JEE Main Mathematics — Calculus previous year questions with solutions.
The function, $f(x)=(3x-7){x}^{\frac{2}{3}},x\in R$, is increasing for all $x$ lying in
The set of all real values $\lambda$ for which the function $f(x)=(1-{\mathrm{cos}}^{2}x).(\lambda +\mathrm{sin}x),x\epsilon (-\frac{\pi }{2},\frac{\pi }{2})$, has exactly one maxima and exactly one minima, is :
If the value of the integral ${\int }_{0}^{\frac{1}{2}}\frac{{x}^{2}}{{(1-{x}^{2})}^{\frac{3}{2}}}dx$ is $\frac{k}{6}$, then $k$ is equal to:
$\underset{x\rightarrow 1}{\mathrm{lim}}(\frac{{\int }_{0}^{(x-1{)}^{2}}t\mathrm{cos}{t}^{2}dt}{(x-1)\mathrm{sin}(x-1)})$
$\underset{x\rightarrow 2}{lim}\frac{{3}^{x}+{3}^{3-x}-12}{{3}^{-\frac{x}{2}}-{3}^{1-x}}$ is equal to
The area (in sq. units) of the region ${(x, y)\in {R}^{2}:{x}^{2}\leq y\leq 3-2x}$, is.
If $x=2\mathrm{sin}\theta -\mathrm{sin}2\theta$ and $y=2\mathrm{cos}\theta -\mathrm{cos}2\theta ,$$\theta \in [0,2\pi ],$ then $\frac{{d}^{2}y}{d{x}^{2}}$ at $\theta =\pi$ is:
Let $[t]$ denote the greatest integer $\leq t$. If $\lambda \epsilon R-{0,1},\underset{x\rightarrow 0}{\mathrm{lim}}|\frac{1-x+|x|}{\lambda -x+[x]}|=L$, then $L$ is equal to
Let $f:(-1,\infty )\rightarrow R$ be defined by $f(0)=1$ and $f(x)=\frac{1}{x}{\mathrm{log}}_{e}(1+x),x\neq 0$. Then the function $f$
If $f(a+b+1-x)=f(x),$ for all $x,$ where $a$ and $b$ are fixed positive real numbers, then $\frac{1}{a+b}\int _{a}^{b}x(f(x)+f(x+1))dx$ is equal to
The area (in sq. units) of the largest rectangle $ABCD$ whose vertices $A$ and $B$ lie on the $x$-axis and vertices $C$ and $D$ lie on the parabola, $y={x}^{2}-1$ below the $x$-axis, is :
If ${y}^{2}+{\mathrm{log}}_{e}({\mathrm{cos}}^{2}x)=y,x\in (-\frac{\pi }{2},\frac{\pi }{2})$ then :
The area (in sq. units) of the region ${(x,y)\in {R}^{2}|4{x}^{2}\leq y\leq 8x+12}$ is
Let a function $f:[0,5]\rightarrow R$ be continuous, $f(1)=3$ and $F$ be defined as: $F(x)={\int }_{1}^{x}{t}^{2}g(t)dt,$ where $g(t)={\int }_{1}^{t}f(u)du.$ Then for the function $F(x),$ the point $x=1$ is:
${\int }_{-\pi }^{\pi }|\pi -|x||dx$ is equal to
Let $f(x)$ be a polynomial of degree $5$ such that $x=\pm 1$ are its critical points. If $\underset{x\rightarrow 0}{lim}(2+\frac{f(x)}{{x}^{3}})=4,$ then which one of the following is not true?
Let $AD$ and $BC$ be two vertical poles at $A\text{and}B$ respectively on a horizontal ground. If $AD=8m$, $\mathrm{BC}=11m$, $\mathrm{AB}=10m;$ then the distance (in meters) of a point $M$ lying in between $\mathrm{AB}$ from the point $A$ such that ${\mathrm{MD}}^{2}+{\mathrm{MC}}^{2}$ is minimums, is__
If $\underset{x\rightarrow 0}{\mathrm{lim}}{\frac{1}{{x}^{8}}(1-\mathrm{cos}\frac{{x}^{2}}{2}-\mathrm{cos}\frac{{x}^{2}}{4}+\mathrm{cos}\frac{{x}^{2}}{2}\mathrm{cos}\frac{{x}^{2}}{4})}={2}^{-k}$ then the value of k is
If $\int \frac{\mathrm{cos}xdx}{{\mathrm{sin}}^{3}x{(1+{\mathrm{sin}}^{6}x)}^{\frac{2}{3}}}=f(x){(1+{\mathrm{sin}}^{6}x)}^{\frac{1}{\lambda }}+c$, where $c$ is a constant of integration, then $\lambda f(\frac{\pi }{3})$ is equal to
If ${x}^{3}dy+xy\cdot dx={x}^{2}dy+2ydx;y(2)=e$ and $x>1$, then $y(4)$ is equal to :
If a curve $y=f(x)$, passing through the point $(1,2)$, is the solution of the differential equation $2{x}^{2}dy=(2xy+{y}^{2})dx$, then $f(\frac{1}{2})$ is equal to
If $y=y(x)$ is the solution of the differential equation $\frac{5+{e}^{x}}{2+y}\cdot \frac{dy}{dx}+{e}^{x}=0$ satisfying $y(0)=1$ then value of $y({\mathrm{log}}_{e}13)$ is
Let $S$ be the set of points where the function $,f(x)=|2-|x-3|,x\in R,$ is not differentiable. Then $\underset{x\in S}{\sum }f(f(x))$ is equal to
The area (in sq. units) of the region $A={(x,y):|x|+|y|\leq 1,2{y}^{2}\geq |x|}$