JEE Main 2019 — Calculus

A curve amongst the family of curves represented by the differential equation, $({x}^{2}-{y}^{2}) dx+2xy dy=0$ which passes through $(1,1)$, is

A $2$m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate $25cm/sec$ , then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is $1$ m above the ground is:

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

A value of $\alpha$ such that $\int _{\alpha }^{\alpha +1}\frac{dx}{(x+\alpha )(x+\alpha +1)}=lo{g}_{e}(\frac{9}{8})$ is

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is ${tan}^{-1}(\frac{1}{2}).$ Water is poured into it at a constant rate of $5cubicm/min.$ Then the rate $($in $m/min),$ at which the level of water is rising at the instant when the depth of water in the tank is $10 m;$ is:

Consider the differential equation, ${y}^{2}dx+(x-\frac{1}{y})dy=0$. If value of $y$ is $1$ when $x=1$, then the value of $x$ for which $y=2,$is

$\underset{x\rightarrow 0}{lim} \frac{{sin}^{2}x}{\sqrt{2}-\sqrt{1+cosx}}$ equals

For each $x\in R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $\underset{x\rightarrow {0}^{-}}{lim}\frac{x([x]+|x|)\mathrm{sin}⁡[x]}{|x|}$ is equal to

For each $t\in R,$ let $[t]$ be the greatest integer less than or equal to $t$. Then, $\underset{x\rightarrow {1}^{+}}{lim}\frac{(1-|x|+sin|1-x|)sin([1-x]\frac{\pi }{2})}{|1-x|[1-x]}$

For $x>1$, if ${(2x)}^{2y}=4{e}^{2x-2y}$, then ${(1+{\mathrm{log}}_{e}⁡2x)}^{2} \frac{dy}{dx}$ is equal to

For, ${x}^{2}\neq n\pi +1,n\in N$ (the set of natural numbers), the integral $\int x\sqrt{\frac{2\mathrm{sin}({x}^{2}-1)-\mathrm{sin}2({x}^{2}-1)}{2\mathrm{sin}({x}^{2}-1)+\mathrm{sin}2({x}^{2}-1)}}dx$, is equal to (where $c$ is a constant of integration).

If a curve passes through the point $(1,-2)$ and has slope of the tangent at any point $(x, y)$ on it as $\frac{{x}^{2}-2y}{x}$, then the curve also passes through the point

If ${S}_{1}$ and ${S}_{2}$ are respectively the sets of local minimum and local maximum points of the function, $f(x)=9{x}^{4}+12{x}^{3}-36{x}^{2}+25,x\in R,$ then

If $\frac{dy}{dx}+\frac{3}{{\mathrm{cos}}^{2}x}y=\frac{1}{{\mathrm{cos}}^{2}x}$ , $x\in (-\frac{\pi }{3},\frac{\pi }{3}),$ and $y(\frac{\pi }{4})=\frac{4}{3},$ then $y(-\frac{\pi }{4})$ equals

If $cosx\frac{dy}{dx}-ysinx=6x$, $(0<x<\frac{\pi }{2})$ and $y(\frac{\pi }{3})=0,$ then $y(\frac{\pi }{6})$ is equal to

If $x=3 tant$ and $y=3sect,$ then the value of $\frac{{d}^{2}y}{d{x}^{2}}$ at $t=\frac{\pi }{4},$ is:

If $f(x)=\int \frac{(5{x}^{8}+7{x}^{6})}{{({x}^{2}+1+2{x}^{7})}^{2}}dx$, $(x\geq 0),$ and $f(0)=0,$ then the value of $f(1)$ is

If $f(x)=\frac{2-xcosx}{2+xcosx}$ and $g(x)={\mathrm{log}}_{e}⁡x,$ then the value of the integral $\int _{-\frac{\pi }{4}}^{\frac{\pi }{4}}g(f(x))dx$ is

If $\int \frac{\sqrt{1-x^{2}}}{x^{4}} d x=A(\mathrm{x})\left(\sqrt{1-x^{2}}\right)^{m}+C,$ for a suitable chosen integer $\mathrm{m}$ and a function $\mathrm{A}(\mathrm{x})$, where $\mathrm{C}$ is a constant of integration, then $(\mathrm{A}(\mathrm{x}))^{\mathrm{m}}$ equals :

If $f:R\rightarrow R$ is a differentiable function and $f(2)=6,$ then $\underset{x\rightarrow 2}{lim}{\int }_{6}^{f(x)}\frac{2tdt}{(x-2)}$ is:

If $f(x)$ is a non-zero polynomial of degree four, having local extreme points at $x= –1, 0, 1;$ then the set $S={x\in R :f(x)=f(0)}$ contains exactly

If $f(x)={\begin{matrix}\frac{sin(p+1)x+sinx}{x} & , & x<0 \\ q & , & x=0 \\ \frac{\sqrt{x+{x}^{2}}-\sqrt{x}}{{x}^{3/2}} & , & x>0\end{matrix}$ is continuous at $x=0$ , then the ordered pair $(p, q)$ is equal to:

If $m$ is the minimum value of $k$ for which the function $f(x)=x\sqrt{kx-{x}^{2}}$ is increasing in the interval $[0,3]$ and $M$ is the maximum value of $f$ in $[0,3]$ when $k=m,$ then the ordered pair $(m, M)$ is equal to:

If $y(x)$ is the solution of the differential equation $\frac{d y}{d x}+\left(\frac{2 x+1}{x}\right) y=e^{-2 x}, x>0,$ where $y(1)=\frac{1}{2} e^{-2},$ then:

If $y=y(x)$ is the solution of the differential equation, $x\frac{dy}{dx}+2y={x}^{2}$ satisfying $y(1)=1,$ then $y(\frac{1}{2})$ is equal to

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}=(tanx-y){sec}^{2}x$ , $x\in (-\frac{\pi }{2}, \frac{\pi }{2})$ , such that $y(0)=0$, then $y(-\frac{\pi }{4})$ is equal to:

If the area enclosed between the curves $y=k{x}^{2}$ and $x=k{y}^{2},(k>0),$ is $1sq.unit.$ Then $k$ is

If the area (in sq. units) bounded by the parabola ${y}^{2}=4\lambda x$ and the line $y=\lambda x, \lambda >0,$ is $\frac{1}{9}$, then $\lambda$ is equal to

If the area (in sq. units) of the region ${(x,y):{y}^{2}\leq 4x,x+y\leq 1,x\geq 0,y\geq 0}$ is $a\sqrt{2}+b,$ then $a-b$ is equal to

If the function $f$ defined on $(\frac{\pi }{6},\frac{\pi }{3})$ by $f(x)={\begin{matrix}\frac{\sqrt{2}cosx-1}{cotx-1},x\neq \frac{\pi }{4} \\ k, x=\frac{\pi }{4}\end{matrix}$ is continuous, then $k$ is equal to

If the function $f$ given by $f(x)={x}^{3}-3(a-2){x}^{2}+3ax+7$, for some $a\in R$ is increasing in $(0, 1]$ and decreasing in $[1, 5)$, then a root of the equation, $\frac{f(x)-14}{{(x-1)}^{2}}=0, (x\neq 1)$ is :

If the function $f(x)={\begin{matrix}a|\pi -x|+1, x\leq 5 \\ b|x-\pi |+3, x>5\end{matrix}$ is continuous at $x=5,$ then the value of $a-b$ is:

If ${e}^{y}+xy=e,$ the ordered pair $(\frac{dy}{dx},\frac{{d}^{2}y}{d{x}^{2}})$ at $x=0$ is equal to

If $\int {e}^{secx}(secx\mathrm{tan}xf(x)+(secx\mathrm{tan}x+se{c}^{2}x))dx={e}^{secx}f(x)+C,$ then a possible choice of $f(x)$ is:

If $x \log _{e}\left(\log _{e} x\right)-x^{2}+y^{2}=4(y>0),$ then $\frac{d y}{d x}$ at $x=e$ is equal to :

If $\underset{x\rightarrow 1}{\mathrm{lim}}\frac{{x}^{2}-ax+b}{x-1}=5,$ then $a+b$ is equal to:

If $\int _{0}^{\frac{\pi }{2}}\frac{cotx}{cotx+\mathrm{cosec}x}dx=m(\pi +n),$ then $mn$ is equal to

If $2y={({\mathrm{cot}}^{-1}(\frac{\sqrt{3}\mathrm{cos}x+\mathrm{sin}x}{\mathrm{cos}x-\sqrt{3}\mathrm{sin}x}))}^{2}\forall x\in (0,\frac{\pi }{2})$, then $\frac{dy}{dx}$is equal to

If $\int _{0}^{x}f(t)dt={x}^{2}+\int _{x}^{1}{t}^{2}f(t)dt,$ then ${f}^{'}(\frac{1}{2})$ is

If $\underset{x\rightarrow 1}{lim}\frac{{x}^{4}-1}{x-1}=\underset{x\rightarrow k}{lim}\frac{{x}^{3}-{k}^{3}}{{x}^{2}-{k}^{2}}$ , then $k$ is

If $f(1)=1,{f}^{'}(1)=3$, then the derivative of $f(f(f(x)))+{(f(x))}^{2}$ at $x=1$ is:

If $\int _{0}^{\pi /3}\frac{\mathrm{tan}⁡\theta }{\sqrt{2k \mathrm{sec}⁡\theta }}d\theta =1-\frac{1}{\sqrt{2}}, (k>0)$ , then the value of $k$ is

If $\int \frac{dx}{{({x}^{2}-2x+10)}^{2}}=A({\mathrm{tan}}^{-1}(\frac{x-1}{3})+\frac{f(x)}{{x}^{2}-2x+10})+C$, then (where $C$ is a constant of integration)

If $f(x)=[x]-[\frac{x}{4}], x\in R,$ where $[x]$ denotes the greatest integer function, then:

If $\int {x}^{5}{e}^{-{x}^{2}}dx=g(x){e}^{-{x}^{2}}+c$, where $c$ is a constant of integration, then $g(-1)$ is equal to

If $\int {x}^{5}{e}^{-4{x}^{3}}dx=\frac{1}{48}{e}^{-4{x}^{3}}f(x)+C$, where $C$ is a constant of integration, then $f(x)$ is equal to

If $\int \frac{x+1}{\sqrt{2 x-1}} \mathrm{~d} x=f(x) \sqrt{2 x-1}+\mathrm{C},$ where $\mathrm{C}$ is a constant of integration, then $f(x)$ is equal to:

If $\int \frac{dx}{{x}^{3}{(1+{x}^{6})}^{\frac{2}{3}} }=xf(x){(1+{x}^{6})}^{\frac{1}{3}}+C$, where $C$ is a constant of integration, then the function $f(x)$ is equal to

$\underset{x\rightarrow \frac{\pi }{4}}{lim}\frac{co{t}^{3}x-tanx}{cos(x+\frac{\pi }{4})}$ is

$\int \frac{sin\frac{5x}{2}}{sin\frac{x}{2}}dx$, is equal to

$\underset{x\rightarrow {1}^{-}}{\mathrm{lim}}\frac{\sqrt{\pi }-\sqrt{2{\mathrm{sin}}^{-1}x}}{\sqrt{1-x}}$ is equal to

$\lim _{x \rightarrow 0} \frac{x \cot (4 x)}{\sin ^{2} x \cot ^{2}(2 x)}$ is equal to:

$\int se{c}^{2}x\cdot {\mathrm{cot}}^{\frac{4}{3}}xdx$ is equal to

$\underset{x\rightarrow 0}{lim}\frac{x+2sinx}{\sqrt{{x}^{2}+2sinx+1} - \sqrt{{sin}^{2}x-x+1}}$ is

Let $f$ and $g$ be continuous functions on $[0,a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$, then ${\int }_{0}^{a}f(x)g(x)dx$ is equal to

Let $f(x)=5-|x-2|$ and $g(x)=|x+1|,$ $x \in R.$ If $f(x)$ attains maximum value at $\alpha$ and $g(x)$ attains minimum value at $\beta ,$ then $\underset{x\rightarrow -\alpha \beta }{lim}\frac{(x-1)({x}^{2}-5x+6)}{{x}^{2}-6x+8}$ is equal to

Let $S(\alpha )={(x,y):{y}^{2}\leq x, 0\leq x\leq \alpha }$ and $A(\alpha )$ is area of the region $S(\alpha ).$ If for a $\lambda ,0<\lambda <4, A(\lambda ):A(4)=2:5,$then $\lambda$ equals:

Let $f(x)=\left\{\begin{array}{cl}-1, & -2 \leq x < 0 \\ x^{2}-1, & 0 \leq x \leq 2\end{array}\right.$ and $g(x)=|\eta(x)|+f(x \mid) .$ Then, in the interval $(-2,2), g$ is:

Let $f(x)={e}^{x}-x$ and $g(x)={x}^{2}-x, \forall x \epsilon R$ . Then the set of all $x \epsilon R$ , where the function $h(x)=(fog)(x)$ is increasing, is:

Let $f:R\rightarrow R$ be a continuous and differentiable function such that $f(2)=6$ and ${f}^{'}(2)=\frac{1}{48}$. If ${\int }_{6}^{f(x)}4{t}^{3}dt=(x-2)g(x),$ then $\underset{x\rightarrow 2}{lim}g(x)$ is equal to

Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x)-f(y)|\leq 2{|x-y|}^{3/2},$ for all $x,y\in R\text{.}$ If $f(0)=1$ then $\int _{0}^{1}{f}^{2}(x)dx$ is equal to

Let $f:R\rightarrow R$ be a differentiable function satisfying ${f}^{'}(3)+{f}^{'}(2)=0.$ Then $\underset{x\rightarrow 0}{lim}{(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)})}^{\frac{1}{x}}$ is equal to

Let $f(x)$ be a differentiable function such that ${f}^{'}(x)=7-\frac{3}{4}\frac{f(x)}{x}, (x>0)$ and $f(1)\neq 4.$ Then $\underset{x\rightarrow {0}^{+}}{\mathrm{lim}}⁡x f(\frac{1}{x})$

Let $f$ be a differentiable function such that $f(1)=2$ and ${f}^{'}(x)=f(x)$ for all $x\in R$. If $h(x)=f(f(x))$, then ${h}^{'}(1)$ is equal to :

Let $f:R\rightarrow R$ be a function defined as $f(x)={\begin{matrix} 5, if x\leq 1 \\ a+bx, if 1<x<3 \\ b+5x, if 3\leq x<5 \\ 30, if x\geq 5\end{matrix}$ Then $f$ is:

Let $f:(-1, 1)\rightarrow R$ be a function defined by $f(x)=max{-|x|, -\sqrt{1-{x}^{2}}}.$ If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly

Let, $f:R\rightarrow R$ be a function such that $f(x)={x}^{3}+{x}^{2}f'(1)+xf''(2)+f'''(3),\forall x\in R.$ Then $f(2)$ equals

Let, $n\geq 2$ be a natural number and $0<\theta <\frac{\pi }{2}.$ Then $\int \frac{{(si{n}^{n}\theta -sin\theta )}^{\frac{1}{n}}cos\theta }{si{n}^{n+1}\theta }d\theta ,$ is equal to

Let $f:[0, 2]\rightarrow R$ be a twice differentiable function such that ${f}^{''}(x)>0,$ for all $x\in [0, 2].$ If $\phi (x)= f(x)+ f(2–x),$ then $\phi$ is

Let $\alpha \in (0,\frac{\pi }{2})$, be constant.If the integral $\int \frac{tanx+tan\alpha }{tanx-\mathrm{tan}\alpha }dx=A(x)cos2\alpha +B(x)sin2\alpha +C$, where C is a constant of integration, then the functions $A(x)$ and $B(x)$ are respectively

