JEE Main 2017 — Calculus

A tangent to the curve, $y=f(x)$ at $P(x, y)$ meets $x$-axis at $A$ and $y$-axis at $B$. If $AP:BP=1:3$ and $f(1)=1,$ then the curve also passes through the point

$\underset{x\rightarrow \frac{\pi }{2}}{\mathrm{lim}}\frac{\mathrm{cot}x-\mathrm{cos}x}{{(\pi -2x)}^{3}}$ equals

If $(2+\mathrm{sin}x)\frac{dy}{dx}+(y+1)\mathrm{cos}x=0$ and $y(0)=1$, then $y(\frac{\pi }{2})$ is equal to

If $2x={y}^{\frac{1}{5}}+{y}^{-\frac{1}{5}}$ and $({x}^{2}-1)\frac{{d}^{2}y}{d{x}^{2}}+\lambda x\frac{dy}{dx}+ky=0$, then $\lambda +k$ is equal to

If $f(\frac{3x-4}{3x+4})=x+2, x\neq -\frac{4}{3}$, and $\int f(x)dx=A\mathrm{log}|1-x|+Bx+C$ , then the ordered pair $(A, B)$ is equal to

If for $x\in (0,\frac{1}{4}),$ the derivative of ${\mathrm{tan}}^{-1}⁡(\frac{6x\sqrt{x}}{1-9{x}^{3}})$ is $\sqrt{x} \cdot g(x)$ , then $g(x)$ equals:

If $y={[x+\sqrt{{x}^{2}-1}]}^{15}+{[x-\sqrt{{x}^{2}-1}]}^{15}$, then $({x}^{2}-1)\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}$ is equal to

If $\int _{1}^{2}\frac{dx}{{({x}^{2}-2x+4)}^{\frac{3}{2}}}=\frac{k}{k+5}$, then $k$ is equal to

$\underset{x\rightarrow 3}{\mathrm{lim}}⁡\frac{\sqrt{3x}-3}{\sqrt{2x-4}- \sqrt{2}}$ is equal to

Let $f$ be a polynomial function such that $f(3x)={f}^{'}(x). {f}^{''}(x),$ for all $x\in R.$ Then :

Let, ${I}_{n}=\int {\mathrm{tan}}^{n}xdx(n>1)$ . If ${I}_{4}+{I}_{6}=a{\mathrm{tan}}^{5}x+b{x}^{5}+c$, then the ordered pair $(a,b)$, is equal to

The area (in sq. units) of the region ${(x, y):x\geq 0, x+y\leq 3, {x}^{2}\leq 4y and y\leq 1+\sqrt{x}}$ is

The area (in sq. units) of the smaller portion enclosed between the curves, ${x}^{2}+{y}^{2}=4$ and ${y}^{2}=3x$, is:

The curve satisfying the differential equation, $ydx-(x+3{y}^{2})dy=0$ and passing through the point $(1,1)$ also passes through the point

The function $f$ defined by $f(x)={x}^{3}-3{x}^{2}+5x+7$ is:

The integral $\int _{\frac{\pi }{12}}^{\frac{\pi }{4}}\frac{8\mathrm{cos}2x}{{(\mathrm{tan}x+\mathrm{cot}x)}^{3}}dx$ equals

The integral $\int \sqrt{1+2\mathrm{cot}x(\mathrm{cosec}x+\mathrm{cot}x)}dx, (0<x<\frac{\pi }{2})$is equal to

The integral $\int _{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{dx}{1+\mathrm{cos}x}$ is equal to

The value of $k$ which the function $f(x)= {\begin{matrix}{(\frac{4}{5})}^{\frac{\mathrm{tan}⁡4x}{\mathrm{tan}⁡5x}}, & 0<x<\frac{\pi }{2} \\ k+\frac{2}{5}, & x=\frac{\pi }{2}\end{matrix}$ is continuous at $x=\frac{\pi }{2},$ is

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is: