JEE Main 2012 — Calculus

If $[x]$ is the greatest integer $\leq x$, then the value of the integral $\int_{-0.9}^{0.9}\left(\left[x^2\right]+\log \left(\frac{2-x}{2+x}\right)\right) d x$ is

If a metallic circular plate of radius $50 \mathrm{~cm}$ is heated so that its radius increases at the rate of $1 \mathrm{~mm}$ per hour, then the rate at which, the area of the plate increases (in $\mathrm{cm}^2 /$ hour) is

$\lim _{x \rightarrow 0}\left(\frac{x-\sin x}{x}\right) \sin \left(\frac{1}{x}\right)$

The parabola $y^2=x$ divides the circle $x^2+y^2=2$ into two parts whose areas are in the ratio

Let $a, b \in R$ be such that the function $f$ given by $f(x)=\ln |x|+b x^2+a x, x \neq 0$ has extreme values at $x=-1$ and $x=2$. Statement 1: $f$ has local maximum at $x=-1$ and at $x=2$. Statement 2: $\mathrm{a}=\frac{1}{2}$ and $\mathrm{b}=\frac{-1}{4}$

Let $y(x)$ be a solution of $\frac{(2+\sin x}{(1+y)} \frac{d y)}{d x}=\cos x$. If $y(0)=2$, then $y\left(\frac{\pi}{2}\right)$ equals

If $f^{\prime}(x)=\sin (\log x)$ and $y=f\left(\frac{2 x+3}{3-2 x}\right)$, then $\frac{d y}{d x}$ equals

If $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ is a function defined by $f(x)=[\mathrm{x}] \cos \left(\frac{2 x-1}{2}\right) \pi$, where $[\mathrm{x}]$ denotes the greatest integer function, then $f$ is

If a circular iron sheet of radius $30 \mathrm{~cm}$ is heated such that its area increases at the uniform rate of $6 \pi \mathrm{cm}^2 / \mathrm{hr}$, then the rate (in $\mathrm{mm} / \mathrm{hr}$ ) at which the radius of the circular sheet increases is

Statement 1: The degrees of the differential equations $\frac{d y}{d x}+y^2=x$ and $\frac{d^2 y}{d x^2}+y=\sin x$ are equal. Statement 2: The degree of a differential equation, when it is a polynomial equation in derivatives, is the highest positive integral power of the highest order derivative involved in the differential equation, otherwise degree is not defined.

Let $f(x)$ be an indefinite integral of $\cos ^3 x$. Statement 1: $f(x)$ is a periodic function of period $\pi$. Statement 2: $\cos ^3 x$ is a periodic function.

If $f(x)=a|\sin x|+b e^{|x|}+c|x|^3$, where $a, b, c \in R$, is differentiable at $x=0$, then

Let $f:[1,3] \rightarrow R$ be a function satisfying $\frac{x}{[x]} \leq f(x) \leq \sqrt{6-x}$, for all $x \neq 2$ and $f(2)=1$, where $R$ is the set of all real numbers and $[x]$ denotes the largest integer less than or equal to $x$. Statement 1: $\lim _{x \rightarrow 2^{-}} f(x)$ exists. Statement 2: $f$ is continuous at $x=2$.

Consider the function $f(x)=|x-2|+|x-5|, x \in R$. Statement $1$: $f^{\prime}(4)=0$ Statement $2$: $f$ is continuous in $[2,5]$, differentiable in $(2,5)$ and $f(2)=f(5)$.

The weight $W$ of a certain stock of fish is given by $W=n w$, where $n$ is the size of stock and $w$ is the average weight of a fish. If $n$ and $w$ change with time $t$ as $n=2 t^2+3$ and $w=t^2-t+2$, then the rate of change of $W$ with respect to $t$ at $t=1$ is

Consider a rectangle whose length is increasing at the uniform rate of $2 \mathrm{~m} / \mathrm{sec}$, breadth is decreasing at the uniform rate of $3 \mathrm{~m} / \mathrm{sec}$ and the area is decreasing at the uniform rate of $5 \mathrm{~m}^2 / \mathrm{sec}$. If after some time the breadth of the rectangle is $2 \mathrm{~m}$ then the length of the rectangle is

If $f(x)=x e^{x(1-x)}, x \in R$, then $f(x)$ is

A spherical balloon is filled with $4500 ~\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 ~\pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is

If $f(x)=\int\left(\frac{x^2+\sin ^2 x}{1+x^2}\right) \sec ^2 x d x$ and $f(0)=0$, then $f(1)$ equals

$f(x)=\int \frac{d x}{\sin ^6 x}$ is a polynomial of degree

If $\int_e^x t f(t) d t=\sin x-x \cos x-\frac{x^2}{2}$, for all $x \in R-\{0\}$, then the value of $f\left(\frac{\pi}{6}\right)$ is

The value of the integral $\int_0^{0.9}[x-2[x]] d x$, where [.] denotes the greatest integer function is

If $g(x)=\int_0^x \cos 4 t ~d t$, then $g(x+\pi)$ equals

The area of the region bounded by the curve $y=x^3$, and the lines, $y=8$, and $x=0$, is

The area bounded by the parabola $y^2=4 x$ and the line $2 x-3 y+4=0$, in square unit, is

If a straight line $y-x=2$ divides the region $x^2+y^2 \leq 4$ into two parts, then the ratio of the area of the smaller part to the area of the greater part is

The area bounded between the parabolas $x^2=\frac{y}{4}$ and $x^2=9 y$, and the straight line $y=2$ is

The integrating factor of the differential equation $\left(x^2-1 \frac{d y}{d x}+2\right) x y=x$ is

The population $\mathrm{p}(\mathrm{t})$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{\mathrm{dp}(\mathrm{t})}{\mathrm{dt}}=0.5 \mathrm{~p}(\mathrm{t})$ $-450$. If $p(0)=850$, then the time at which the population becomes zero is

The integral of $\frac{x^2-x}{x^3-x^2+x-1}$ w.r.t. $x$ is

The general solution of the differential equation $\frac{d y}{d x}+\frac{2}{x} y=x^2$ is

If $f(x)=3 x^{10}-7 x^8+5 x^6-21 x^3+3 x^2-7$, then $\lim _{\alpha \rightarrow 0} \frac{f(1-\alpha)-f(1)}{\alpha^3+3 \alpha}$ is

The area enclosed by the curves $y=x^2, y=x^3$, $x=0$ and $x=p$, where $p>1$, is $1 / 6$. The $\mathrm{p}$ equals

If $x+|y|=2 y$, then $y$ as a function of $x$, at $x=0$ is

If $\frac{d}{d x} G(x)=\frac{e^{\tan x}}{x}, x \in(0, \pi / 2)$, then $\int_{1 / 4}^{1 / 2} \frac{2}{x} \cdot e^{\tan \left(\pi x^2\right)} d x$ is equal to

If the integral $\int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|+k$, then $a$ is equal to

Let $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$ be defined by $f(x)=x^3+1$ Statement 1: The function fhas a local extremum at $x=0$ Statement 2: The function $f$ is continuous and differentiable on $(-\infty, \infty)$ and $f^{\prime}(0)=0$

Statement 1: A function $f: R \rightarrow R$ is continuous at $x_0$ if and only if $\lim _{x \rightarrow x_0} f(x)$ exists and $\lim _{x \rightarrow x_0} f(x)=f\left(x_0 \cdot\right)$ Statement 2: A function $f: R \rightarrow R$ is discontinuous at $x_0$ if and only if, $\lim _{x \rightarrow x_0} f(x)$ exists and $\lim _{x \rightarrow x_0} f(x) \neq f\left(x_0.\right)$

$\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2}$ equals