JEE Main 2025 — Calculus

The integral ∫cos(x)dx equals:

Let $y=f(x)$ be the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{x y}{x^2-1}=\frac{x^6+4 x}{\sqrt{1-x^2}},-1 \lt x \lt 1$ such that $f(0)=0$. If $6 \int_{-1 / 2}^{1 / 2} f(x) \mathrm{d} x=2 \pi-\alpha$ then $\alpha^2$ is equal to _______ .

Let $y=y(x)$ be the solution curve of the differential equation $x\left(x^2+e^x\right) d y+\left(e^x(x-2) y-x^3\right) d x=0, x \gt 0$ passing through the point $(1,0)$. Then $y(2)$ is equal to :

Let $\mathrm{I}(x)=\int \frac{d x}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $\mathrm{I}(37)-\mathrm{I}(24)=\frac{1}{4}\left(\frac{1}{\mathrm{~b}^{\frac{1}{13}}}-\frac{1}{\mathrm{c}^{\frac{1}{13}}}\right), \mathrm{b}, \mathrm{c} \in \mathrm{N}$, then $3(\mathrm{~b}+\mathrm{c})$ is equal to

Let the function $f(x)=\left(x^2+1\right)\left|x^2-a x+2\right|+\cos |x|$ be not differentiable at the two points $x=\alpha=2$ and $x=\beta$. Then the distance of the point $(\alpha, \beta)$ from the line $12 x+5 y+10=0$ is equal to :

If the area of the region $\left\{(x, y):\left|4-x^2\right| \leq y \leq x^2, y \leq 4, x \geq 0\right\}$ is $\left(\frac{80 \sqrt{2}}{\alpha}-\beta\right), \boldsymbol{\alpha}, \boldsymbol{\beta} \in \mathbf{N}$, then $\alpha+\beta$ is equal to _______.

Let the area of the bounded region $\left\{(x, y): 0 \leq 9 x \leq y^2, y \geq 3 x-6\right\}$ be A. Then 6 A is equal to ________

Let $m$ and $n$ be the number of points at which the function $f(\mathrm{x})=\max \left\{\mathrm{x}, \mathrm{x}^3, \mathrm{x}^5, \ldots ., \mathrm{x}^{21}\right\}, \mathrm{x} \in \mathbb{R}$, is not differentiable and not continuous, respectively. Then $\mathrm{m}+\mathrm{n}$ is equal to ________ .

The area of the region enclosed by the curves $y=x^2-4 x+4$ and $y^2=16-8 x$ is :

Let $\mathrm{f}(x)=\left\{\begin{array}{lc}3 x, & x \lt 0 \\ \min \{1+x+[x], x+2[x]\}, & 0 \leq x \leq 2 \\ 5, & x\gt2,\end{array}\right.$ where [.] denotes 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 [.] denote the greatest integer function. If $\int_0^{e^3}\left[\frac{1}{\mathrm{e}^{\mathrm{x}-1}}\right] \mathrm{dx}=\alpha-\log _{\mathrm{e}} 2$, then $\alpha^3$ is equal to _______ .

Given below are two statements : Statement I : $\lim _{x \rightarrow 0}\left(\frac{\tan ^{-1} x+\log _e \sqrt{\frac{1+x}{1-x}}-2 x}{x^5}\right)=\frac{2}{5}$ Statement II : $\lim _{\mathrm{x} \rightarrow 1}\left(\mathrm{x}^{\frac{2}{1-\mathrm{x}}}\right)=\frac{1}{\mathrm{e}^2}$ In the light of the above statements, choose the correct answer from the options given below :

If the set of all values of a, for which the equation $5 x^3-15 x-a=0$ has three distinct real roots, is the interval $(\alpha, \beta)$, then $\beta-2 \alpha$ is equal to ______

If $\operatorname{Lim}_{x \rightarrow 0}\left(\frac{\tan x}{x}\right)^{\frac{1}{x^2}}=p$, then $96 \log _e p$ is equal to ______

If $\lim _{x \rightarrow 0} \frac{\cos (2 x)+a \cos (4 x)-b}{x^4}$ is finite, then $(a+b)$ is equal to :

Let [t] be the greatest integer less than or equal to t. Then the least value of \(\mathrm{p} \in \mathbf{N}\) for which \(\lim _{x \rightarrow 0^{+}}\left(x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+\ldots+\left[\frac{\mathrm{p}}{x}\right]\right)-x^2\left(\left[\frac{1}{x^2}\right]+\left[\frac{2^2}{x^2}\right]+\ldots+\left[\frac{9^2}{x^2}\right]\right)\right) \geq 1\) is equal to ________.