Let $f:[-1,3]\rightarrow R$ be defined as $f(x)={\begin{matrix}|x|+[x], \\ x+|x|, \\ x+[x],\end{matrix}\begin{matrix}-1\leq x<1 \\ 1\leq x<2 \\ 2\leq x\leq 3,\end{matrix}$ Where $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is discontinuous at:

Let $f:R\rightarrow R$ be differentiable at $c\in R$ and $f(c)=0$. If $g(x)=|f(x)|$, then at $x=c, g$ is:

Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $\frac{x^{\mathrm{m}} y^{\mathrm{n}}}{\left(1+x^{2 \mathrm{~m}}\right)\left(1+y^{2 \mathrm{n}}\right)}$ is :

Let $S$ be the set of all points in $(-\pi ,\pi )$ at which the function, $f(x)=\mathrm{min}{\mathrm{sin}x,\mathrm{cos}x}$ is not differentiable. Then $S$ is a subset of which of the following?

Let $\mathrm{K}$ be the set of all real values of $x$ where the function $f(x)=\sin |x|-|x|+2(x-\pi) \cos |x|$ is not differentiable. Then the set $K$ is equal to :

Let $y=y(x)$ be the solution of the differential equation, $x\frac{dy}{dx}+y=x{\mathrm{log}}_{e}x, (x>1)$. If $2y(2)={\mathrm{log}}_{e}4-1$, then $y(e)$ is equal to

Let $y=y(x)$ be the solution of the differential equation, $\frac{dy}{dx}+y\mathrm{tan}x=2x+{x}^{2}\mathrm{tan}x,x\in (-\frac{\pi }{2},\frac{\pi }{2}),$ such that $y(0)= 1.$ Then

Let $y=y(x)$ be the solution of the differential equation, ${({x}^{2}+1)}^{2} \frac{dy}{dx}+2x({x}^{2}+1)y=1$ such that $y(0)=0.$ If $\sqrt{a} y(1)=\frac{\pi }{32},$ then the value of $a$ is

Let $[\mathrm{x}]$ denote the greatest integer less than or equal to $\mathrm{X}$. Then : $\lim _{x \rightarrow 0} \frac{\tan \left(\pi \sin ^{2} x\right)+(|\mathrm{x}|-\sin (x[x]))^{2}}{x^{2}}$

Let $I={\int }_{a}^{b}({x}^{4}-2{x}^{2})dx.$ If $I$ is minimum then the ordered pair $(a, b)$ is

Let $f(x)={\begin{matrix}max(|x|,{x}^{2}), & |x|\leq 2 \\ 8-2|x|, & 2<|x|\leq 4\end{matrix}.$ Let $S$ be the set of points in the interval $(-4,4)$ at which $f$ is not differentiable. Then $S$

Let $f(x)=15–|x –10|;x\in R.$ Then the set of all values of $x$, at which the function $g(x)=f(f(x))$ is not differentiable, is:

Let $f(x)=\int _{0}^{x}g(t)dt,$ where $g$ is a non-zero even function. If $f(x+5)=g(x),$ then $\int _{0}^{x}f(t)dt$ equals

Let $\sum _{k=1}^{10}f(a+k)=16({2}^{10}-1),$ where the function $f$ satisfies $f(x+y)=f(x)f(y)$ for all natural numbers $x, y$ and $f(1)=2.$ Then the natural number $'a'$ is:

Let $f(x)=\frac{x}{\sqrt{a^{2}+x^{2}}}-\frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}}, x \in \mathbb{R}$ wherea, b and d are non-zero real constants. Then :

The area (in sq. units) bounded by the parabola $y={x}^{2}-1,$ the tangent at the point $(2,3)$ to it and the $y$-axis is

The area (in sq. units) in the first quadrant bounded by the parabola, $y=x^{2}+1$, the tangent to it at the point (2,5) and the coordinate axes is :

