Calculus PYQ — Page 34

The area (in sq. units) of the region bounded by the curves ${x}^{2}+2y-1=0,{y}^{2}+4x-4=0$ and ${y}^{2}-4x-4=0$ in the upper half plane is _________.

Let $M$ and $m$ respectively be the maximum and minimum values of the function $f(x)={\mathrm{tan}}^{-1}(\mathrm{sin}x+\mathrm{cos}x)$ in $[0,\frac{\pi }{2}]$. Then the value of $\mathrm{tan}(M-m)$ is equal to:

Let $T$ be the tangent to the ellipse $E:{x}^{2}+4{y}^{2}=5$ at the point $P(1,1)$. If the area of the region bounded by the tangent $T$, ellipse $E$, lines $x=1$ and $x=\sqrt{5}$ is $\alpha \sqrt{5}+\beta +\gamma {\mathrm{cos}}^{-1}(\frac{1}{\sqrt{5}})$, then $|\alpha +\beta +\gamma |$ is equal to______.

If the solution curve of the differential equation $(2x-10{y}^{3})dy+ydx=0$, passes through the points $(0,1)$ and $(2,\beta )$, then $\beta$ is a root of the equation?

Let us consider a curve, $y=f(x)$ passing through the point $(-2,2)$ and the slope of the tangent to the curve at any point $(x,f(x))$ is given by $f(x)+x{f}^{'}(x)={x}^{2}.$ Then

Let $y=y(x)$ be solution of the following differential equation ${e}^{y}\frac{dy}{dx}-2{e}^{y}\mathrm{sin}x+\mathrm{sin}x{\mathrm{cos}}^{2}x=0,y(\frac{\pi }{2})=0$. If $y(0)={\mathrm{log}}_{e}(\alpha +\beta {e}^{-2})$, then $4(\alpha +\beta )$ is equal to .

Let $y=y(x)$ be the solution of the differential equation ${cosec}^{2}xdy+2dx=(1+y\mathrm{cos}2x){cosec}^{2}xdx,$ with $y(\frac{\pi }{4})=0.$ Then, the value of ${(y(0)+1)}^{2}$ is equal to:

Let a curve $y=y(x)$ be given by the solution of the differential equation $\mathrm{cos}(\frac{1}{2}{\mathrm{cos}}^{-1}({e}^{-x}))dx=(\sqrt{{e}^{2x}-1})dy$. If it intersects $y$-axis at $y=-1$, and the intersection point of the curve with $x-$axis is $(\alpha ,0)$, then ${e}^{\alpha }$ is equal to

Let $y=y(x)$ be the solution of the differential equation $x\mathrm{tan}(\frac{y}{x})dy=(y\mathrm{tan}(\frac{y}{x})-x)dx$, $-1\leq x\leq 1,y(\frac{1}{2})=\frac{\pi }{6}.$ Then the area of the region bounded by the curves $x=0,x=\frac{1}{\sqrt{2}}$ and $y=y(x)$ in the upper half plane is:

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+(\mathrm{tan}x)y=\mathrm{sin}x,0\leq x\leq \frac{\pi }{3},$ with $y(0)=0,$ then $y(\frac{\pi }{4})$ is equal to

The difference between degree and order of a differential equation that represents the family of curves given by ${y}^{2}=a(x+\frac{\sqrt{a}}{2}),a>0$ is _______.

If the curve, $y=y(x)$ represented by the solution of the differential equation $(2x{y}^{2}-y)dx+xdy=0$, passes through the intersection of the lines, $2x-3y=1$ and $3x+2y=8$, then $|y(1)|$ is equal to ___ .

Let $f:R\rightarrow R$ satisfy the equation $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in R$ and $f(x)\neq 0$ for any $x\in R$. If the function $f$ is differentiable at $x=0$ and ${f}^{'}(0)=3$, then $\underset{h\rightarrow 0}{\mathrm{lim}}\frac{1}{h}(f(h)-1)$ is equal to ___ .

The value of ${\int }_{-\pi /2}^{\pi /2}\frac{{\mathrm{cos}}^{2}x}{1+{3}^{x}}dx$ is:

If $y=y(x)$ is the solution of the differential equation, $\frac{dy}{dx}+2y\mathrm{tan}x=\mathrm{sin}x,y(\frac{\pi }{3})=0$, then the maximum value of the function $y(x)$ over $R$ is equal to :

The area, enclosed by the curves $y=\mathrm{sin}x+\mathrm{cos}x$ and $y=|\mathrm{cos}x-\mathrm{sin}x|$ and the lines $x=0,x=\frac{\pi }{2},$ is :

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2(y+2\mathrm{sin}x-5)x-2\mathrm{cos}x$ such that $y(0)=7.$ Then $y(\pi )$ is equal to

Let $A=[{a}_{ij}]$ be a $3\times 3$ matrix, where ${a}_{ij}={\begin{matrix}1 & , & \mathrm{if}i=j \\ -x & , & \mathrm{if}|i-j|=1 \\ 2x+1 & , & \mathrm{otherwise}\end{matrix}$ Let a function $f:R\rightarrow R$ be defined as $f(x)=det(A)$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to:

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

Let $y=y(x)$ be the solution of the differential equation $(x-{x}^{3})dy=(y+y{x}^{2}-3{x}^{4})dx,x>2$ If $y(3)=3,$ then $y(4)$ is equal to:

The value of the integral $\int \frac{\mathrm{sin}\theta \cdot \mathrm{sin}2\theta ({\mathrm{sin}}^{6}\theta +{\mathrm{sin}}^{4}\theta +{\mathrm{sin}}^{2}\theta )\sqrt{2{\mathrm{sin}}^{4}\theta +3{\mathrm{sin}}^{2}\theta +6}}{1-\mathrm{cos}2\theta }d\theta$ is (where $c$ is a constant of integration)

If the value of the integral ${\int }_{0}^{5}\frac{x+[x]}{{e}^{x-[x]}}dx=\alpha {e}^{-1}+\beta ,$ where $\alpha ,\beta \in R,5\alpha +6\beta =0,$ and $[x]$ denotes the greatest integer less than or equal to $x;$ then the value of $(\alpha +\beta {)}^{2}$ is equal to :

The value of ${\int }_{\frac{-1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}}{({(\frac{x+1}{x-1})}^{2}+{(\frac{x-1}{x+1})}^{2}-2)}^{\frac{1}{2}}dx$ is:

The value of $\underset{x\rightarrow {0}^{+}}{\mathrm{lim}}\frac{{\mathrm{cos}}^{-1}(x-[x{]}^{2})\cdot {\mathrm{sin}}^{-1}(x-[x{]}^{2})}{x-{x}^{3}},$ where $[x]$ denotes the greatest integer $\leq x$ is: