JEE Main 2014 — Calculus

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \mathrm{log}|x|+\beta {x}^{2}+x$, then

If $\frac{dy}{dx}+y\mathrm{tan}x=sin2x$ and $y(0)=1$, then $y(\pi )$ is equal to

If [ ] denotes the greatest integer function, then the integral $\int_0^\pi[\cos x d x$ is equal to:

If for a continuous function $\mathrm{f}(\mathrm{x})$, $\int_{-\pi}^t(f(x)+x d x)=\pi^2-t^2$, for all $t \geq-\pi$, then $\mathrm{f}\left(-\frac{\pi}{3}\right)$ is equal to:

If for $\mathrm{n} \geq 1, \mathrm{P}_{\mathrm{n}}=\int_1^{\mathrm{e}}\left(\log \mathrm{x}^{\mathrm{n}}\right) \mathrm{dx}$, then $\mathrm{P}_{10}-90 \mathrm{P}_8$ is equal to:

If $m$ is a non-zero number and $\int \frac{{x}^{5m-1}+2{x}^{4m-1}}{{({x}^{2m}+{x}^{m}+1)}^{3}}dx=f(x)+c,$ then $f(x)$ is equal to

If $f(x)$ is continuous and $f(\frac{9}{2})=\frac{2}{9},$ then $\underset{x\rightarrow 0}{\mathrm{lim}}f(\frac{1-\mathrm{cos}3x}{{x}^{2}})$ equals to

If non-zero real numbers $b$ and $c$ are such that $minf(x)>maxg(x)$, where $f(x)={x}^{2}+2bx+2{c}^{2}$ and $g(x)=-{x}^{2}-2cx+{b}^{2},(x\in R)$; then $|\frac{c}{b}|$ lies in the interval

If the function $f(x)={\begin{matrix}\frac{\sqrt{2+cosx}-1}{{(\pi -x)}^{2}}, & x\neq \pi \\ k, & x=\pi \end{matrix}$ is continuous at $x=\pi$, then $k$ equals

If the general solution of the differential equation $y^{\prime}=\frac{y}{x}+\Phi\left(\frac{x}{y}\right)$, for some function $\Phi$, is given by $y \ln |c x|=x$, where $\mathrm{c}$ is an arbitrary constant, then $\Phi(2)$ is equal to:

If the volume of a spherical ball is increasing at the rate of $4\pi \mathrm{cc}/\mathrm{sec}$ then the rate of increase of its radius $(\mathrm{in}\mathrm{cm}/\mathrm{sec}),$ when the volume is $288\pi \mathrm{cc}$ is

If $\lim _{x \rightarrow 2} \frac{\tan \left(x-2\left\{x^2+k+2 x-2 k\right\}\right.}{x^2-4 x+4}=5$, then $\mathrm{k}$ is equal to:

If $y={e}^{nx}$, then $\frac{{d}^{2}y}{d{x}^{2}}.\frac{{d}^{2}x}{d{y}^{2}}$ is equal to :

If $f(x)={(\frac{3}{5})}^{x}+{(\frac{4}{5})}^{x}-1,x\in R,$ then the equation $f(x)=0$ has :

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{\mathrm{sin}({\pi cos}^{2}x)}{{x}^{2}}$ is equal to

$\int \frac{{\mathrm{sin}}^{8}x-{\mathrm{cos}}^{8}x}{(1-2{\mathrm{sin}}^{2}x{\mathrm{cos}}^{2}x)}dx$ is equal to

Let $f$ and $g$ be two differentiable functions on $R$ such that $f^{\prime}(x)>0$ and $g^{\prime}(x) < 0$ for all $x \in R$. Then for all $\mathrm{x}$ :

Let $f(x)=x|x|, g(x)=\sin x$ and $h(x)=(g \circ f)(x)$. Then

Let $f:R\rightarrow R$ be a function such that $|f(x)|\leq {x}^{2},$ for all $x\in R.$ Then, at $x=0,$ $f$ is

Let $\mathrm{f}, \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be two functions defined by $f(x)= \begin{cases}x \sin \left(\frac{1}{x}\right) & , x \neq 0 \\ 0, & , x=0\end{cases}$ and $g(x)=x f(x)$ Statement I: $f$ is a continuous function at $\mathrm{x}=0$. Statement II: $g$ is a differentiable function at $x=0$.

Let $A={(x,y):{y}^{2}\leq 4x,y-2x\geq -4}$. The area of the region $A$ in square units is

Let, the function $F$ be defined as $F(x)={\int }_{1}^{x}\frac{{e}^{t}}{t}dt,x>0,$ then the value of the integral ${\int }_{1}^{x}\frac{{e}^{t}}{t+a}dt,$ where $a>0,$ is

Let the population of rabbits surviving at a time $t$ be governed by the differential equation $\frac{dp(t)}{dt}=\frac{1}{2}{p(t)-400}$. If $p(0)=100$, then $p(t)$ equals

$$ \text { The integral } \int \frac{\sin ^2 x \cos ^2 x}{\left(\sin ^3 x+\cos ^3 x\right)^2} d x \text { is equal to: } $$

The area (in sq. unit) of the region described by $A={(x,y):{x}^{2}+{y}^{2}\leq 1\mathrm{and}{y}^{2}\leq 1-x}$ is

The area of the region $($in square units$)$ above the $x$-axis bounded by the curve $y=\mathrm{tan}x,0\leq x\leq \frac{\pi }{2}$ and the tangent to the curve at $x=\frac{\pi }{4}$ is

The general solution of the differential equation, $\sin 2 x\left(\frac{d y}{d x}-\sqrt{\tan x}\right)-y=0$, is :

The integral $\int _{0}^{\pi }\sqrt{1+4 {\text{sin}}^{2}\frac{x}{2}-4 \text{sin}\frac{x}{2}}dx$ equals

The integral $\int _{0}^{\frac{1}{2}}\frac{\mathrm{ln}(1+2x)}{1+4{x}^{2}}dx$ equals

The integral $\int (1+x-\frac{1}{x}){e}^{x+\frac{1}{x}}dx$, is equal to

The integral $\int x \cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right) d x(x>0)$ is equal to:

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $=\sqrt{3}$ is: