JEE Main 2020 — Calculus

If y = x³ + x², then dy/dx at x = 1 is:

The area bounded by y = x², x-axis, and x = 2 is:

If $f(x)={\begin{matrix}\frac{sin(a+2)x+sinx}{x} & ;x<0 \\ b & ;x=0 \\ \frac{{(x+3{x}^{2})}^{1/3}-{x}^{1/3}}{{x}^{1/3}} & ;x>0\end{matrix}$ is continuous at $x=0$ , then $a+2b$ is equal to:

The value of $\alpha$ for which $4\alpha \int _{-1}^{2}{e}^{-\alpha |x|}dx=5$ , is

Let $f:(0,\infty )\rightarrow (0,\infty )$ be a differentiable function such that $f(1)=e$ and $\underset{t\rightarrow x}{\mathrm{lim}}\frac{{t}^{2}{f}^{2}(x)-{x}^{2}{f}^{2}(t)}{t-x}=0.$ If$f(x)=1$, then $x$ is equal to:

If $(a+\sqrt{2}b\mathrm{cos}x)(a-\sqrt{2}b\mathrm{cos}y)={a}^{2}-{b}^{2}$, where $a>b>0,$ then$\frac{dx}{dy}$ at $(\frac{\pi }{4},\frac{\pi }{4})$ is:

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of ${\int }_{1}^{2}|2x-[3x]|dx$ is

$\underset{x\rightarrow 0}{lim}\frac{{\int }_{0}^{x}tsin(10t)dt}{x}$, is equal to

Consider a region $R={(x,y)\in {R}^{2}:{x}^{2}\leq y\leq 2x}$. If a line $y=\alpha$ divides the area of region $R$ into two equal parts, then which of the following is true ?

If $\frac{dy}{dx}=\frac{xy}{{x}^{2}+{y}^{2}};y(1)=1;$ then a value of $x$ satisfying $y(x)=e$ is:

The function, $f(x)=(3x-7){x}^{\frac{2}{3}},x\in R$, is increasing for all $x$ lying in

$\underset{x\rightarrow 1}{\mathrm{lim}}(\frac{{\int }_{0}^{(x-1{)}^{2}}t\mathrm{cos}{t}^{2}dt}{(x-1)\mathrm{sin}(x-1)})$

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 $\alpha$ is the positive root of the equation, $p(x)={x}^{2}-x-2=0$, then $\underset{x\rightarrow {\alpha }^{+}}{\mathrm{lim}}\frac{\sqrt{1-\mathrm{cos}p(x)}}{x+\alpha -4}$is equal to

$\underset{x\rightarrow a}{\mathrm{lim}}\frac{{(a+2x)}^{\frac{1}{3}}-{(3x)}^{\frac{1}{3}}}{{(3a+x)}^{\frac{1}{3}}-{(4x)}^{\frac{1}{3}}}(a\neq 0)$ is equal to:

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 $\underset{x\rightarrow 1}{\mathrm{lim}}\frac{x+{x}^{2}+{x}^{3}+...+{x}^{n}-n}{x-1}=820,(n\in N)$ then the value of $n$ is equal to....

$\underset{x\rightarrow 0}{\mathrm{lim}}{(\frac{3{x}^{2}+2}{7{x}^{2}+2})}^{\frac{1}{{x}^{2}}}$ is equal to

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 $f$ be any function continuous on $[a,b]$ and twice differentiable on $(a,b)$ . If all $x\in (a,b),{f}^{'}(x)>0$ and ${f}^{''}(x)<0$ , then for any $c\in (a,b),\frac{f(c)-f(a)}{f(b)-f(c)}$

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 $f$

The solution curve of the differential equation, $(1+{e}^{-x})(1+{y}^{2})\frac{dy}{dx}={y}^{2}$ which passes through the point $(0,1)$, is

Let $y=y(x)$ be the solution curve of the differential equation, $({y}^{2}-x)\frac{dy}{dx}=1$ , satisfying $y(0)=1$ . This curve intersects the $X-$axis at a point whose abscissa is

The integral ${\int }_{0}^{2}||x-1|-x|dx$ is equal to

The area (in sq. units) of the region enclosed by the curves $y={x}^{2}-1$ and $y=1-{x}^{2}$ is equal to:

Let $f:R\rightarrow R$ be a function defined by $f(x)=\mathrm{max}{x,{x}^{2}}$.Let $S$ denote the set of all points in $R$,where $f$ is not differentiable.Then :

If the function $f(x)={\begin{matrix}{k}_{1}(x-\pi {)}^{2}-1, & x\leq \pi \\ {k}_{2}\mathrm{cos}x, & x>\pi \end{matrix}$ is twice differentiable, then the ordered pair $({k}_{1},{k}_{2})$ is equal to:

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}{x}^{5}\mathrm{sin}(\frac{1}{x})+5{x}^{2} & , & x<0 \\ 0 & , & x=0 \\ {x}^{5}\mathrm{cos}(\frac{1}{x})+\lambda {x}^{2} & , & x>0\end{matrix}$. The value of $\lambda$ for which ${f}^{"}(0)$ exists, is___.

Let $f(x)=x\cdot [\frac{x}{2}],$ for $-10<x<10,$ where $[t]$ denotes the greatest integer function. Then the number of points of discontinuity of $f(x)$ is equal to

Let $[t]$ denote the greatest integer $\leq t$ and $\underset{x\rightarrow 0}{\mathrm{lim}}⁡x[\frac{4}{x}]=A.$ Then the function, $f(x)=[{x}^{2}]\mathrm{sin}⁡(\pi x)$ is discontinuous, when $x$ is equal to:

If a function $f(x)$ defined by $f(x)={\begin{matrix}a{e}^{x}+b{e}^{-x}, & -1\leq x<1 \\ c{x}^{2}, & 1\leq x\leq 3 \\ a{x}^{2}+2cx, & 3<x\leq 4\end{matrix}$be continuous for some $a,b,c\in R$ and ${f}^{'}(0)+{f}^{'}(2)=e$, then the value of $a$ is

Let $S$, be the set of all functions $f:[0,1]\rightarrow R$, which are continuous on $[0,1]$, and differentiable on $(0,1)$. Then for every $f$ in $S$, there exists $c\in (0,1)$, depending on $f$, such that.

If the function $f$ defined on $(-\frac{1}{3},1/3)$ by $f(x)={\begin{matrix}\frac{1}{x}{log}_{e}(\frac{1+3x}{1-2x}), & \text{when }x\neq 0 \\ k & ,\text{when }x=0\end{matrix}$, is continuous, then $k$ is equal to.

$\underset{x\rightarrow 2}{lim}\frac{{3}^{x}+{3}^{3-x}-12}{{3}^{-\frac{x}{2}}-{3}^{1-x}}$ is equal to

If ${y}^{2}+{\mathrm{log}}_{e}({\mathrm{cos}}^{2}x)=y,x\in (-\frac{\pi }{2},\frac{\pi }{2})$ then :

The derivative of ${\mathrm{tan}}^{-1}(\frac{\sqrt{1+{x}^{2}}-1}{x})$ with respect to ${\mathrm{tan}}^{-1}(\frac{2x\sqrt{1-{x}^{2}}}{1-2{x}^{2}})$ at $x=\frac{1}{2}$ is :

Let $f$ and $g$ be differentiable functions on $R$ such that $fog$ is the identity function. If for some $a,b\in R,{g}^{'}(a)=5$ and $g(a)=b,$ then ${f}^{'}(b)$ is equal to:

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:

If $y(\alpha )=\sqrt{2(\frac{tan\alpha +cot\alpha }{1+ta{n}^{2}\alpha })+\frac{1}{si{n}^{2}\alpha }},\alpha \in (\frac{3\pi }{4},\pi )$, then $\frac{dy}{d\alpha }$ at $\alpha =\frac{5\pi }{6}$ is

Let ${x}^{k}+{y}^{k}={a}^{k},(a,k>0)$ and $\frac{dy}{dx}+{(\frac{y}{x})}^{\frac{1}{3}}=0,$ then $k$ is

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 :

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 $x=1$ is a critical point of the function $f(x)=(3{x}^{2}+ax-2-a){e}^{x},$ then

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 the surface area of a cube is increasing at a rate of $3.6c{m}^{2}/sec$, retaining its shape; then the rate of change of its volume (in $c{m}^{3}/sec$), when the length of a side of the cube is $10cm$, is:

Suppose $f(x)$ is a polynomial of degree four having critical points at $-1,0,1$. If $T={x\in R|f(x)=f(0)}$, then the sum of squares of all the elements of $T$ is :

The position of a moving car at time $t$ is given by $f(t)=a{t}^{2}+bt+c,t>0$, where $a,b\text{and}c$ are real numbers greater than $1$. Then the average speed of the car over the time interval $[{t}_{1},{t}_{2}]$ is attained at the point:

If $p(x)$ be a polynomial of degree three that has a local maximum value $8$ at $x=1$ and a local minimum value $4$ at $x=2$ then $p(0)$ is equal to

A spherical iron ball of $10cm$ radius is coated with a layer of ice of uniform thickness that melts at a rate of $50{cm}^{3}/min$ . When the thickness of ice is $5cm$ , then the rate (in $cm/min$ .) at which of the thickness of ice decreases, is:

Let $f(x)$, be a polynomial of degree $3$, such that $f(-1)=10, f(1)=-6, f(x)$, has a critical point at $x=-1$ and $f'(x)$, has a critical point at $x=1.$ Then $f(x)$, has local minima at $x=$

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?

