JEE Main Mathematics — Calculus previous year questions with solutions.
If $f(t)=\int_0^\pi \frac{2 x \mathrm{~d} x}{1-\cos ^2 \mathrm{t} \sin ^2 x}, 0 < \mathrm{t} < \pi$, then the value of $\int_0^{\frac{\pi}{2}} \frac{\pi^2 \mathrm{dt}}{f(\mathrm{t})}$ equals_________
The solution curve of the differential equation $y\frac{dx}{dy}=x({\mathrm{log}}_{e}x-{\mathrm{log}}_{e}y+1),x>0,y>0$ passing through the point $(e,1)$ is
Let $y=y(x)$ be the solution of the differential equation $(1-{x}^{2})dy=[xy+({x}^{3}+2)\sqrt{3(1-{x}^{2})}]dx$, $-1<x<1,y(0)=0$. If $y(\frac{1}{2})=\frac{m}{n},m$ and $n$ are coprime numbers, then $m+n$ is equal to __________.
Let $\alpha|x|=|y| \mathrm{e}^{x y-\beta}, \alpha, \beta \in \mathbf{N}$ be the solution of the differential equation $x \mathrm{~d} y-y \mathrm{~d} x+x y(x \mathrm{~d} y+y \mathrm{~d} x)=0$, $y(1)=2$. Then $\alpha+\beta$ is equal to ________
The area of the region in the first quadrant inside the circle $x^2+y^2=8$ and outside the parabola $y^2=2 x$ is equal to :
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{{4}^{x}}{{4}^{x}+2}$ and $M=\int _{f(a)}^{f(1-a)}x{\mathrm{sin}}^{4}(x(1-x))dx$, $N=\int _{f(a)}^{f(1-a)}{\mathrm{sin}}^{4}(x(1-x))dx;a\neq \frac{1}{2}$. If $\alpha M=\beta N,\alpha ,\beta \in \mathbb{N}$, then the least value of ${\alpha }^{2}+{\beta }^{2}$ is equal to ______
If $\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cossc}^2 x+\frac{3}{2}\right)+\beta \log _\epsilon\left|\tan \frac{x}{2}\right|+C$ where $\alpha, \beta \in \mathbb{R}$ and $\mathrm{C}$ is the constant of integration, then the value of $8(\alpha+\beta)$ equals _______
Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y-y={Y}^{'}(x)(X-x)$ and the co-ordinate axes, where $(x,y)$ is any point on the curve, is always $\frac{-{y}^{2}}{2{Y}^{'}(x)}+1,{Y}^{'}(x)\neq 0$. If $Y(1)=1$, then $12Y(2)$ equals ________.
Let the solution $y=y(x)$ of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+10$ is equal to ______
Let $f:\rightarrow R\rightarrow (0,\infty )$ be strictly increasing function such that $\underset{x\rightarrow \infty }{\mathrm{lim}}\frac{f(7x)}{f(x)}=1$. Then, the value of $\underset{x\rightarrow \infty }{\mathrm{lim}}[\frac{f(5x)}{f(x)}-1]$ is equal to
If the function $f(x)=2 x^3-9 x^2+12 \mathrm{a}^2 x+1, \mathrm{a}>0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^2$, then $\alpha$ and $\alpha^2$ are the roots of the equation :
Let $\lim _{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right.$ $\left.+\ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right)$ be $\frac{\pi}{k}$, using only the principal values of the inverse trigonometric functions. Then $\mathrm{k}^2$ is equal to ________
If $\int \frac{1}{\mathrm{a}^2 \sin ^2 x+\mathrm{b}^2 \cos ^2 x} \mathrm{~d} x=\frac{1}{12} \tan ^{-1}(3 \tan x)+$ constant, then the maximum value of $\mathrm{a} \sin x+\mathrm{b} \cos x$, is :
The integral $\int \frac{({x}^{8}-{x}^{2})\mathrm{dx}}{({x}^{12}+3{x}^{6}+1){\mathrm{tan}}^{-1}({x}^{3}+\frac{1}{{x}^{3}})}$ is equal to :
Let $y=y(x)$ be the solution of the differential equation $\left(1+x^2\right) \frac{d y}{d x}+y=e^{\tan ^{-1} x}$, $y(1)=0$. Then $y(0)$ is
Suppose the solution of the differential equation $\frac{d y}{d x}=\frac{(2+\alpha) x-\beta y+2}{\beta x-2 \alpha y-(\beta \gamma-4 \alpha)}$ represents a circle passing through origin. Then the radius of this circle is :
The function $f(x)=2x+3{x}^{\frac{2}{3}},x\in R$, has
The area of the region enclosed by the parabola $y=4x-{x}^{2}$ and $3y={(x-4)}^{2}$ is equal to
Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 x}{\left(1+x^2\right)^2} y=x \mathrm{e}^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0 .$ Then the area enclosed by the curve $f(x)=y(x) \mathrm{e}^{-\frac{1}{\left(1+x^2\right)}}$ and the line $y-x=4$ is__________
$\int_0^{\pi / 4} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x$ is equal to
The area of the region enclosed by the parabolas $y=x^2-5 x$ and $y=7 x-x^2$ is
Let $f:R\rightarrow R$ be defined $f(x)=a{e}^{2x}+b{e}^{x}+cx$. If $f(0)=-1,{f}^{'}({\mathrm{log}}_{e}2)=21$ and ${\int }_{0}^{\mathrm{log}4}(f(x)-cx)dx=\frac{39}{2}$, then the value of $|a+b+c|$ equals:
If the solution of the differential equation $(2x+3y-2)dx+(4x+6y-7)dy=0,y(0)=3$, is $\alpha x+\beta y+3{\mathrm{log}}_{e}|2x+3y-\gamma |=6$, then $\alpha +2\beta +3\gamma$ is equal to ______.
If $a=\underset{x\rightarrow 0}{\mathrm{lim}}\frac{\sqrt{1+\sqrt{1+{x}^{4}}}-\sqrt{2}}{{x}^{4}}$ and $b=\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\mathrm{sin}}^{2}x}{\sqrt{2}-\sqrt{1+\mathrm{cos}x}}$, then the value of $a{b}^{3}$ is :