Calculus PYQ — Page 28

$\underset{x\rightarrow \frac{\pi }{4}}{\mathrm{lim}}\frac{8\sqrt{2}-{(\mathrm{cos}x+\mathrm{sin}x)}^{7}}{\sqrt{2}-\sqrt{2}\mathrm{sin}2x}$ is equal to

The area of the smaller region enclosed by the curves ${y}^{2}=8x+4$ and ${x}^{2}+{y}^{2}+4\sqrt{3}x-4=0$ is equal to

Let $\frac{dy}{dx}=\frac{ax-by+a}{bx+cy+a}$, where $a,b,c$ are constants. represent a circle passing through the point $(2,5)$. Then the shortest distance of the point $(11,6)$ from this circle is

The integral $\frac{24}{\pi }{\int }_{0}^{\sqrt{2}}\frac{(2-{x}^{2})\mathrm{dx}}{(2+{x}^{2})\sqrt{4+{x}^{4}}}$ is equal to ______.

If $n(2n+1){\int }_{0}^{1}{(1-{x}^{n})}^{2n}dx=1177{\int }_{0}^{1}{(1-{x}^{n})}^{2n+1}dx$, then $n\in N$ is equal to _______.

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,

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

$\underset{x\rightarrow \frac{1}{\sqrt{2}}}{\mathrm{lim}}\frac{\mathrm{sin}({\mathrm{cos}}^{-1}x)-x}{1-\mathrm{tan}({\mathrm{cos}}^{-1}x)}$ is equal to

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

Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area ${A}_{1}$ is twice the area ${A}_{2}$. Then the normal to the curve perpendicular to the line $2x-12y=15$ does NOT pass through the point __ 

Let a function $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined as: $f(x)={\begin{matrix}{\int }_{0}^{x}(5-|t-3|)dt, & x>4 \\ {x}^{2}+bx, & x\leq 4\end{matrix}$ where $b\in \mathbb{R}$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

The integral ${\int }_{0}^{1}\frac{1}{{7}^{[\frac{1}{x}]}}dx$, where $[\cdot ]$ denotes the greatest integer function, is equal to

The area of the region bounded by ${y}^{2}=8x$ and ${y}^{2}=16(3-x)$ is equal to

The value of ${\mathrm{log}}_{e}2\frac{d}{\mathrm{dx}}({\mathrm{log}}_{\mathrm{cos}x}cosecx)$ at $x=\frac{\pi }{4}$ is

If the solution curve of the differential equation $(({\mathrm{tan}}^{-1}y)-x)dy=(1+{y}^{2})dx$ passes through the point $(1,0)$ then the abscissa of the point on the curve whose ordinate is $\mathrm{tan}(1)$ is

Let $f(x)$ be a polynomial function such that $f(x)+{f}^{'}(x)+{f}^{''}(x)={x}^{5}+64$. Then, the value of $\underset{x\rightarrow 1}{\mathrm{lim}}\frac{f(x)}{x-1}$ is equal to

Let $P$ and $Q$ be any points on the curves ${(x-1)}^{2}+{(y+1)}^{2}=1$ and $y={x}^{2}$, respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of $P$ in the interval

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

If $f(x)={\begin{matrix}x+a, & x\leq 0 \\ |x-4|, & x>0\end{matrix}$ and $g(x)={\begin{matrix}x+1, & x<0 \\ {(x-4)}^{2}+b, & x\geq 0\end{matrix}$ are continuous on $R$, then $(gof)(2)+(fog)(-2)$ is equal to:

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:

If $y=y(x),x\in (0,\frac{\pi }{2})$ be the solution curve of the differential equation $({\mathrm{sin}}^{2}2x)\frac{dy}{dx}+(8{\mathrm{sin}}^{2}2x+2\mathrm{sin}4x)y=$ $2{e}^{-4x}(2\mathrm{sin}2x+\mathrm{cos}2x),$ with $y(\frac{\pi }{4})={e}^{-\pi }$, then $y(\frac{\pi }{6})$ is equal to

If $\frac{dy}{dx}+\frac{{2}^{x-y}({2}^{y}-1)}{{2}^{x}-1}=0,x,y>0,y(1)=1$, then $y(2)$ is equal to

Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point $(x,y)$ on it is directly proportional to $(\frac{-y}{x})$. If the curve passes through the points $(1,2)$ and $(8,1)$, then $|y(\frac{1}{8})|$ is equal to

If ${\int }_{0}^{\sqrt{3}}\frac{15{x}^{3}}{\sqrt{1+{x}^{2}+\sqrt{{(1+{x}^{2})}^{3}}}}dx=\alpha \sqrt{2}+\beta \sqrt{3}$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to