If$\int \frac{\mathrm{cos}\theta }{5+7\mathrm{sin}\theta -2{\mathrm{cos}}^{2}\theta }d\theta ={Alog}_{e}|B(\theta )|+C,$ where $C$ is a constant of integration, then $\frac{B(\theta )}{A}$ can be:

If ${I}_{1}={\int }_{0}^{1}{(1-{x}^{50})}^{100}dx$ and ${I}_{2}={\int }_{0}^{1}{(1-{x}^{50})}^{101}dx$ such that ${I}_{2}=\alpha {I}_{1}$ then $\alpha$ equals to :

If $\int ({e}^{2x}+2{e}^{x}-{e}^{-x}-1){e}^{({e}^{x}+{e}^{-x})}dx=g(x){e}^{({e}^{x}+{e}^{-x})}+c,$ where $c$ is a constant of integration, then $g(0)$ is

If $\int {\mathrm{sin}}^{-1}(\frac{\sqrt{x}}{1+x})dx=A(x){\mathrm{tan}}^{-1}(\sqrt{x})+B(x)+C$, where $C$ is a constant of integration, then the ordered pair $(A(x),B(x))$ can be :

If $\int \frac{d\theta }{{\mathrm{cos}}^{2}\theta (\mathrm{tan}2\theta +\mathrm{sec}2\theta )}=$ $\lambda \mathrm{tan}\theta +2{\mathrm{log}}_{e}|f(\theta )|+C$ where $C$ is a constant of integration, then the ordered pair $(\lambda ,f(\theta ))$ is equal to:

Let $f(x)=\int \frac{\sqrt{x}}{{(1+x)}^{2}}dx(x\geq 0)$. Then $f(3)-f(1)$ is equal to :

The integral ${\int }_{1}^{2}{e}^{x}.{x}^{x}(2+{\mathrm{log}}_{e}x)\mathrm{dx}$ equals :

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:

Let $f(x)=|x-2|$ and $g(x)=f(f(x)),x\in [0,4]$. Then ${\int }_{0}^{3}(g(x)-f(x))dx$ is equal to

Let ${x}$ and $[x]$ denote the fractional part of $x$ and the greatest integer $\leq x$ respectively of a real number $x$. if ${\int }_{0}^{n}{x}dx,{\int }_{0}^{n}[x]dx$ and $10({n}^{2}-n),(n\in N,n>1)$ are three consecutive terms of a G.P. then $n$ is equal to__

${\int }_{-\pi }^{\pi }|\pi -|x||dx$ is equal to

The value of ${\int }_{0}^{2\pi }\frac{x{\mathrm{sin}}^{8}⁡x}{{\mathrm{sin}}^{8}⁡x+{\mathrm{cos}}^{8}⁡x}dx$ is equal to:

If for all real triplets $(a,b,c),f(x)=a+bx+c{x}^{2};$ then $\int _{0}^{1}f(x)dx$ is equal to:

If $I=\int _{1}^{2}\frac{dx}{\sqrt{2{x}^{3}-9{x}^{2}+12x+4}}$, then

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 region $A={(x,y):(x-1)[x]\leq y\leq 2\sqrt{x},0\leq x\leq 2},$ where $[t]$ denotes the greatest integer function, is :

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

The area (in sq. units) of the region ${(x,y)\in {R}^{2}|4{x}^{2}\leq y\leq 8x+12}$ is

Given: $f(x)={\begin{matrix}\begin{matrix}x,0\leq x<\frac{1}{2} \\ \frac{1}{2},x=\frac{1}{2}\end{matrix} \\ 1-x,\frac{1}{2}<x\leq 1\end{matrix}$ and $g(x)={(x-\frac{1}{2})}^{2},x\in R.$ Then, the area (in sq. units) of the region bounded by the curves, $y=f(x)$ and $y=g(x)$ between the lines $2x=1$ and $2x=\sqrt{3},$ is:

The area of the region (in sq. units), enclosed by the circle ${x}^{2}+{y}^{2}=2$ which is not common to the region bounded by the parabola ${y}^{2}=x$ and the straight line $y=x$, is

The solution of the differential equation $\frac{dy}{dx}-\frac{y+3x}{{\mathrm{log}}_{e}(y+3x)}+3=0$ is (where $C$ is a constant of integration)

