Calculus PYQ — Page 45

If $\int \frac{\sqrt{1-x^{2}}}{x^{4}} d x=A(\mathrm{x})\left(\sqrt{1-x^{2}}\right)^{m}+C,$ for a suitable chosen integer $\mathrm{m}$ and a function $\mathrm{A}(\mathrm{x})$, where $\mathrm{C}$ is a constant of integration, then $(\mathrm{A}(\mathrm{x}))^{\mathrm{m}}$ equals :

Let $y=y(x)$ be the solution of the differential equation, ${({x}^{2}+1)}^{2} \frac{dy}{dx}+2x({x}^{2}+1)y=1$ such that $y(0)=0.$ If $\sqrt{a} y(1)=\frac{\pi }{32},$ then the value of $a$ is

If ${e}^{y}+xy=e,$ the ordered pair $(\frac{dy}{dx},\frac{{d}^{2}y}{d{x}^{2}})$ at $x=0$ is equal to

The value of $\int _{0}^{2\pi }[\mathrm{sin}⁡2x(1+\mathrm{cos}⁡3x)]dx$ , where $[t]$ denotes the greatest integer function is

If ${S}_{1}$ and ${S}_{2}$ are respectively the sets of local minimum and local maximum points of the function, $f(x)=9{x}^{4}+12{x}^{3}-36{x}^{2}+25,x\in R,$ then

$\lim _{x \rightarrow 0} \frac{x \cot (4 x)}{\sin ^{2} x \cot ^{2}(2 x)}$ is equal to:

If $2y={({\mathrm{cot}}^{-1}(\frac{\sqrt{3}\mathrm{cos}x+\mathrm{sin}x}{\mathrm{cos}x-\sqrt{3}\mathrm{sin}x}))}^{2}\forall x\in (0,\frac{\pi }{2})$, then $\frac{dy}{dx}$is equal to

If $x=3 tant$ and $y=3sect,$ then the value of $\frac{{d}^{2}y}{d{x}^{2}}$ at $t=\frac{\pi }{4},$ is:

Let $S$ be the set of all points in $(-\pi ,\pi )$ at which the function, $f(x)=\mathrm{min}{\mathrm{sin}x,\mathrm{cos}x}$ is not differentiable. Then $S$ is a subset of which of the following?

If $\int _{0}^{\pi /3}\frac{\mathrm{tan}⁡\theta }{\sqrt{2k \mathrm{sec}⁡\theta }}d\theta =1-\frac{1}{\sqrt{2}}, (k>0)$ , then the value of $k$ is

If the area enclosed between the curves $y=k{x}^{2}$ and $x=k{y}^{2},(k>0),$ is $1sq.unit.$ Then $k$ is

Let $f(x)$ be a differentiable function such that ${f}^{'}(x)=7-\frac{3}{4}\frac{f(x)}{x}, (x>0)$ and $f(1)\neq 4.$ Then $\underset{x\rightarrow {0}^{+}}{\mathrm{lim}}⁡x f(\frac{1}{x})$

The integral $\int _{1}^{e}{{(\frac{x}{e})}^{2x}-{(\frac{e}{x})}^{x}}lo{g}_{e} x dx$ is equal to

Let, $n\geq 2$ be a natural number and $0<\theta <\frac{\pi }{2}.$ Then $\int \frac{{(si{n}^{n}\theta -sin\theta )}^{\frac{1}{n}}cos\theta }{si{n}^{n+1}\theta }d\theta ,$ is equal to

The solution of the differential equation $x\frac{dy}{dx}+2y={x}^{2},(x\neq 0)$ with $y(1)=1$, is

If $\int \frac{x+1}{\sqrt{2 x-1}} \mathrm{~d} x=f(x) \sqrt{2 x-1}+\mathrm{C},$ where $\mathrm{C}$ is a constant of integration, then $f(x)$ is equal to:

Let, $f:R\rightarrow R$ be a function such that $f(x)={x}^{3}+{x}^{2}f'(1)+xf''(2)+f'''(3),\forall x\in R.$ Then $f(2)$ equals

Let $y=y(x)$ be the solution of the differential equation, $\frac{dy}{dx}+y\mathrm{tan}x=2x+{x}^{2}\mathrm{tan}x,x\in (-\frac{\pi }{2},\frac{\pi }{2}),$ such that $y(0)= 1.$ Then

If $f(x)=\int \frac{(5{x}^{8}+7{x}^{6})}{{({x}^{2}+1+2{x}^{7})}^{2}}dx$, $(x\geq 0),$ and $f(0)=0,$ then the value of $f(1)$ is

If $f(x)=[x]-[\frac{x}{4}], x\in R,$ where $[x]$ denotes the greatest integer function, then:

The area (in sq. units) in the first quadrant bounded by the parabola, $y=x^{2}+1$, the tangent to it at the point (2,5) and the coordinate axes is :

If $\int \frac{dx}{{x}^{3}{(1+{x}^{6})}^{\frac{2}{3}} }=xf(x){(1+{x}^{6})}^{\frac{1}{3}}+C$, where $C$ is a constant of integration, then the function $f(x)$ is equal to

For each $t\in R,$ let $[t]$ be the greatest integer less than or equal to $t$. Then, $\underset{x\rightarrow {1}^{+}}{lim}\frac{(1-|x|+sin|1-x|)sin([1-x]\frac{\pi }{2})}{|1-x|[1-x]}$

$\int se{c}^{2}x\cdot {\mathrm{cot}}^{\frac{4}{3}}xdx$ is equal to