JEE Main Mathematics — Calculus previous year questions with solutions.
If the functions $f(x)=\frac{{x}^{3}}{3}+2bx+\frac{a{x}^{2}}{2}$ and $g(x)=\frac{{x}^{3}}{3}+ax+b{x}^{2},a\neq 2b$ have a common extreme point, then $a+2b+7$ is equal to
$\underset{x\rightarrow \infty }{\mathrm{lim}}\frac{{(\sqrt{3x+1}+\sqrt{3x-1})}^{6}+{(\sqrt{3x+1}-\sqrt{3x-1})}^{6}}{{(x+\sqrt{{x}^{2}-1})}^{6}+{(x-\sqrt{{x}^{2}-1})}^{6}}{x}^{3}$
Let $A$ be the area bounded by the curve $y=x|x-3|$, the $x$-axis and the ordinates $x=-1$ and $x=2$. Then $12A$ is equal to _____ .
Let $f$ and $g$ be two functions defined by $f(x)={\begin{matrix}x+1,x<0 \\ |x-1|,x\geq 0\end{matrix}$and $g(x)={\begin{matrix}x+1,x<0 \\ 1,x\geq 0\end{matrix}$.Then $(gof)(x)$ is
Let $f(x)=\frac{x}{{(1+{x}^{n})}^{\frac{1}{n}}},x\in \mathbb{R}-{-1},n\in \mathbb{N},n>2$. If ${f}^{n}(x)=(fofof....$ upto $n$ times) $(x)$, then $\underset{n\rightarrow \infty }{\mathrm{lim}}{\int }_{0}^{1}{x}^{n-2}({f}^{n}(x))dx$ is equal to
Let the tangent at any point $P$ on a curve passing through the points $(1,1)$ and $(\frac{1}{10},100)$, intersect positive $x$-axis and $y$-axis at the points $A$ and $B$ respectively. If $PA:PB=1:k$ and $y=y(x)$ is the solution of the differential equation ${e}^{\frac{dy}{dx}}=kx+\frac{k}{2},y(0)=k$, then $4y(1)-5{\mathrm{log}}_{e}3$ is equal to _______________
Let $[t]$ denote the greatest integer $\leq t$. Then $\frac{2}{\pi }{\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}(8[\mathrm{cosec}x]-5[\mathrm{cot}x])dx$ is equal to $_______$
The minimum value of the function $f(x)={\int }_{0}^{2}{e}^{|x-t|}dt$ is
The value of $\frac{{e}^{-\frac{\pi }{4}}+{\int }_{0}^{\frac{\pi }{4}}{e}^{-x}\mathrm{tan}{}^{50}xdx}{{\int }_{0}^{\frac{\pi }{4}}{e}^{-x}({\mathrm{tan}}^{49}x+{\mathrm{tan}}^{51}x)dx}$
If ${\int }_{0}^{1}\frac{1}{(5+2x-2{x}^{2})(1+{e}^{(2-4x)})}dx=\frac{1}{\alpha }{\mathrm{log}}_{e}(\frac{\alpha +1}{\beta }),\alpha ,\beta >0$, then ${\alpha }^{4}-{\beta }^{4}$ is equal to
The absolute minimum value, of the function $f(x)=|{x}^{2}-x+1|+[{x}^{2}-x+1]$, where $[t]$ denotes the greatest integer function, in the interval $[-1,2]$, is
For $\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$, if $\int ({(\frac{x}{e})}^{2x}+{(\frac{e}{x})}^{2x}){\mathrm{log}}_{e}xdx=\frac{1}{\alpha }{(\frac{x}{e})}^{\beta x}-\frac{1}{\gamma }{(\frac{e}{x})}^{\delta x}+C$, where $e=\sum _{n=0}^{\infty }\frac{1}{n!}$ and $C$ is constant of integration, then $\alpha +2\beta +3\gamma -4\delta$ is equal to
${\int }_{0}^{\infty }\frac{6}{{e}^{3x}+6{e}^{2x}+11{e}^{x}+6}dx=$
Let $y=p(x)$ be the parabola passing through the points $(–1,0),(0,1)$ and $(1,0)$. If the area of the region ${(x,y):{(x+1)}^{2}+{(y-1)}^{2}\leq 1,y\leq p(x)}$ is $A$, then $12(\pi -4A)$ is equal to ________ .
If the area of the region $S={(x,y):2y-{y}^{2}\leq {x}^{2}\leq 2y,x\geq y}$ is equal to $\frac{n+2}{n+1}-\frac{\pi }{n-1},$ then the natural number $n$ is equal to $_______$
The value of the integral ${\int }_{1}^{2}(\frac{{t}^{4}+1}{{t}^{6}+1})dt$ is :
Area of the region ${(x,y):{x}^{2}+{(y-2)}^{2}\leq 4,{x}^{2}\geq 2y}$ is
A square piece of tin of side $30\mathrm{cm}$ is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in ${\mathrm{cm}}^{2}$) is equal to
Let $g(x)=f(x)+f(1-x)$ and ${f}^{"}(x)>0,x\in (0,1)$. If $g$ is decreasing in the interval $(0,\alpha )$ and increasing in the interval $(\alpha ,1)$, then ${\mathrm{tan}}^{-1}2\alpha +{\mathrm{tan}}^{-1}(\frac{1}{\alpha })+{\mathrm{tan}}^{-1}(\frac{\alpha +1}{\alpha })$ is equal to
Let $A={(x,y)\in {\mathbb{R}}^{2}:y\geq 0,2x\leq y\leq \sqrt{4-{(x-1)}^{2}}}$ and $B={(x,y)\in \mathbb{R}\times \mathbb{R}:0\leq y\leq \mathrm{min}{2x,\sqrt{4-{(x-1)}^{2}}}}$. Then the ratio of the area of $A$ to the area of $B$ is
Let $[x]$ be the greatest integer $\leq x$. Then the number of points in the interval $(–2,1)$ where the function $f(x)=|[x]|+\sqrt{x-[x]}$ is discontinuous, is _____.
Let $x=x(y)$ be the solution of the differential equation $2(y+2){\mathrm{log}}_{e}(y+2)dx+(x+4-2{\mathrm{log}}_{e}(y+2))dy=0$, $y>-1$ with $x({e}^{4}-2)=1$. Then $x({e}^{9}-2)$ is equal to
The set of all $a\in \mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is
Let $f(x)=\frac{\mathrm{sin}x+\mathrm{cos}-\sqrt{2}}{\mathrm{sin}x-\mathrm{cos}x},x\in [0,\pi ]-{\frac{\pi }{4}}$, then $f(\frac{7\pi }{12}){f}^{"}(\frac{7\pi }{12})$ is equal to