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

Given \[ y = -3x - 3 + m e^{2x} \] Find the Orthogonal Trajectories (O.T.).
If the function \( f(x) \) satisfies \[ f'(x) = f(x) - \pi x + \pi, \] then the possible value of \( f(1) \) is
If \[ f(x) = \big( f(x) - \pi x \big) + \pi, \] then the possible value(s) of \( f(3) - f(2) \) is/are:

Suppose that \( f, g : \mathbb{R} \to \mathbb{R} \) are differentiable functions such that \( f \) is strictly increasing and \( g \) is strictly decreasing. Define \( p(x) = f(g(x)) \) and \( q(x) = g(f(x)) \), \( \forall x \in \mathbb{R} \). Then, for \( t > 0 \), the sign of \( \int_0^t p'(x) (q'(x)-3) \, dx \) is 
 

Let \( a \) be a positive real number. If \( f \) is a continuous and even function defined on the interval \( [-a, a] \), then \( \int_{-a}^{a} \dfrac{f(x)}{1+e^x} dx \) is equal to 
 

If \[ y(x) = \int_{\sqrt{x}}^x e^t \, dt, \quad x>0 \] then \( y'(1) = \) .............
The radius of convergence of the power series \[ \sum_{n=0}^{\infty} n! x^{n^2} \] is ...........
For \( x>1 \), let \[ f(x) = \int_1^x \left( \sqrt{\log t - \frac{1}{2} \log \sqrt{t}} \right) dt. \] The number of tangents to the curve \( y = f(x) \) parallel to the line \( x + y = 0 \) is ..............
For a real number \( x \), define \( \left\lceil x \right\rceil \) to be the smallest integer greater than or equal to \( x \). Then \[ \int_0^1 \int_0^1 \left( \left\lceil x \right\rceil + \left\lceil y \right\rceil + \left| z \right| \right) dx \, dy \, dz = \text{.............}. \]
Let \( T \) be the smallest positive real number such that the tangent to the helix \[ x = \cos t, \quad y = \sin t, \quad z = \frac{t}{\sqrt{2}} \] at \( t = T \) is orthogonal to the tangent at \( t = 0 \). Then the line integral of \( \mathbf{F} = xj - y\mathbf{i} \) along the section of the helix from \( t = 0 \) to \( t = T \) is ..........
Let \( y(x), x>0 \) be the solution of the differential equation \[ x^2 \frac{d^2y}{dx^2} + 5x \frac{dy}{dx} + 4y = 0 \] satisfying the conditions \( y(1) = 1 \) and \( y'(1) = 0 \). Then the value of \( e^2y(e) \) is .............
For \( a, \beta \in \mathbb{R} \), define the map \( \varphi_{a,\beta}: \mathbb{R} \to \mathbb{R} \) by \[ \varphi_{a,\beta}(x) = ax + \beta. \] Let \[ G = \{ \varphi_{a,\beta} \mid (a, \beta) \in \mathbb{R}^2 \}. \] For \( f, g \in G \), define \( g \circ f \in G \) by \[ (g \circ f)(x) = g(f(x)). \] Then which of the following statements is/are TRUE?
The volume of the solid \[ \left\{ (x, y, z) \in \mathbb{R}^3 \mid 1 \leq x \leq 2, 0 \leq y \leq 2/x, 0 \leq z \leq x \right\} \] is expressible as
Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be a function. Then which of the following statements is/are TRUE?
Let \( S \) be an infinite subset of \( \mathbb{R} \) such that \( S \setminus \{a\} \) is compact for some \( a \in S \). Then which one of the following is TRUE?
Let \( f: \mathbb{R} \to [0, \infty) \) be a continuous function. Then which one of the following is NOT TRUE? \[ \text{There exists } x \in \mathbb{R} \text{ such that } f(x) = \frac{f(0) + f(1)}{2}. \]
Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(2) = 2 \) and \[ |f(x) - f(y)| \leq 5| |x - y|^{3/2}. \] For all \( x \in \mathbb{R}, y \in \mathbb{R} \), let \( g(x) = x^3 f(x) \). Then} \[ g'(2) = \]
The flux of \[ \mathbf{F} = y \hat{i} - x \hat{j} + 2z \hat{k} \] along the outward normal, across the surface of the solid \[ \left\{ (x, y, z) \in \mathbb{R}^3 \mid 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq \sqrt{2 - x^2 - y^2} \right\} \] is equal to \[ \iint_S \mathbf{F} \cdot \hat{n} \, dS = \]
The line integral of the vector field \[ \mathbf{F} = zx \hat{i} + xy \hat{j} + yz \hat{k} \] along the boundary of the triangle with vertices \( (1,0,0), (0,1,0) \) and \( (0,0,1) \), oriented anti-clockwise, when viewed from the point \( (2,2,2) \), is \[ \int_C \mathbf{F} \cdot d\mathbf{r} = \]
Let \[ \mathbf{F} = (3 + 2xy) \hat{i} + (x^2 - 3y^2) \hat{j} \] and let \( \mathbf{L} \) be the curve} \[ \mathbf{r}(t) = e^t \sin t \, \hat{i} + e^t \cos t \, \hat{j}, \quad 0 \leq t \leq \pi. \] Then \[ \int_{\mathbf{L}} \mathbf{F} \cdot d\mathbf{r} = \]
If
\[ f(x) = \begin{cases} 1 + x & \text{if } x < 0, \\ (1 - x)(px + q) & \text{if } x \geq 0, \end{cases} \] satisfies the assumptions of Rolle’s theorem in the interval \( [-1, 1] \), then the ordered pair \( (p, q) \) is
Let \( f_1(x), f_2(x), g_1(x), g_2(x) \) be differentiable functions on \( \mathbb{R} \). Let
\[ F(x) = \begin{vmatrix} f_1(x) & f_2(x) \\ g_1(x) & g_2(x) \end{vmatrix}. \] Then \( F'(x) \) is equal to
Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function. If \( g(u, v) = f(u^2 - v^2) \), then
\[ \frac{\partial^2 g}{\partial u^2} + \frac{\partial^2 g}{\partial v^2} = \]
Let \( \varphi: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( \varphi'(1) = 0 \). Let \( \alpha \) and \( \beta \) denote the minimum and maximum values of \( \varphi(x) \) on the interval [2, 3], respectively. Then which one of the following is TRUE?
Let \[ f(x, y) = \frac{xz y}{x^2 + y^2 + z^2}, \quad (x, y) \neq (0, 0). \] Then \[ \frac{\partial f}{\partial x} \text{ and } f \text{ are bounded and unbounded.} \]