Calculus PYQ — Page 54

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

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

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

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 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

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 $x=2$.

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

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

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 $f^{\prime}(0)=0$

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

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

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

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 $\lim _{x \rightarrow x_0} f(x) \neq f\left(x_0.\right)$

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 $y(x)$ 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

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

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

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 $\mathrm{b}=\frac{-1}{4}$

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

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 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 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 $f(2)=f(5)$.