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 $y=y(x)$ be the solution of the differential equation, $x{y}^{'}-y={x}^{2}(x\mathrm{cos}x+\mathrm{sin}x)$,$x>0$. If $y(\pi )=\pi ,$ then $y''(\frac{\pi }{2})+y(\frac{\pi }{2})$ 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 ${x}^{3}dy+xy\cdot dx={x}^{2}dy+2ydx;y(2)=e$ and $x>1$, then $y(4)$ is equal to :

Let $y=y(x)$ be a solution of the differential equation, $\sqrt{1-{x}^{2}}\frac{dy}{dx}+\sqrt{1-{y}^{2}}=0,|x|<1.$ If $y(\frac{1}{2})=\frac{\sqrt{3}}{2},$ then $y(\frac{-1}{\sqrt{2}})$ 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 for $x\geq 0,y=y(x)$ is the solution of the differential equation, $(x+1)dy=({(x+1)}^{2}+y-3)dx,y(2)=0$ then $y(3)$ is equal to ________

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 $A={(x,y):|x|+|y|\leq 1,2{y}^{2}\geq |x|}$

If ${\theta }_{1}$ and ${\theta }_{2}$ be respectively the smallest and the largest values of $\theta$ in $(0,2\pi )-{\pi }$ which satisfy the equation, $2{\mathrm{cot}}^{2}⁡\theta -\frac{5}{\mathrm{sin}⁡\theta }+4=0$ , then $\int _{{\theta }_{1}}^{{\theta }_{2}}{\mathrm{cos}}^{2}⁡3\theta d\theta$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $\mathrm{cos}x\frac{dy}{dx}+2y\mathrm{sin}x=\mathrm{sin}2x,x\in (0,\frac{\pi }{2})$ If $y(\pi /3)=0,$ then $y(\pi /4)$ is equal to :

The integral $\int \frac{dx}{{(x+4)}^{\frac{8}{7}}{(x-3)}^{\frac{6}{7}}}$ is equal to: (where $C$ is a constant of integration)

Let $y=y(x)$ be the solution of the differential equation, $\frac{2+\mathrm{sin}x}{y+1}.\frac{dy}{dx}=-\mathrm{cos}x$, $y>0,y(0)=1$. If $y(\pi )=a$ and $\frac{dy}{dx}$ at $x=\pi$ is $b$, then the ordered pair $(a,b)$ is equal to

Area (in sq. units) of the region outside $\frac{|x|}{2}+\frac{|y|}{3}=1$ and inside the ellipse $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$ is

Let $y=y(x)$ be a function of $x$ satisfying $y\sqrt{1-{x}^{2}}=k-x\sqrt{1-{y}^{2}}$ where $k$ is a constant and $y(\frac{1}{2})=-\frac{1}{4}$.Then $\frac{dy}{dx}$ at $x=\frac{1}{2}$ , is equal to

The general solution of the differential equation $\sqrt{1+{x}^{2}+{y}^{2}+{x}^{2}{y}^{2}}+xy\frac{dy}{dx}=0$ (where C is a constant of integration)

The function $f(x)={\begin{matrix}\frac{\pi }{4}+{\mathrm{tan}}^{-1}x, & |x|\leq 1 \\ \frac{1}{2}(|x|-1), & |x|>1\end{matrix}$ is :

If ${f}^{'}(x)={\mathrm{tan}}^{-1}⁡(\mathrm{sec}⁡x+\mathrm{tan}⁡x),-\frac{\pi }{2}<x<\frac{\pi }{2}$ and $f(0)=0$ , then $f(1)$ is equal to:

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{x({e}^{(\sqrt{1+{x}^{2}+{x}^{4}}-1)/x}-1)}{\sqrt{1+{x}^{2}+{x}^{4}}-1}$

The integral $\int {(\frac{x}{x\mathrm{sin}x+\mathrm{cos}x})}^{2}dx$ is equal to, (where $C$ is a constant of integration):

The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{1}{1+{e}^{\mathrm{sin}x}}dx$ is :

$\underset{x\rightarrow 0}{\mathrm{lim}}{(\mathrm{tan}(\frac{\pi }{4}+x))}^{1/x}$ is equal to

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:

Let $f(x)=x{\mathrm{cos}}^{-1}⁡(-\mathrm{sin}⁡|x|),x\in [-\frac{\pi }{2},\frac{\pi }{2}],$ then which of the following is true?

The integral ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}{\mathrm{tan}}^{3}x\cdot {\mathrm{sin}}^{2}3x(2{\mathrm{sec}}^{2}x\cdot {\mathrm{sin}}^{2}3x+3\mathrm{tan}x\cdot \mathrm{sin}6x)dx$ is equal to: