List of top Mathematics Questions on Calculus asked in JEE Main

Let $E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1$ be an ellipse. Ellipses $E_i$ are constructed such that their centers and eccentricities are the same as that of $E_1$, and the length of the minor axis of $E_{i+1}$ is the length of the major axis of $E_i$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi} \sum_{i=1}^{\infty} A_i$ is equal to:

If $ f(x) = \frac{2^x}{2^x + \sqrt{2}} $, $x \in \mathbb{R}$, then $\sum_{k=1}^{81} f\left(\frac{k}{82}\right)$ is equal to:

The area (in sq. units) of the region $(x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3$ is:

Let $ f : \mathbb{R} \to \mathbb{R} $ be a twice-differentiable function such that $ f(2) = 1 $. If $ F(x) = x f(x) $ for all $ x \in \mathbb{R} $, and the integrals $ \int_0^2 x F'(x) \, dx = 6 $ and $ \int_0^2 x^2 F''(x) \, dx = 40 $, then $ F'(2) + \int_0^2 F(x) \, dx $ is equal to:

Let $ f $ be a real-valued continuous function defined on the positive real axis such that $ g(x) = \int_0^x t f(t) \, dt $. If $ g(x^3) = x^6 + x^7 $, then the value of $ \sum_{r=1}^{15} f(r^3) $ is:

The area of the region bounded by the curves $ x(1 + y^2) = 1 $ and $ y^2 = 2x $ is:

If \[ \int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left( \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| \right) + C, \] where $ C $ is the constant of integration, then $ \alpha + 2\beta $ is equal to ……..

Let $ P $ be the image of the point $ Q(7, -2, 5) $ in the line $ L: \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4} $, and let $ R(5, p, q) $ be a point on $ L $. Then the square of the area of $ \triangle PQR $ is:

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word "MATHS" such that any letter that appears in the word must appear at least twice is:

Let \[ S = \left\{ x : \cos^{-1}x = \pi + \sin^{-1}x + \sin^{-1}(2x+1) \right\}. \] Then \[ \sum_{x \in S} (2x - 1)^2 \] is equal to:

Let $ \mathbf{a} = 2\hat{i} - \hat{j} + 3\hat{k} $, $ \mathbf{b} = 3\hat{i} - 5\hat{j} + \hat{k} $, and $ \mathbf{c} $ be a vector such that $ \mathbf{a} \times \mathbf{c} = \mathbf{a} \times \mathbf{b} $ and \[ (\mathbf{a} + \mathbf{c}) \cdot (\mathbf{b} + \mathbf{c}) = 168. \] Then the maximum value of $ | \mathbf{c} |^2 $ is:

Let $ M $ and $ m $ respectively be the maximum and the minimum values of \[ f(x) = \frac{1 + \sin^2 x}{\cos^2 x} + \frac{4 \sin 4x}{\sin^2 x \cos^2 x} \quad {for} \quad x \in \mathbb{R} \] Then $ M^4 - m^4 $ is equal to:

Let $ f(x) $ be defined as follows:

\[ f(x) = \begin{cases} 3x, & \text{if } x < 0 \\ \min(1 + x + \lfloor x \rfloor, 2 + x \lfloor x \rfloor), & \text{if } 0 \leq x \leq 2 \\ 5, & \text{if } x > 2 \end{cases} \]

where $ \lfloor . \rfloor $ denotes the greatest integer function. If $ \alpha $ and $ \beta $ are the number of points where $ f $ is not continuous and is not differentiable, respectively, then $ \alpha + \beta $ equals:

Let $E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1$ be an ellipse. Ellipses $E_i$ are constructed such that their centers and eccentricities are the same as that of $E_1$, and the length of the minor axis of $E_{i+1}$ is the length of the major axis of $E_i$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi} \sum_{i=1}^{\infty} A_i$ is equal to:

The area (in sq. units) of the region $(x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3$ is:

Let $ f(x) = 2^x - x^2, \, x \in \mathbb{R} $. If $ m $ and $ n $ are respectively the number of points at which the curves $ y = f(x) $ and $ y = f'(x) $ intersect the x-axis, then the value of $ m + n $ is

For the function $f(x) = (\cos x) - x + 1, x \in \mathbb{R}$, find the correct relationship between the following two statements
(S1) $f(x) = 0$ for only one value of x is $[0, \pi]$.
(S2) $f(x)$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.

The area of the region enclosed by the parabolas $ y = x^2 - 5x $ and $ y = 7x - x^2 $ is _________.