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

The area (in sq. units) of the region $A={(x,y)\in R\times R|0\leq x\leq 3, 0\leq y\leq 4,y\leq {x}^{2}+3x}$ is

The area (in sq. units) of the region bounded by the curves $y={2}^{x}$ and $y=|x+1|$, in the first quadrant is

The area (in sq. units) of the region $A={(x,y):\frac{{y}^{2}}{2}\leq x\leq y+4}$ is:

The area (in sq. units) of the region bounded by the parabola, $y={x}^{2}+2$ and the lines, $y=x+1, x=0$ and $x=3$, is

The area (in sq. units) of the region bounded by the curve $x^{2}=4 y$ and the straight line $x=4 y-2$ is :

The area of the region $A={(x, y): 0\leq y\leq x|x|+1\mathrm{and}-1\leq x\leq 1}$ in sq. units, is

The derivative of ${tan}^{-1}(\frac{sinx-cosx}{sinx+cosx})$ with respect to $\frac{x}{2},$ where $x\in (0,\frac{\pi }{2})$, is

The general solution of the differential equation $({y}^{2}-{x}^{3})dx-xydy=0,(x\neq 0)$ is (where c is a constant of integration)

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius $3$ is:

The integral $\int_{\pi / 6}^{\pi / 4} \frac{\mathrm{d} x}{\sin 2 x\left(\tan ^{5} x+\cot ^{5} x\right)}$ equals:

The integral ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}{sec}^{\frac{2}{3}}x\cdot cose{c}^{\frac{4}{3}}xdx$ is equal to

The integral $\int \frac{2{x}^{3}-1}{{x}^{4}+x}dx$, is equal to

The integral $\int _{1}^{e}{{(\frac{x}{e})}^{2x}-{(\frac{e}{x})}^{x}}lo{g}_{e} x dx$ is equal to

The integral $\int cos(\mathrm{lnx})dx$, is equal to

The integral $\int \frac{3{x}^{13}+2{x}^{11}}{{(2{x}^{4}+3{x}^{2}+1)}^{4}}dx$, is equal to

The maximum area (in sq. units) of a rectangle having its base on the $x-$ axis and its other two vertices on the parabola, $y=12-{x}^{2}$ such that the rectangle lies inside the parabola, is :

The maximum volume $(in cu.m)$ of the right circular cone having slant height $3 m$ is:

The region represented by $|x-y|\leq 2$ and $|x+y|\leq 2$ is bounded by a

The solution of the differential equation, $\frac{\mathrm{d} y}{\mathrm{~d} x}=(x-y)^{2}$, when $y(1)=1,$ is:

The solution of the differential equation $x\frac{dy}{dx}+2y={x}^{2},(x\neq 0)$ with $y(1)=1$, is

The value of $\underset{y\rightarrow 0}{lim}\frac{\sqrt{1+\sqrt{1+{y}^{4}}}-\sqrt{2}}{{y}^{4}}$

The value of $\int _{0}^{\pi /2}\frac{{sin}^{3}x}{sinx+cosx}dx$ is:

The value of ${\int }_{0}^{\pi }{|\mathrm{cos}x|}^{3}dx$ is

The value of the integral ${\int }_{0}^{1}x{cot}^{-1}(1-{x}^{2}+{x}^{4})dx$ is

The value of the integral $\int_{-2}^{2} \frac{\sin ^{2} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}} d x$ (where $[x]$ denotes the greatest integer less than or equal to x) is

The value of $\int _{0}^{2\pi }[\mathrm{sin}⁡2x(1+\mathrm{cos}⁡3x)]dx$ , where $[t]$ denotes the greatest integer function is

The value of $\int _{-\pi /2}^{\pi /2}\frac{dx}{[x]+[\mathrm{sin}⁡x] + 4},$ where $[t]$ denotes the greatest integer less than or equal to $t,$ is

Themaximum value of the finction $f(x)=3 x^{3}-18 x^{2}+27 x-40$ on the set $\mathrm{S}=\left\{x \in R: x^{2}+30 \leq 11 x\right\}$ is :