Calculus PYQ — Page 33

Let $a$ be a positive real number such that ${\int }_{0}^{a}{e}^{x-[x]}dx=10e-9$ where, $[x]$ is the greatest integer less than or equal to $x$. Then, $a$ is equal to:

Let $f:R\rightarrow R$ be defined as $f(x)={e}^{-x}\mathrm{sin}x.$ If $F:[0,1]\rightarrow R$ is a differentiable function such that $F(x)={\int }_{0}^{x}f(t)dt,$ then the value of ${\int }_{0}^{1}({F}^{'}(x)+f(x)){e}^{x}dx$ lies in the interval

If ${I}_{m,n}={\int }_{0}^{1}{x}^{m-1}{(1-x)}^{n-1}dx$, for $m,n\geqslant 1$, and ${\int }_{0}^{1}\frac{{x}^{m-1}+{x}^{n-1}}{{(1+x)}^{m+n}}dx=\alpha {I}_{m,n},\alpha \in R$, then $\alpha$ equals ________.

If the normal to the curve $y(x)={\int }_{0}^{x}(2{t}^{2}-15t+10)dt$ at a point $(a,b)$ is parallel to the line $x+3y=-5,a>1$, then the value of $|a+6b|$ is equal to ________.

The value of the integral ${\int }_{0}^{\pi }|\mathrm{sin}2x|dx$ is ________.

The area of the region $S={(x,y):3{x}^{2}\leq 4y\leq 6x+24}$ is______.

If the line $y=mx$ bisects the area enclosed by the lines $x=0,y=0,x=\frac{3}{2}$ and the curve $y=1+4x-{x}^{2},$ then $12m$ is equal to .

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 _________.

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 $y=y(x)$ 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 $y=y(x)$ 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

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 ___ .

If the area of the bounded region $R={(x,y):\mathrm{max}{0,{\mathrm{log}}_{e}x}\leq y\leq {2}^{x},\frac{1}{2}\leq x\leq 2}$ is, $\alpha {({\mathrm{log}}_{e}2)}^{-1}+\beta ({\mathrm{log}}_{e}2)+\gamma$ then the value of ${(\alpha +\beta -2\gamma )}^{2}$ 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 ___ .

$\underset{n\rightarrow \infty }{\mathrm{lim}}{(1+\frac{1+\frac{1}{2}+\ldots \ldots +\frac{1}{n}}{{n}^{2}})}^{n}$ is equal to

Let ${A}_{1}$ be the area of the region bounded by the curves $y=\mathrm{sin}x,y=\mathrm{cos}x$ and $y$-axis in the first quadrant. Also, let ${A}_{2}$ be the area of the region bounded by the curves $y=\mathrm{sin}x,y=\mathrm{cos}x$, $x$-axis and $x=\frac{\pi }{2}$ in the first quadrant. Then,

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 function $f(x)=|{x}^{2}-2x-3|\cdot {e}^{9{x}^{2}-12x+4}$ is not differentiable at exactly :

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:

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:

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 :