) = x \sin\!\left(\frac{1}{y}\right). \] Then

Define \[ S = \lim_{n \to \infty} \left(1 - \frac{1}{2^2}\right)\left(1 - \frac{1}{3^2}\right)\cdots\left(1 - \frac{1}{n^2}\right). \] Then

Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \( f(x) = f(x+1) \) for all \(x \in \mathbb{R}\). Then

Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function such that for all \(x \in \mathbb{R}\), \[ \int_0^1 f(xt) \, dt = 0. \quad \text{(*)} \] Then

Let \(p\) and \(t\) be positive real numbers. Let \(D_t\) be the closed disc of radius \(t\) centered at \((0,0)\), i.e., \[ D_t = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \le t^2 \}. \] Define \[ I(p,t) = \iint_{D_t} \frac{dx\,dy}{(p^2 + x^2 + y^2)^2}. \] Then \(\lim_{t \to \infty} I(p,t)\) is finite

Let

\[ f(x, y) = \begin{cases} \frac{x^3 + y^3}{x^2 - y^2}, & x^2 - y^2 \neq 0 \\ 0, & x^2 - y^2 = 0 \end{cases} \]

Then the directional derivative of \( f \) at \( (0, 0) \) in the direction of \( \frac{4}{5} \hat{i} + \frac{3}{5} \hat{j} \) is ............

The coefficient of \( (x - \frac{\pi}{2}) \) in the Taylor series expansion of the function

\[ f(x) = \begin{cases} \frac{4(1 - \sin(x))}{2x - \pi}, & x \neq \frac{\pi}{2} \\ 0, & x = \frac{\pi}{2} \end{cases} \]

about \( x = \frac{\pi}{2} \) is ............