Calculus PYQ — Page 23

Let $x(t)=2\sqrt{2}\mathrm{cos}t\sqrt{\mathrm{sin}2t}$ and $y(t)=2\sqrt{2}\mathrm{sin}t\sqrt{\mathrm{sin}2t},t\in (0,\frac{\pi }{2})$. Then $\frac{1+{(\frac{dy}{dx})}^{2}}{\frac{{d}^{2}y}{{\mathrm{dx}}^{2}}}$ at $t=\frac{\pi }{4}$ is equal to

∫ eˣ dx is equal to:

The maximum value of sin x + cos x is:

$\underset{x\rightarrow \frac{\pi }{2}}{\mathrm{lim}}({\mathrm{tan}}^{2}x({(2{\mathrm{sin}}^{2}x+3\mathrm{sin}x+4)}^{\frac{1}{2}}-{({\mathrm{sin}}^{2}x+6\mathrm{sin}x+2)}^{\frac{1}{2}}))$ is equal to

Let the slope of the tangent to a curve $y=f(x)$ at $(x,y)$ be given by $2\mathrm{tan}x(\mathrm{cos}x-y)$. if the curve passes through the point $(\frac{\pi }{4},0)$, then the value of ${\int }_{0}^{\frac{\pi }{2}}ydx$ is equal to

The odd natural number a, such that the area of the region bounded by $y=1,y=3,x=0,x={y}^{a}$ is $\frac{364}{3}$, equal to:

Let $f(x)={\begin{matrix}|4{x}^{2}-8x+5|, & \mathrm{if}8{x}^{2}-6x+1\geq 0 \\ [4{x}^{2}-8x+5], & \mathrm{if}8{x}^{2}-6x+1<0\end{matrix}$, where $[\alpha ]$ denotes the greatest integer less than or equal to $\alpha$. Then the number of points in $R$ where $f$ is not differentiable is _____ .

$f,g:R\rightarrow R$ be two real valued function defined as $f(x)={\begin{matrix}-|x+3| & , & x<0 \\ {e}^{x} & , & x\geq 0\end{matrix}$ and $g(x)={\begin{matrix}{x}^{2}+{k}_{1}x & , & x<0 \\ 4x+{k}_{2} & , & x\geq 0\end{matrix}$, where ${k}_{1}$ and ${k}_{2}$ are real constants. If $gof$ is differentiable at $x=0$, then $gof(-4)+$$gof(4)$ is equal to

If the solution curve of the differential equation $\frac{dy}{dx}=\frac{x+y-2}{x-y}$ passes through the point $(2,1)$ and $(k+1,2),k>0$, then

Let $f(x)={\begin{matrix}{x}^{3}-{x}^{2}+10x-7, & x\leq 1 \\ -2x+{\mathrm{log}}_{2}({b}^{2}-4), & x>1\end{matrix}$ Then the set of all values of $b$, for which $f(x)$ has maximum value at $x=1$, is:

Let $f(x)=2+|x|-|x-1|+|x+1|,x\in R$. Consider $(S1):{f}^{'}(-\frac{3}{2})+{f}^{'}(-\frac{1}{2})+{f}^{'}(\frac{1}{2})+{f}^{'}(\frac{3}{2})=2$ $(S2):{\int }_{-2}^{2}f(x)dx=12$ Then,

If the area of the region ${(x,y):{x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}\leq 1,x+y\geq 0,y\geq 0}$ is $A$, then $\frac{256A}{\pi }$ is

If the maximum value of $a$, for which the function ${f}_{a}(x)={\mathrm{tan}}^{-1}2x-3ax+7$ is non-decreasing in $(-\frac{\pi }{6},\frac{\pi }{6})$, is $\bar{a}$, then ${f}_{\bar{a}}(\frac{\pi }{8})$ is equal to

The area enclosed by the curves $y={\mathrm{log}}_{e}(x+{e}^{2}),x={\mathrm{log}}_{e}(\frac{2}{y})$ and $x={\mathrm{log}}_{e}2$, above the line $y=1$ is

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is $3$ units and after $5$ seconds, it becomes $7$ units, then its radius after $9$ seconds is

Let $y=y(x)$ be the solution of the differential equation $x(1-{x}^{2})\frac{dy}{dx}+(3{x}^{2}y-y-4{x}^{3})=0,x>1$ with $y(2)=-2$. Then $y(3)$ is equal to

Let $f(x)=[2{x}^{2}+1]$ and $g(x)={\begin{matrix}2x-3, & x<0 \\ 2x+3, & x\geq 0\end{matrix}$, where $[t]$ is the greatest integer $\leq t$. Then, in the open interval $(-1,1)$, the number of points where fog is discontinuous is equal to ______.

If $\underset{n\rightarrow \infty }{\mathrm{lim}}(\sqrt{{n}^{2}-n-1}+n\alpha +\beta )=0$ then $8(\alpha +\beta )$ is equal to

$\underset{x\rightarrow 0}{\mathrm{lim}}{(\frac{{(x+2\mathrm{cos}x)}^{3}+2{(x+2\mathrm{cos}x)}^{2}+3\mathrm{sin}(x+2\mathrm{cos}x)}{{(x+2)}^{3}+2{(x+2)}^{2}+3\mathrm{sin}(x+2)})}^{\frac{100}{x}}$ is equal to

The number of distinct real roots of ${x}^{4}-4x+1=0$ is

Let $f$ be a real valued continuous function on $[0,1]$ and $f(x)=x+{\int }_{0}^{1}(x-t)f(t)dt$. Then which of the following points $(x,y)$ lies on the curve $y=f(x)$?

Let $f(x)={\begin{matrix}\frac{\mathrm{sin}(x-[x])}{x-[x]}, & x\in (-2,-1) \\ \mathrm{max}(2x,3[|x|]), & |x|<1 \\ 1, & \mathrm{otherwise}\end{matrix}$ where $[t]$ denotes greatest integer $\leq t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, the ordered pair $(m,n)$ is:

The number of points where the function $f(x)={\begin{matrix}|2{x}^{2}-3x-7| & \mathrm{if}x\leqslant -1 \\ [4{x}^{2}-1] & \mathrm{if}-1<x<1 \\ |x+1|+|x-2| & \mathrm{if}x\geqslant 1\end{matrix}$, where $[t]$ denotes the greatest integer $\leqslant t$, is discontinuous is ______

${\int }_{0}^{20\pi }{(|\mathrm{sin}x|+|\mathrm{cos}x|)}^{2}dx$ is equal to: