Calculus PYQ — Page 17

The solution curve of the differential equation $y\frac{dx}{dy}=x({\mathrm{log}}_{e}x-{\mathrm{log}}_{e}y+1),x>0,y>0$ passing through the point $(e,1)$ is

Let $I(x)=\int \frac{{x}^{2}(x{\mathrm{sec}}^{2}+\mathrm{tan}x)}{(x\mathrm{tan}x+1{)}^{2}}dx$ If $I(0)=0,$ then $I(\frac{\pi }{4})$ is equal to

Let $y=y(x)$ be the solution of the differential equation $(3{y}^{2}-5{x}^{2})ydx+2x({x}^{2}-{y}^{2})dy=0$ such that $y(1)=1$. Then $|(y(2){)}^{3}-12y(2)|$ is equal to :

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

If $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function satisfying ${\int }_{0}^{\frac{\pi }{2}}f(\mathrm{sin}2x)\mathrm{sin}xdx+\alpha {\int }_{0}^{\frac{\pi }{4}}f(\mathrm{cos}2x)\mathrm{cos}xdx=0$, then the value of $\alpha$ is

Let $\alpha \in (0,1)$ and $\beta ={\mathrm{log}}_{e}(1-\alpha )$. Let ${P}_{n}(x)=x+\frac{{x}^{2}}{2}+\frac{{x}^{3}}{3}+\ldots .+\frac{{x}^{n}}{n},x\in (0,1)$. Then the integral ${\int }_{0}^{\alpha }\frac{{t}^{50}}{1-t}dt$ is equal to

Let for $x\in R$, $f(x)=\frac{x+|x|}{2}$ and $g(x)={\begin{matrix}x, \\ {x}^{2},\end{matrix}\begin{matrix}x<0 \\ x\geq 0\end{matrix}$. Then area bounded by the curve $y=(fog)(x)$ and the lines $y=0,2y-x=15$ is equal to _____ .

Let $y=p(x)$ be the parabola passing through the points $(–1,0),(0,1)$ and $(1,0)$. If the area of the region ${(x,y):{(x+1)}^{2}+{(y-1)}^{2}\leq 1,y\leq p(x)}$ is $A$, then $12(\pi -4A)$ is equal to ________ .

If ${\int }_{0}^{1}\frac{1}{(5+2x-2{x}^{2})(1+{e}^{(2-4x)})}dx=\frac{1}{\alpha }{\mathrm{log}}_{e}(\frac{\alpha +1}{\beta }),\alpha ,\beta >0$, then ${\alpha }^{4}-{\beta }^{4}$ is equal to

Let $x=2$ be a local minima of the function $f(x)=2{x}^{4}-18{x}^{2}+8x+12,x\in (-4,4)$. If $M$ is local maximum value of the function $f$ in $(-4,4)$, then $M=$

The sum of the abosolute maximum and minimum values of the function $f(x)=|{x}^{2}-5x+6|-3x+2$ in the interval $[-1,3]$ is equal to :

If the functions $f(x)=\frac{{x}^{3}}{3}+2bx+\frac{a{x}^{2}}{2}$ and $g(x)=\frac{{x}^{3}}{3}+ax+b{x}^{2},a\neq 2b$ have a common extreme point, then $a+2b+7$ is equal to

$\underset{t\rightarrow 0}{\mathrm{lim}}{({1}^{\frac{1}{{\mathrm{sin}}^{2}t}}+{2}^{\frac{1}{{\mathrm{sin}}^{2}t}}+{3}^{\frac{1}{{\mathrm{sin}}^{2}t}}......{n}^{\frac{1}{{\mathrm{sin}}^{2}t}})}^{{\mathrm{sin}}^{2}t}$ is equal to

If $2{x}^{y}+3{y}^{x}=20,$ then $\frac{dy}{dx}$ at $(2,2)$ is equal to:

Let the solution curve $x=x(y),0<y<\frac{\pi }{2}$, of the differential equation ${({\mathrm{log}}_{e}(\mathrm{cos}y))}^{2}\mathrm{cos}ydx-(1+3x{\mathrm{log}}_{e}(\mathrm{cos}y))\mathrm{sin}ydy=0$ satisfy $x(\frac{\pi }{3})=\frac{1}{2{\mathrm{log}}_{e}2}$. If $x(\frac{\pi }{6})=\frac{1}{{\mathrm{log}}_{e}m-{\mathrm{log}}_{e}n}$, where $m$ and $n$ are coprime, then $mn$ is equal to

Let $\Delta$ be the area of the region ${(x,y)\in {\mathbb{R}}^{2}:{x}^{2}+{y}^{2}\leq 21,{y}^{2}\leq 4x,x\geq 1}$. Then $\frac{1}{2}(\Delta -21{\mathrm{sin}}^{-1}\frac{2}{\sqrt{7}})$ is equal to

Let [x] denote the greatest integer $\leq x$. Consider the function $f(x)=\mathrm{max}{{x}^{2},1+[x]}$. Then the value of the integral ${\int }_{0}^{2}f(x)dx$ is :

$\underset{x\rightarrow 0}{\mathrm{lim}}((\frac{1-{\mathrm{cos}}^{2}(3x)}{{\mathrm{cos}}^{3}(4x)})(\frac{{\mathrm{sin}}^{3}(4x)}{{({\mathrm{log}}_{e}(2x+1))}^{5}}))$ is equal to

The derivative of sin(x²) with respect to x is:

The area of the region given by ${(x,y):xy\leq 8,1\leq y\leq {x}^{2}}$ is :

∫₀¹ x² dx is equal to:

If $\alpha >\beta >0$ are the roots of the equation $a{x}^{2}+bx+1=0$, and $\underset{x\rightarrow \frac{1}{\alpha }}{\mathrm{lim}}{(\frac{1-\mathrm{cos}({x}^{2}+bx+a)}{2(1-\alpha x{)}^{2}})}^{\frac{1}{2}}=\frac{1}{k}(\frac{1}{\beta }-\frac{1}{\alpha })$, then $k$ is equal to

If the total maximum value of the function $f(x)={(\frac{\sqrt{3e}}{2\mathrm{sin}x})}^{{\mathrm{sin}}^{2}x},x\in (0,\frac{\pi }{2}),$ is $\frac{k}{e},$ then ${(\frac{k}{e})}^{8}+\frac{{k}^{8}}{{e}^{5}}+{k}^{8}$ is equal to

The value of the integral ${\int }_{-{\mathrm{log}}_{e}2}^{{\mathrm{log}}_{e}2}{e}^{x}({\mathrm{log}}_{e}({e}^{x}+\sqrt{1+{e}^{2x}}))dx$ is equal to