Calculus Questions - JEE

Consider curves $C_1: y^2-x=0$; $C_2: y-x^2=0$; $0\leq x\leq\frac{\sqrt{3}}{2}$ and $C_3: y=f(x)$; $f(x)<0$ $\forall x\in\left(0,\frac{\sqrt{3}}{2}\right)$. From any point $P$ on $C_2$, lines parallel to coordinate axes intersect $C_1$ at $Q$ and $C_3$ at $R$. If area of region $OPRO$ is twice the area of region $OPQO$ (O = origin), then $\left|32f\!\left(\frac{1}{2}\right)\right|$ is

Let a function $f(x)$ be defined in $[-2, 2]$ as $f(x) = \begin{cases} \{x\}, & -2 \leq x < -1 \\ |\text{sgn } x|, & -1 \leq x \leq 1 \\ \{-x\}, & 1 < x \leq 2 \end{cases}$, where $\{x\}$ denotes fractional part, then area bounded by graph of $f(x)$ and $x$-axis is:

A strictly increasing continuous function $f(x)$ intersects with its inverse $f^{-1}(x)$ at $x = \alpha$ and $x = \beta$. If $\int_{\alpha}^{\beta}(f(x) + f^{-1}(x))\,dx = 13$ where $\alpha, \beta \in N$, then the value of $|\alpha\beta|$ equals:

A strictly increasing continuous function $f(x)$ intersects its inverse $f^{-1}(x)$ at $x=\alpha$ and $x=\beta$, $\displaystyle\int_\alpha^\beta (f(x)+f^{-1}(x))\,dx=13$, where $\alpha,\beta\in\mathbb{N}$. Then $|\alpha\beta|$ equals

Let $f:[-1,0] \to R$ be a function differentiable within the domain and that $\displaystyle\int_{-1}^{0}(f(x))^2\,dx = 10$ and $f(-1) = 2$. The value of the integral $\displaystyle\int_{-1}^{0} x f'(x) f(x)\,dx$, is:

If $f(x)$ and $g(x)$ are both continuous functions then the value of $\displaystyle\int_{\ln\lambda}^{\ln(1/\lambda)} \dfrac{f\!\left(\dfrac{x^2}{4}\right)(f(x) - f(-x))}{g\!\left(\dfrac{x^2}{4}\right)(g(x) + g(-x))}\,dx$ is equal to:

If $x=\cos\theta$ and $y=\sin^3\theta$, then $\left|y\dfrac{d^2y}{dx^2}+\left(\dfrac{dy}{dx}\right)^2\right|$ at $\theta=\dfrac{\pi}{2}$ is

If $\displaystyle\int (x^2+1)(x+1)^2 e^x\,dx = A(f(x))^2 + C$ and $f(-1) = \frac{2}{e}$, then $2A + f(0)$ equals

Let $I(x)=\displaystyle\int\frac{x+1}{x(x^2e^{2x}-1)}\,dx=\frac{1}{4}\ln\frac{(xe^x)^\alpha-2(xe^x)^\beta+\gamma}{x^4e^{4x}}+c$, then $\alpha+\beta+\gamma$ equals

The area bounded by the $x$-axis, part of the curve $y=1+x^{-2}$ and the ordinates $x=1$, $x=2$ is divided into equal parts by the ordinate at $x=a$ such that $a=\dfrac{3+\sqrt{p}}{8}$. The value of $p$ is

If $\displaystyle\int e^{\frac{1}{2}\left(x^2+\frac{1}{x^2}\right)}\cdot\frac{x^4+x^2-1}{x^2}\,dx = f(x)+c$, then $\bigl(f(\sqrt{2})\bigr)^4$ is

Value of $\displaystyle\int_0^1 x^6(x^3-1)^{2022}\,dx$ is

For positive integer $n$, let $I_n=\displaystyle\int_{-\pi}^{\pi}\!\left(\frac{\pi}{2}-|x|\right)\cos nx\,dx$. Find $[I_1+I_2+I_3+I_4]$ (GIF).

The area enclosed between the curves $y=ax^2$ and $x=ay^2$ ($a>0$) is 1 sq. unit. Then the value of $a$ is

If $x=\cos\theta$ and $y=\sin^3\theta$, then $\left|y\dfrac{d^2y}{dx^2}+\left(\dfrac{dy}{dx}\right)^2\right|$ at $\theta=\dfrac{\pi}{2}$ is

Area enclosed by $y=g(x)$, $x=1$ and $x=37$, where $g(x)$ is the inverse of $f(x)=x^3+3x+1$, is

If $f(x) = \sin x + \displaystyle\int_{-\pi/2}^{\pi/2}(\sin x + t\cos x)f(t)\,dt$, then $f(x)$ may be equal to $\left(-\dfrac{1}{k}\sin x - \dfrac{2}{k}\cos x\right)$, where $k$ is a numerical quantity which equals

The area of the region bounded by $y=x^2$ and $y=\sec^{-1}[-\sin^2 x]$, where $[\cdot]$ is the GIF, is

If $f(x)$ is even and periodic with period $T$, $\int_0^a f(x)dx=3$ and $\int_{-T/2}^{3T/2}f(x)dx=18$, then $\int_{-a}^{a+5T}f(x)dx$ is

Value of $\displaystyle\int_0^1 \frac{\sin x}{x}\,dx$ lies in the interval

The equation $1012x^{2023}-12138x^{2022}-119x+714=0$ has a root in $(a^{1/2022},b^{1/3})$; $a,b\in\mathbb{N}\geq2$. The value of $4\displaystyle\int_{\sqrt{a}}^{b^{1/3}}\frac{x\cos x^2}{\cos x^2+\cos(263-x^2)}\,dx$ is

A wire $20cm$ long be divided into two parts, if one part is to be bent into a circle, the other part is to be bent into a square and the two plane figures are to have areas the sum of which is maximum, then side length of square is_____.

Let $f: [0, \infty) \to \mathbb{R}$ be a continuous strictly increasing function such that $f'(x) = \int_{0}^{x} tf^2(t)dt$ for every $x \geq 0$, then value of $f(6)$ is _____.

$$\int \frac{x^2 - 1}{\left(x^4 + 3x^2 + 1\right)\tan^{-1}\left(x + \frac{1}{x}\right)} dx =$$

Suppose $f(x)$ is a function satisfying the following conditions:\n(i) $f(0) = 2, f(1) = 1$,\n(ii) $f$ has a minimum value at $x = 5/2$\n(iii) For all $x, f'(x) = \begin{vmatrix} 2ax & 2ax - 1 & 2ax + b + 1 \\ b & b + 1 & -1 \\ 2(ax + b) & 2ax + 2b + 1 & 2ax + b \end{vmatrix}$\nThe value of $f(2)$ is

If $f(x+h) - f(x) + h f'(x+\theta h), 0 < \theta < 1$, the value of $40$, when $f(x) = Ax^2 + Bx + C$ is_____

If $I = \int \frac{dx}{(x-2)\left(1+\sqrt{7x-10-x^2}\right)} = f(t) + c$ (Where $t = \sqrt{\frac{5-x}{x-2}}$ and $f(0) = k\ln\frac{3-\sqrt{5}}{3+\sqrt{5}}$ $(k > 0)$ then $k^2$ is equal to ____.

If $\int \frac{\sqrt{2-x-x^2}}{x^2} dx = \frac{A\sqrt{2-x-x^2}}{x} + \frac{B}{4\sqrt{2}} \ln\left|\frac{4-x+4\sqrt{2-x-x^2}}{x}\right| - \sin^{-1}\left(\frac{2x+1}{3}\right) + c$ then $|A+B|$ is equal to ____.

The value of $\lim_{x \to \frac{\pi}{2}} \sqrt{\frac{\tan x - \sin\left[\tan^{-1}(\tan x)\right]}{\tan x + \cos^2(\tan x)}}$ is______.

The figure shows two regions in the first quadrant. $A(t)$ is the area under the curve $y = \sin x^2$ from $0$ to $t$ and $B(t)$ is the area of the triangle with vertices $O$, $P$ and $M(t, 0)$. If $\lim_{t \to 0} \frac{A(t)}{B(t)} = \frac{1}{k}$, then $k$ is______.

The figure shows a right triangle with its hypotenuse $OB$ along the $y$-axis and its vertex $A$ on the parabola $y = x^2$. Let $h$ represents the length of the hypotenuse which depends on the $x$-coordinate of the point $A$. The value of $\lim_{x \to 0} (h)$ equals

Let $k(x) = \int \frac{(x^2 + 1)dx}{\sqrt[4]{x^4 + 3x + 6}}$ and $k(-1) = \frac{1}{3\sqrt{2}}$, then the value of $k(-2)$ is

Suppose $\begin{vmatrix} f'(x) & f(x) \\ f''(x) & f'(x) \end{vmatrix} = 0$ where $f(x)$ is differentiable function with $f'(x) \neq 0$ & satisfies $f(0) = 1, f'(0) = 2$. If $f(x) = e^{\lambda x} + \mu$ then $\lambda + \mu$ is ____.

Let $P(x)$ be a quadratic polynomial with $P(1)=-1$. If $\displaystyle\int\frac{P(x)\,dx}{(2x-3)^2(3x-2)^2}=\frac{1}{5}\ln|f(x)|+C$ where $f(x)$ is rational and $\lim_{x\to\infty}f(x)=\frac{4}{3}$, then $|f(1)|$ is

Let $f$ be a twice differentiable function defined in $[-3,3]$ such that $f(0) = -4, f'(3) = 0, f'(-3) = 12$ and $f''(x) \geq -2\forall x \in [-3,3]$. If $g(x) = \int_0^x f(t)dt$ then find maximum value of $g(x), x \in [-3,3]$

Let $f(x) = x\tan^{-1}(x^2) + x^4$. Let $f^k(x)$ denotes kth derivative of $f(x)$ w.r.t. $x$, $k \in \mathbb{N}$. If $f^{2m}(0) \neq 0$, $m \in \mathbb{N}$, then $m$ equals to______.

If $\int_x^e f(t)dt = \sin x - \cos x - \frac{x^2}{2}$ for all $x \in R$ then value of $-2f\left(\frac{\pi}{6}\right)$ is _____.

y = $\tan^{-1}(\cot x) + \cot^{-1}(\tan x)$, $\frac{\pi}{2} < x < \pi$, then $\left(-\frac{dy}{dx}\right)$ is equal to_____.

Let $f(x) = \tan^{-1} x, |x| \leq 1 = \frac{\pi}{4} \operatorname{sgn} x + \frac{x-1}{2}, |x| > 1$, (where $\operatorname{sgn}$ denotes signum function). Then the value of $4f\left(1^{+}\right)$ equals.

If $a^2 + b^2 = 1$ and $u$ is the minimum value of $\frac{b+1}{a+b-2}$, then find the value of $u^2$.

If $\int \frac{\sqrt{1+x^{2n}}\ln(1+x^{2n}) - 2n\ln x}{x^{2n+1}} dx = \frac{\alpha p}{\beta n}[1-3\ln p] + c$ where $p = \sqrt{1+\frac{1}{x^{2n}}}, \alpha,\beta \in \mathbb{N}$ then $\alpha + \beta = $

If $\int \frac{2\cos x - \sin x + \lambda}{\cos x + \sin x + 2} dx = A\ln|\cos x + \sin x - 2| + Bx + C$. Then the value of $A + B + |\lambda|$ is

Let $f$ be a differentiable function on $R$ and satisfying $f(x) = -(x^2 - x + 1)e^x + \int_0^x e^{xy} f'(y)dy$. If $f(1) + f'(1) + f''(1) = ke$, where $k \in N$, then find $k$.

Let $f : R \to R$ be a differentiable function satisfying $f(x) = f(x)f(x-y), \forall x, y \in R$ and $f'(0) = \int_0^{3}\{2x\}dx$, where $\{.\}$ denotes the fractional part function and $f'(-3) = \alpha e^\beta$. Then, $|\alpha + \beta|$ is equal to_____.

In a $\triangle ABC$, angles $A, B, C$ are in A.P. If $f(C) = \lim_{A \to C} \frac{\sqrt{3} - 4\sin A \sin C}{A - C}$, then $f\left(\frac{\pi}{12}\right)$ is equal to______.

Let $a = \min(x^2 + 2x + 3, x \in \mathbb{R})$ and $b = \lim_{x \to 0} \frac{\sin x \cos x}{e^x - e^{-x}}$. Then the value of $\sum_{r=0}^{n} a^r b^{n-r}$ is :

$\lim_{n \to \infty} \left(\frac{\sqrt[n]{p} + \sqrt[n]{q}}{2}\right)^n$, $p, q > 0$ equals :

Consider a real valued continuous function $f$ such that $f(x) = \sin x + \int_{-\pi/2}^{\pi/2} (\sin x + tf(t))dt$. If $M$ and $m$ are maximum and minimum value of the function $f$, then the value of $M/m$ is _____.

The values of $\text{Lim}_{x \to 0^+} \frac{f(-x)-x^2}{1-\cos x}$ where $[\cdot]$ denote greatest integer function and $(\cdot)$ denote fraction part function.

Calculus Questions (384)