For $\alpha, \beta, \gamma, \in \mathbf{R}$, if $\lim _{x \rightarrow 0} \frac{x^2 \sin \alpha x+(\gamma-1) e^{x^2}}{\sin 2 x-\beta x}=3$, then $\beta+\gamma-\alpha$ is equal to:

Let $f(x)=\lim _{\mathrm{n} \rightarrow \infty} \sum_{\mathrm{r}=0}^{\mathrm{n}}\left(\frac{\tan \left(x / 2^{r+1}\right)+\tan ^3\left(x / 2^{r+1}\right)}{1-\tan ^2\left(x / 2^{r+1}\right)}\right)$. Then $\lim _{x \rightarrow 0} \frac{\mathrm{e}^x-\mathrm{e}^{f(x)}}{(x-f(x))}$ is equal to

$\lim _{x \rightarrow 0} \operatorname{cosec} x\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right)$ is:

$\lim _{x \rightarrow \infty} \frac{\left(2 x^2-3 x+5\right)(3 x-1)^{\frac{x}{2}}}{\left(3 x^2+5 x+4\right) \sqrt{(3 x+2)^x}}$ is equal to :

If $\lim _{x \rightarrow \infty}\left(\left(\frac{\mathrm{e}}{1-\mathrm{e}}\right)\left(\frac{1}{\mathrm{e}}-\frac{x}{1+x}\right)\right)^x=\alpha$, then the value of $\frac{\log _{\mathrm{e}} \alpha}{1+\log _{\mathrm{e}} \alpha}$ equals :

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

If the area of the region $\left\{(\mathrm{x}, \mathrm{y}): 1+\mathrm{x}^2 \leq \mathrm{y} \leq \min \{\mathrm{x}+7,11-3 \mathrm{x}\}\right\}$ is A , then 3 A is equal to

If $\begin{aligned} \int\left(\frac{1}{x}+\frac{1}{x^3}\right) & \left(\sqrt[23]{3 x^{-24}+x^{-26}}\right) d x \\ & =-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^\gamma\right)^{\frac{\alpha+1}{\alpha}}+C, x \gt 0,\end{aligned}$ $(\alpha, \beta, \gamma \in Z)$, where $C$ is the constant of integration, then $\alpha+\beta+\gamma$ is equal to ________ .

Let $\int x^3 \sin x \mathrm{~d} x=g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\right)+g^{\prime}\left(\frac{\pi}{2}\right)\right)=\alpha \pi^3+\beta \pi^2+\gamma, \alpha, \beta, \gamma \in Z$, then $\alpha+\beta-\gamma$ equals :

Let $f$ be a differentiable function such that $2(x+2)^2 f(x)-3(x+2)^2=10 \int_0^x(t+2) f(t) d t, x \geq 0$. Then $f(2)$ is equal to ______.

If the function $f(x)=\frac{\tan (\tan x)-\sin (\sin x)}{\tan x-\sin x}$ is continuous at $\mathrm{x}=0$, then $f(0)$ is equal to ________

$\operatorname{IfI}(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n\gt0$, then $I(9,14)+I(10,13)$ is

The number of points of discontinuity of the function $f(\mathrm{x})=\left[\frac{\mathrm{x}^2}{2}\right]-[\sqrt{\mathrm{x}}], \mathrm{x} \in[0,4]$, where $[\cdot]$ denotes the greatest integer function is ________

Let $f(\mathrm{x})= \begin{cases}(1+\mathrm{ax})^{1 / \mathrm{x}} & , \quad \mathrm{x} \lt 0 \\ 1+\mathrm{b} & , \quad \mathrm{x}=0 \\ \frac{(\mathrm{x}+4)^{1 / 2}-2}{(\mathrm{x}+\mathrm{c})^{1 / 3}-2} & ,\end{cases}$ be continuous at $x=0$. Then $e^a b c$ is equal to

