JEE Main Mathematics — Calculus previous year questions with solutions.
The value of the integral ${\int }_{-2}^{2}\frac{|{x}^{3}+x|}{({e}^{x|x|}+1)}dx$ is equal to
If $f(\theta )=\mathrm{sin}\theta +{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(\mathrm{sin}\theta +t\mathrm{cos}\theta )\cdot f(t)dt$, then $|{\int }_{0}^{\frac{\pi }{2}}f(\theta )d\theta |$ is
If $f(x)={\begin{matrix}x+a, & x\leq 0 \\ |x-4|, & x>0\end{matrix}$ and $g(x)={\begin{matrix}x+1, & x<0 \\ {(x-4)}^{2}+b, & x\geq 0\end{matrix}$ are continuous on $R$, then $(gof)(2)+(fog)(-2)$ is equal to:
The area of the region given by $A={(x,y):{x}^{2}\leq y\leq \mathrm{min}{x+2,4-3x}}$ is
The area bounded by the curves $y=|{x}^{2}-1|$ and $y=1$ is
The area of the region $S={(x,y):{y}^{2}\leq 8x,y\geq \sqrt{2}x,x\geq 1}$ is
For real numbers $a,b(a>b>0)$, let Area ${(x,y):{x}^{2}+{y}^{2}\leq {a}^{2}$ and $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}\geq 1}=30\pi$ and Area ${(x,y):{x}^{2}+{y}^{2}\geq {b}^{2}$ and $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}\leq 1}=18\pi$ Then the value of $(a-b){}^{2}$ is equal to _____.
The area of the bounded region enclosed by the curve $y=3-|x-\frac{1}{2}|-|x+1|$ and the $x$-axis is
The area bounded by the curve $y=|{x}^{2}-9|$ and the line $y=3$ is
Let $f:[0,1]\rightarrow R$ be a twice differentiable function in $(0,1)$ such that $f(0)=3$ and $f(1)=5$. If the line $y=2x+3$ intersects the graph of $f$ at only two distinct points in $(0,1)$, then the least number of points $x\in (0,1)$, at which ${f}^{''}(x)=0$, is
Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+(\frac{2{x}^{2}+11x+13}{{x}^{3}+6{x}^{2}+11x+6})y=\frac{(x+3)}{x+1},x>-1$, which passes through the point $(0,1)$. Then $y(1)$ is equal to
Let $y={y}_{1}(x)$ and $y={y}_{2}(x)$ be two distinct solutions of the differential equation $\frac{dy}{dx}=x+y$, with ${y}_{1}(0)=0$ and ${y}_{2}(0)=1$ respectively. Then, the number of points of intersection of $y={y}_{1}(x)$ and $y={y}_{2}(x)$ is
If $\frac{dy}{dx}+2y\mathrm{tan}x=\mathrm{sin}x,0<x<\frac{\pi }{2}$ and $y(\frac{\pi }{3})=0$, then the maximum value of $y(x)$ is
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{4{y}^{3}+2y{x}^{2}}{3x{y}^{2}+{x}^{3}},y(1)=1$. If for some $n\in N,y(2)\in [n-1,n)$, then $n$ is equal to _______.
Let the solution curve of the differential equation $x\frac{dy}{dx}-y=\sqrt{{y}^{2}+16{x}^{2}},y(1)=3$ be $y=y(x)$. Then $y(2)$ is equal to
If $y=y(x)$ is the solution of the differential equation $(1+{e}^{2x})\frac{dy}{dx}+2(1+{y}^{2}){e}^{x}=0$ and $y(0)=0$, then $6(y'(0)+{(y({\mathrm{log}}_{c}\sqrt{3}))}^{2})$ is equal to:
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{\sqrt{2}y}{2{\mathrm{cos}}^{4}x-\mathrm{cos}2x}=x{e}^{{\mathrm{tan}}^{-1}(\sqrt{2}\mathrm{cot}2x)},0<x<\frac{\pi }{2}$ with $y(\frac{\pi }{4})=\frac{{\pi }^{2}}{32}$. If $y(\frac{\pi }{3})=\frac{{\pi }^{2}}{18}{e}^{-{\mathrm{tan}}^{-1}(\alpha )}$, then the value of $3{\alpha }^{2}$ is equal to ______.
Let $y=y(x)$ be the solution of the differential equation $(1-{x}^{2})dy=(xy+({x}^{3}+2)\sqrt{1-{x}^{2}})dx,-1<x<1$ and $y(0)=0$. If ${\int }_{-\frac{1}{2}}^{\frac{1}{2}}\sqrt{1-{x}^{2}}y(x)dx=k$ then ${k}^{-1}$ is equal to
Let the solution curve $y=y(x)$ of the differential equation $(4+{x}^{2})dy-2x({x}^{2}+3y+4)dx=0$ pass through the origin. Then $y(2)$ is equal to _____.
Let $y=y(x)$ be the solution of the differential equation $(x+1){y}^{'}-y={e}^{3x}{(x+1)}^{2}$, with $y(0)=\frac{1}{3}$. Then, the point $x=-\frac{4}{3}$ for the curve $y=y(x)$ is
Let ${a}_{n}={\int }_{-1}^{n}(1+\frac{x}{2}+\frac{{x}^{2}}{3}+\ldots +\frac{{x}^{n-1}}{n})dx$ for every $n\in N$. Then the sum of all the elements of the set ${n\in N:{a}_{n}\in (2,30)}$ is _________.
Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+\frac{1}{{x}^{2}-1}y={(\frac{x-1}{x+1})}^{\frac{1}{2}}$, $x>1$ passing through the point $(2,\sqrt{\frac{1}{3}})$. Then $\sqrt{7}y(8)$ is equal to
Let $f(x)=\mathrm{max}{|x+1|,|x+2|,\ldots ,|x+5|}$. Then ${\int }_{-6}^{0}f(x)dx$ is equal to ______.
Consider a cuboid of sides $2x,4x$ and $5x$ and a closed hemisphere of radius $r$. If the sum of their surface areas is constant $k$, then the ratio $x:r$, for which the sum of their volumes is maximum, is