Calculus PYQ — Page 53

If $\int \frac{x^2-x+1}{x^2+1} e^{\cot ^{-1} x} d x=A(x) e^{\cot ^{-1} x}+C$, then $A(x)$ is equal to :

If $\int f(x)dx=\psi (x)$, then $\int {x}^{5}f({x}^{3})dx$, is equal to

If $\int \frac{d x}{x+x^7}=p(x)$ then, $\int \frac{x^6}{x+x^7} d x$ is equal to:

Let $f(x)=\frac{x^2-x}{x^2+2 x} x \neq 0,-2$. Then $\frac{d}{d x}\left[f^{-1}(x)\right]$ (wherever it is defined) is equal to:

Let $f(x)=-1+|x-2|$, and $g(x)=1-|x|$; then the set of all points where $f_{o g}$ is discontinuous is :

The area of the region (in sq. units), in the first quadrant bounded by the parabola $y=9 x^2$ and the lines $x=0, y=1$ and $y=4$, is :

The maximum area of a right angled triangle with hypotenuse $h$ is :

If the surface area of a sphere of radius $r$ is increasing uniformly at the rate $8 \mathrm{~cm}^2 / \mathrm{s}$, then the rate of change of its volume is:

The area bounded by the curve $y=\ln (x)$ and the lines $y=0, y=\ln (3)$ and $x=0$ is equal to:

If $y=\mathrm{sec}({\mathrm{tan}}^{-1}x)$, then $\frac{dy}{dx}$ at $x=1$ is equal to

Statement-1: The equation $x \log x=2-x$ is satisfied by at least one value of $x$ lying between 1 and 2. Statement-2: The function $f(x)=x \log x$ is an increasing function in $[1,2]$ and $g(x)=2-x$ is a decreasing function in $[1,2]$ and the graphs represented by these functions intersect at a point in $[1,2]$

If $f(x)=\sin (\sin x)$ and $f^{\prime \prime}(x)+\tan x f^{\prime}(x)+g(x)$ $=0$, then $g(x)$ is :

The equation of the curve passing through the origin and satisfying the differential equation $\left(1+x^2\right) \frac{d y}{d x}+2 x y=4 x^2$ 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

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

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

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

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

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

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

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 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