Calculus PYQ — Page 21

The area of the region enclosed by the curve $f(x)=\mathrm{max}{\mathrm{sin}x,\mathrm{cos}x},-\pi \leq x\leq \pi$ and the $x-$axis is

Let $k$ and $m$ be positive real numbers such that the function $f(x)={\begin{matrix}3{x}^{2}+k\sqrt{x+1}, & 0<x<1 \\ m{x}^{2}+{k}^{2}, & x\geq 1\end{matrix}$ is differentiable for all $x>0$. Then $\frac{8{f}^{'}(8)}{{f}^{'}(\frac{1}{8})}$ is equal to

If the area bounded by the curve $2{y}^{2}=3x$, lines $x+y=3,y=0$ and outside the circle ${(x-3)}^{2}+{y}^{2}=2$ is A, then $4(\pi +4A)$ is equal to __________.

Let $\alpha x=\mathrm{exp}({x}^{\beta }{y}^{\gamma })$ be the solution of the differential equation $2{x}^{2}ydy-(1-x{y}^{2})dx=0$, $x>0,y(2)=\sqrt{{\mathrm{log}}_{e}2}$. Then $\alpha +\beta -\gamma$ equals :

$\underset{x\rightarrow \infty }{\mathrm{lim}}\frac{{(\sqrt{3x+1}+\sqrt{3x-1})}^{6}+{(\sqrt{3x+1}-\sqrt{3x-1})}^{6}}{{(x+\sqrt{{x}^{2}-1})}^{6}+{(x-\sqrt{{x}^{2}-1})}^{6}}{x}^{3}$

Let $A$ be the area bounded by the curve $y=x|x-3|$, the $x$-axis and the ordinates $x=-1$ and $x=2$. Then $12A$ is equal to _____ .

For $m,n>0$, let $\alpha (m,n)={\int }_{0}^{2}{t}^{m}{(1+3t)}^{n}dt$. If ,$11\alpha (10,6)+18\alpha (11,5)=p{(14)}^{6}$, then $p$ is equal to

Let $f$ be a differentiable function defined on $[0,\frac{\pi }{2}]$ such that $f(x)>0$ and $f(x)+{\int }_{0}^{x}f(t)\sqrt{1-{({\mathrm{log}}_{e}(f(t)))}^{2}}dt=e$ $\forall x\in [0,\frac{\pi }{2}]$, then ${{6{\mathrm{log}}_{e}(f(\frac{\pi }{6}))}}^{2}$ is equal to

Let $f(x)=\frac{x}{{(1+{x}^{n})}^{\frac{1}{n}}},x\in \mathbb{R}-{-1},n\in \mathbb{N},n>2$. If ${f}^{n}(x)=(fofof....$ upto $n$ times) $(x)$, then $\underset{n\rightarrow \infty }{\mathrm{lim}}{\int }_{0}^{1}{x}^{n-2}({f}^{n}(x))dx$ is equal to

Let the tangent at any point $P$ on a curve passing through the points $(1,1)$ and $(\frac{1}{10},100)$, intersect positive $x$-axis and $y$-axis at the points $A$ and $B$ respectively. If $PA:PB=1:k$ and $y=y(x)$ is the solution of the differential equation ${e}^{\frac{dy}{dx}}=kx+\frac{k}{2},y(0)=k$, then $4y(1)-5{\mathrm{log}}_{e}3$ is equal to _______________

Let $\alpha >0$. If ${\int }_{0}^{\alpha }\frac{x}{\sqrt{x+\alpha }-\sqrt{x}}dx=\frac{16+20\sqrt{2}}{15}$ then $\alpha$ is equal to :

Let $[t]$ denote the greatest integer $\leq t$. Then $\frac{2}{\pi }{\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}(8[\mathrm{cosec}x]-5[\mathrm{cot}x])dx$ is equal to $_______$

Let $y=f(x)$ be the solution of the differential equation $y(x+1)dx-{x}^{2}dy=0,y(1)=e$. Then $\underset{x\rightarrow {0}^{+}}{\mathrm{lim}}f(x)$ is equal to

The minimum value of the function $f(x)={\int }_{0}^{2}{e}^{|x-t|}dt$ is

If the solution curve $f(x,y)=0$ of the differential equation $(1+{\mathrm{log}}_{e}x)\frac{dx}{dy}-x{\mathrm{log}}_{e}x={e}^{y},x>0,$ passes through the points $(1,0)$ and $(a,2)$, then ${a}^{a}$ is equal to

The value of $\frac{{e}^{-\frac{\pi }{4}}+{\int }_{0}^{\frac{\pi }{4}}{e}^{-x}\mathrm{tan}{}^{50}xdx}{{\int }_{0}^{\frac{\pi }{4}}{e}^{-x}({\mathrm{tan}}^{49}x+{\mathrm{tan}}^{51}x)dx}$

Let $f$ and $g$ be twice differentiable functions on $R$ such that ${f}^{"}(x)={g}^{"}(x)+6x$ ${f}^{'}(1)=4{g}^{'}(1)-3=9$ $f(2)=3g(2)=12$ Then which of the following is NOT true ?

The absolute minimum value, of the function $f(x)=|{x}^{2}-x+1|+[{x}^{2}-x+1]$, where $[t]$ denotes the greatest integer function, in the interval $[-1,2]$, is

For $\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$, if $\int ({(\frac{x}{e})}^{2x}+{(\frac{e}{x})}^{2x}){\mathrm{log}}_{e}xdx=\frac{1}{\alpha }{(\frac{x}{e})}^{\beta x}-\frac{1}{\gamma }{(\frac{e}{x})}^{\delta x}+C$, where $e=\sum _{n=0}^{\infty }\frac{1}{n!}$ and $C$ is constant of integration, then $\alpha +2\beta +3\gamma -4\delta$ is equal to

The solution of the differential equation $\frac{dy}{dx}=-(\frac{{x}^{2}+3{y}^{2}}{3{x}^{2}+{y}^{2}}),y(1)=0$ is

Let $f$ and $g$ be two functions defined by $f(x)={\begin{matrix}x+1,x<0 \\ |x-1|,x\geq 0\end{matrix}$and $g(x)={\begin{matrix}x+1,x<0 \\ 1,x\geq 0\end{matrix}$.Then $(gof)(x)$ is

${\int }_{0}^{\infty }\frac{6}{{e}^{3x}+6{e}^{2x}+11{e}^{x}+6}dx=$

If $f(x)={x}^{2}+{g}^{'}(1)x+{g}^{"}(2)$ and $g(x)=f(1){x}^{2}+x{f}^{'}(x)+{f}^{"}(x)$, then the value of $f(4)-g(4)$ is equal to _____ .

If the area of the region $S={(x,y):2y-{y}^{2}\leq {x}^{2}\leq 2y,x\geq y}$ is equal to $\frac{n+2}{n+1}-\frac{\pi }{n-1},$ then the natural number $n$ is equal to $_______$