Calculus PYQ — Page 53

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

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

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

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:

If the integral $$ \int \frac{\cos 8 x+1}{\cot 2 x-\tan 2 x} d x=A \cos 8 x+k $$ where $k$ is an arbitrary constant, then $\mathrm{A}$ is equal to:

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

For $0 \leq x \leq \frac{\pi}{2}$, the value of $$ \int_0^{\sin ^2 x} \sin ^{-1}(\sqrt{t}) d t+\int_0^{\cos ^2 x} \cos ^{-1}(\sqrt{t}) d t \text { equals : } $$

For $a>0, t \in\left(0, \frac{\pi}{2}\right)$, let $x=\sqrt{a^{\sin ^{-1} t}}$ and $y=\sqrt{a^{\cos ^{-1} t}}$, Then, $1+\left(\frac{d y}{d x}\right)^2$ equals :

Consider the function : $f(x)=[x]+|1-x|,-1 \leq x \leq 3$ where $[x]$ is the greatest integer function. Statement 1: $f$ is not continuous at $x=0,1,2$ and 3 Statement 2:f(x)= $=\begin{array}{cc}-x, & -1 \leq x < 0 \\ 1-x, & 0 \leq x < 1 \\ 1+x, & 1 \leq x < 2 \\ 2+x, & 2 \leq x \leq 3\end{array}$

Consider the differential equation : $$ \frac{d y}{d x}=\frac{y^3}{2\left(x y^2-x^2\right)} $$ Statement-1: The substitution $z=y^2$ transforms the above equation into a first order homogenous differential equation. Statement-2: The solution of this differential equation is $y^2 e^{-y^2} / x=C$.

At present, a firm is manufacturing $2 00 0$ items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx}=100-12\sqrt{x}$. If the firm employs $2 5$ more workers, then the new level of production of items is

A spherical balloon is being inflated at the rate of $35 \mathrm{cc} / \mathrm{min}$. The rate of increase in the surface area (in $\mathrm{cm}^2 / \mathrm{min}$.) of the balloon when its diameter is $14 \mathrm{~cm}$, 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

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

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

The area of the region bounded by the curve $y=x^3$, and the lines, $y=8$, and $x=0$, 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 area bounded by the parabola $y^2=4 x$ and the line $2 x-3 y+4=0$, in square unit, is

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