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x)=[x]+|x-2|,-2 \lt x \lt 3$, is not continuous and not differentiable. Then $\mathrm{m}+\mathrm{n}$ is equal to :

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for all $x, y \in \mathbf{R}$. Then $\sum_{\mathrm{n}=1}^{100} \log _{\mathrm{e}} f(\mathrm{n})$ is equal to :

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x$ $\sin \mathrm{y})(f(2 \mathrm{x}+2 \mathrm{y})+f(2 \mathrm{x}-2 \mathrm{y}))$, for all $\mathrm{x}, \mathrm{y} \in \mathbf{R}$. If $f^{\prime}(0)=\frac{1}{2}$, then the value of $24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)$ is:

The integral $\int_{-1}^{\frac{3}{2}}\left(\left|\pi^2 x \sin (\pi x)\right|\right) d x$ is equal to:

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a polynomial function of degree four having extreme values at $x=4$ and $x=5$. If $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}=5$, then $f(2)$ is equal to :

Let $x=-1$ and $x=2$ be the critical points of the function $\mathrm{f}(\mathrm{x})=\mathrm{x}^3+\mathrm{ax}^2+\mathrm{b} \log _{\mathrm{c}}|\mathrm{x}|+1, \mathrm{x} \neq 0$. Let $m$ and $M$ respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2,-\frac{1}{2}\right]$. Then $|\mathrm{M}+m|$ is equal to (Take $\log _{\mathrm{c}} 2=0.7$ ):

Let the function $f(x)=\frac{x}{3}+\frac{3}{x}+3, x \neq 0$ be strictly increasing in $\left(-\infty, \alpha_1\right) \mathrm{U}\left(\alpha_2, \infty\right)$ and strictly decreasing in $\left(\alpha_3, \alpha_4\right) \mathrm{U}\left(\alpha_4, \alpha_5\right)$. Then $\sum_{\mathrm{i}=1}^5 \alpha_{\mathrm{i}}^2$ is equal to :-

The shortest distance between the curves $y^2=8 \mathrm{x}$ and $x^2+y^2+12 y+35=0$ is :

Let $\mathrm{a} \gt 0$. If the function $\mathrm{f}(\mathrm{x})=6 \mathrm{x}^3-45 \mathrm{a} \mathrm{x}^2+108 \mathrm{a}^2 \mathrm{x}+1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1 x_2=54$, then $\mathrm{a}+\mathrm{x}_1+\mathrm{x}_2$ is equal to :-

Let f be a differentiable function on $\mathbf{R}$ such that $\mathrm{f}(2) = 1$, $f^{\prime}(2)=4$. Let $\lim _{x \rightarrow 0}(f(2+x))^{3 / x}=e^\alpha$. Then the number of times the curve $y=4 x^3-4 x^2-4(\alpha-7) x-\alpha$ meets x -axis is :-

Let $(2,3)$ be the largest open interval in which the function $f(x)=2 \log _{\mathrm{e}}(x-2)-x^2+a x+1$ is strictly increasing and (b, c) be the largest open interval, in which the function $\mathrm{g}(x)=(x-1)^3(x+2-\mathrm{a})^2$ is strictly decreasing. Then $100(a+b-c)$ is equal to :

The area of the region, inside the circle $(x-2 \sqrt{3})^2+y^2=12$ and outside the parabola $y^2=2 \sqrt{3} x$ is :

Let $f(x)=\int_0^t t\left(t^2-9 t+20\right) d t, 1 \leq x \leq 5$. If the range of $f$ is $[\alpha, \beta]$, then $4(\alpha+\beta)$ equals :

If $\int \frac{\left(\sqrt{1+x^2}+x\right)^{10}}{\left(\sqrt{1+x^2}-x\right)^9} d x=$ $\frac{1}{m}\left(\left(\sqrt{1+x^2}+x\right)^n\left(n \sqrt{1+x^2}-x\right)\right)+C \text { where } C$ is the constant of integration and $m, n \in N$, then $\mathrm{m}+\mathrm{n}$ is equal to

Let $f(x)=\int x^3 \sqrt{3-x^2} d x$. If $5 f(\sqrt{2})=-4$, then $f(1)$ is equal to

If $\int \frac{2 x^2+5 x+9}{\sqrt{x^2+x+1}} \mathrm{~d} x=x \sqrt{x^2+x+1}+\alpha \sqrt{x^2+x+1}+\beta \log _e\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|+\mathrm{C}$, where $C$ is the constant of integration, then $\alpha+2 \beta$ is equal to $\qquad$ _______.

If $f(x)=\int \frac{1}{x^{1 / 4}\left(1+x^{1 / 4}\right)} \mathrm{d} x, f(0)=-6$, then $f(1)$ is equal to :

If $\int \mathrm{e}^x\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}+\frac{x}{1-x^2}\right) \mathrm{d} x=\mathrm{g}(x)+\mathrm{C}$, where C is the constant of integration, then $g\left(\frac{1}{2}\right)$ equals :

Let $f(x)$ be a positive function and $I_1=\int_{-\frac{1}{2}}^1 2 x f(2 x(1-2 x)) d x$ and $I_2=\int_{-1}^2 f(x(1-x)) d x$. Then the value of $\frac{I_2}{I_1}$ is equal to ________

$4 \int_0^1\left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right) d x-3 \log _e(\sqrt{3})$ is equal to :

Let the domain of the function $f(\mathrm{x})=\log _2 \log _4 \log _6\left(3+4 x-x^2\right)$ be $(\mathrm{a}, \mathrm{~b})$. If $\int_0^{\mathrm{b}-\mathrm{a}}\left[\mathrm{x}^2\right] \mathrm{dx}=\mathrm{p}-\sqrt{\mathrm{q}}-\sqrt{\mathrm{r}}, \mathrm{p}, \mathrm{q},$ $\mathrm{r} \in \mathbb{N}, \operatorname{gcd}(\mathrm{p}, \mathrm{q}, \mathrm{r})=1,$ where [$\cdot]$ is the greatest integer function, then $\mathrm{p}+\mathrm{q}+\mathrm{r}$ is equal to

The integral $\int_0^\pi \frac{8 x d x}{4 \cos ^2 x+\sin ^2 x}$ is equal to

The integral \(80 \int_0^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta\) is equal to :

If $24 \int_0^{\frac{\pi}{4}}\left(\sin \left|4 x-\frac{\pi}{12}\right|+[2 \sin x]\right) \mathrm{d} x=2 \pi+\alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to _______.

Let $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f(2)=1$. If $\mathrm{F}(x)=x f(x)$ for all $x \in \mathbf{R}$, $\int_0^2 x \mathrm{~F}^{\prime}(x) \mathrm{d} x=6$ and $\int_0^2 x^2 \mathrm{~F}^{\prime \prime}(x) \mathrm{d} x=40$, then $\mathrm{F}^{\prime}(2)+\int_0^2 \mathrm{~F}(x) \mathrm{d} x$ is equal to :

If $\mathrm{I}=\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} \mathrm{~d} x$, then $\int_0^{21} \frac{x \sin x \cos x}{\sin ^4 x+\cos ^4 x} \mathrm{~d} x$ equals :

Let for some function $\mathrm{y}=f(x), \int_0^x t f(t) d t=x^2 f(x), x\gt0$ and $f(2)=3$. Then $f(6)$ is equal to

The value of $\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}}{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}+e^{\left(\left(6-\log _e x\right)^2+1\right)^{-1}}}\right) d x$ is

If the area of the region bounded by the curves \(y=4-\frac{x^2}{4}\) and \(y=\frac{x-4}{2}\) is equal to \(\alpha\), then \(6 \alpha\) equals

If the area of the region $\{(x, y):|x-5| \leq y \leq 4 \sqrt{x}\}$ is A , then 3 A is equal to _____ .

The area of the region $\{(x, y):|x-y| \leq y \leq 4 \sqrt{x}\}$ is

Let the area enclosed between the curves $|y|=1-x^2$ and $x^2+y^2=1$ be $\alpha$. If $9 \alpha=\beta \pi+\gamma ; \beta, \gamma$ are integers, then the value of $|\beta-\gamma|$ equals.

Let the area of the region \(\left\{(x, y): 2 y \leq x^2+3, y+|x| \leq 3, y \geqslant|x-1|\right\}\) be A. Then 6 A is equal to :

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

The area of the region enclosed by the curves $y=\mathrm{e}^x, y=\left|\mathrm{e}^x-1\right|$ and $y$-axis is:

Consider the region $R=\left\{(x, y): x \leq y \leq 9-\frac{11}{3} x^2, x \geq 0\right\}$. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is:

The area of the region $\left\{(x, y): x^2+4 x+2 \leq y \leq|x+2|\right\}$ is equal to

If the area of the larger portion bounded between the curves $x^2+y^2=25$ and $y=|x-1|$ is $\frac{1}{4}(b \pi+c), b, c \in N$, then $b+c$ is equal to

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f(x+y)=f(x) f(y)$ for all $x, y \in \mathbf{R}$. If $f^{\prime}(0)=4 \mathrm{a}$ and $f$ satisfies $f^{\prime \prime}(x)-3 \mathrm{a} f^{\prime}(x)-f(x)=0$, $\mathrm{a}\gt0$, then the area of the region $\mathrm{R}=\{(x, y) \mid 0 \leq y \leq f(\mathrm{a} x), 0 \leq x \leq 2\}$ is:

If a curve $y=y(x)$ passes through the point $\left(1, \frac{\pi}{2}\right)$ and satisfies the differential equation $\left(7 x^4 \cot y-e^x \operatorname{cosec} y\right) \frac{d x}{d y}=x^5, x \geq 1$, then at $x=2$, the value of cosy is:

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+1\right) y^{\prime}-2 x y=\left(x^4+2 x^2+1\right) \cos x$, $y(0)=1$. Then $\int_{-3}^3 y(x) d x$ is :

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+2 y \sec ^2 x=2 \sec ^2 x+3 \tan x \cdot \sec ^2 x$ such that $\mathrm{y}(0)=\frac{5}{4}$. Then $12\left(\mathrm{y}\left(\frac{\pi}{4}\right)-\mathrm{e}^{-2}\right)$ is equal to _______.

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+3\left(\tan ^2 x\right) y+3 y=\sec ^2 x$ $y(0)=\frac{1}{3}+e^3$. Then $y\left(\frac{\pi}{4}\right)$ is equal to

Let $f:[0, \infty) \rightarrow \mathbb{R}$ be differentiable function such that $f(\mathrm{x})=1-2 \mathrm{x}+\int_0^x e^{x-t} f(t) \mathrm{dt}$ for all $\mathrm{x} \in[0, \infty)$. Then the area of the region bounded by $\mathrm{y}=f(\mathrm{x})$ and the coordinate axes is

Let $f:[1, \infty) \rightarrow[2, \infty)$ be a differentiable function, If $10 \int_1^{\mathrm{x}} f(\mathrm{t}) \mathrm{dt}=5 \mathrm{x} f(\mathrm{x})-\mathrm{x}^5-9$ for all $\mathrm{x} \geq 1$, then the value of $f(3)$ is :

Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{\mathrm{~d} y}{\mathrm{~d} x}=\sin 2 x-4 y \sin x, x \in\left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right)=0$, then $y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)$ is equal to $\qquad$ ________.

If $y=y(x)$ is the solution of the differential equation, $\sqrt{4-x^2} \frac{\mathrm{~d} y}{\mathrm{~d} x}=\left(\left(\sin ^{-1}\left(\frac{x}{2}\right)\right)^2-y\right) \sin ^{-1}\left(\frac{x}{2}\right),-2 \leq x \leq 2, y(2)=\frac{\pi^2-8}{4}$, then $y^2(0)$ is equal to

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a thrice differentiable odd function satisfying $f^{\prime}(\mathrm{x}) \geq 0, f^{\prime}(\mathrm{x})=f(\mathrm{x}), f(0)=0, f^{\prime}(0)=3$. Then $9 f\left(\log _{\mathrm{c}} 3\right)$ is equal to _______.

Let \(y=y(x)\) be the solution of the differential equation \(\cos x\left(\log _{\mathrm{e}}(\cos x)\right)^2 \mathrm{dy}+\left(\sin x-3 y \sin x \log _{\mathrm{e}}(\cos x)\right) \mathrm{d} x=0, x \in\left(0, \frac{\pi}{2}\right)\). If \(y\left(\frac{\pi}{4}\right)=\frac{-1}{\log _{\mathrm{e}} 2}\), then \(y\left(\frac{\pi}{6}\right)\) is equal to :

If $x=f(y)$ is the solution of the differential equation $\left(1+y^2\right)+\left(x-2 \mathrm{e}^{\tan ^{-1} y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with $f(0)=1$, then $f\left(\frac{1}{\sqrt{3}}\right)$ is equal to :

Let a curve $y=f(x)$ pass through the points $(0,5)$ and $\left(\log _e 2, k\right)$. If the curve satisfies the differential equation $2(3+y) e^{2 x} d x-\left(7+e^{2 x}\right) d y=0$, then $k$ is equal to

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(x y-5 x^2 \sqrt{1+x^2}\right) d x+\left(1+x^2\right) d y=0, y(0)=0$. Then $y(\sqrt{3})$ is equal to

Let $x=x(y)$ be the solution of the differential equation $y^2 \mathrm{~d} x+\left(x-\frac{1}{y}\right) \mathrm{d} y=0$. If $x(1)=1$, then $x\left(\frac{1}{2}\right)$ is :

If $\lim _{\mathrm{t} \rightarrow 0}\left(\int_0^1(3 x+5)^{\mathrm{t}} \mathrm{d} x\right)^{\frac{1}{t}}=\frac{\alpha}{5 \mathrm{e}}\left(\frac{8}{5}\right)^{\frac{2}{3}}$, then $\alpha$ is equal to ________

The area of the region bounded by the curve $y=\max \{|x|, x|x-2|\}$, then $x$-axis and the lines $x=-2$ and $x=4$ is equal to _______ .

Let the function, $f(x)= \begin{cases}-3 a x^2-2, & x \lt 1 \\ a^2+b x, & x \geqslant 1\end{cases}$ be differentiable for all $x \in \mathbf{R}$, where $\mathbf{a}\gt1, \mathbf{b} \in \mathbf{R}$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha+\beta \sqrt{3}, \alpha, \beta \in Z$, then the value of $\alpha+\beta$ is ________

If for the solution curve $y=f(x)$ of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+(\tan x) y=\frac{2+\sec x}{(1+2 \sec x)^2}$, $x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right), f\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{10}$, then $f\left(\frac{\pi}{4}\right)$ is equal to :

Let for $f(x)=7 \tan ^8 x+7 \tan ^6 x-3 \tan ^4 x-3 \tan ^2 x, \quad \mathrm{I}_1=\int_0^{\pi / 4} f(x) \mathrm{d} x$ and $\mathrm{I}_2=\int_0^{\pi / 4} x f(x) \mathrm{d} x$. Then $7 \mathrm{I}_1+12 \mathrm{I}_2$ is equal to :

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

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of $81 \mathrm{~cm}^3 / \mathrm{min}$ and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4 \pi} \mathrm{~cm} / \mathrm{min}$. The surface area (in $\mathrm{cm}^2$ ) of the chocolate ball (without the ice-cream layer) is :

Let \(f:(0, \infty) \rightarrow \mathbf{R}\) be a twice differentiable function. If for some \(\mathrm{a} \neq 0, \int_0^1 f(\lambda x) \mathrm{d} \lambda=\mathrm{a} f(x), f(1)=1\) and \(f(16)=\frac{1}{8}\), then \(16-f^{\prime}\left(\frac{1}{16}\right)\) is equal to _______.

$\lim _{x \rightarrow 0^{+}} \frac{\tan \left(5(x)^{\frac{1}{3}}\right) \log _e\left(1+3 x^2\right)}{\left(\tan ^{-1} 3 \sqrt{x}\right)^2\left(e^{5(x)^{\frac{4}{3}}}-1\right)}$ is equal to

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined by $f(x)=\|x+2|-2| x\|$. If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$, then $m+n$ is

If $\quad \lim _{x \rightarrow 1^{+}} \frac{(x-1)(6+\lambda \cos (x-1))+\mu \sin (1-x)}{(x-1)^3}=-1$, where $\lambda, \mu \in \mathbb{R}$, then $\lambda+\mu$ is equal to

Let $g$ be a differentiable function such that $\int_0^x g(t) d t=x-\int_0^x \operatorname{tg}(t) d t, x \geq 0$ and let $y=y(x)$ satisfy the differential equation $\frac{d y}{d x}-y \tan x=$ $2(x+1) \sec x g(x), x \in\left[0, \frac{\pi}{2}\right)$. If $y(0)=0$, then $y\left(\frac{\pi}{3}\right)$ is equal to

The integral $\int_0^\pi \frac{(x+3) \sin x}{1+3 \cos ^2 x} d x$ is equal to :

If the area of the region $\left\{(x, y):-1 \leq x \leq 1,0 \leq y \leq a+\mathrm{e}^{|x|}-\mathrm{e}^{-x}, \mathrm{a}\gt0\right\}$ is $\frac{\mathrm{e}^2+8 \mathrm{e}+1}{\mathrm{e}}$, then the value of $a$ is :

Let $(a, b)$ be the point of intersection of the curve $x^2=2 y$ and the straight line $y-2 x-6=0$ in the second quadrant. Then the integral $I=\int_a^b \frac{9 x^2}{1+5^x} d x$ is equal to :

If the function $f(x)=2 x^3-9 \mathrm{ax}^2+12 \mathrm{a}^2 \mathrm{x}+1$, where $\mathrm{a} \gt 0$, attains its local maximum and local minimum values at $p$ and $q$, respectively, such that $\mathrm{p}^2=\mathrm{q}$, then $f(3)$ is equal to:

If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96 x^2 \cos ^2 x}{\left(1+e^x\right)} \mathrm{d} x=\pi\left(\alpha \pi^2+\beta\right), \alpha, \beta \in \mathbb{Z}$, then $(\alpha+\beta)^2$ equals

The sum of all local minimum values of the function $$ f(x)=\left\{\begin{array}{lr} 1-2 x, & x \lt -1 \\ \frac{1}{3}(7+2|x|), & -1 \leq x \leq 2 \\ \frac{11}{18}(x-4)(x-5), & x\gt2 \end{array}\right. $$ is

The value of $\int_{-1}^1 \frac{(1+\sqrt{|x|-x}) e^x+(\sqrt{|x|-x}) e^{-x}}{e^x+e^{-x}} d x$ is equal to

If the function $f(x)=\left\{\begin{array}{l}\frac{2}{x}\left\{\sin \left(k_1+1\right) x+\sin \left(k_2-1\right) x\right\}, \quad x \lt 0 \\ 4, \quad x=0 \\ \frac{2}{x} \log _e\left(\frac{2+k_1 x}{2+k_2 x}\right), \quad x\gt0\end{array}\right.$ is continuous at $\mathrm{x}=0$, then $\mathrm{k}_1^2+\mathrm{k}_2^2$ is equal to

Let $f(x)=\int_0^{x^2} \frac{\mathrm{t}^2-8 \mathrm{t}+15}{\mathrm{e}^{\mathrm{t}}} \mathrm{dt}, x \in \mathbf{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :

Let $f:(0, \infty) \rightarrow \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f^{\prime}(x)=2 x f(x)+3$, with $f(1)=4$. Then $2 f(2)$ is equal to :

Let $f(x)=x-1$ and $g(x)=e^x$ for $x \in \mathbb{R}$. If $\frac{d y}{d x}=\left(e^{-2 \sqrt{x}} g(f(f(x)))-\frac{y}{\sqrt{x}}\right), y(0)=0$, then $y(1)$ is :-

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y\gt0$ and $x(1)=\frac{\pi}{2}$. Then $\cos (x(2))$ is equal to :