Calculus PYQ — Page 13

$|\frac{120}{{\pi }^{3}}\int _{0}^{\pi }\frac{{x}^{2}\mathrm{sin}x\mathrm{cos}x}{{\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x}dx|$ is equal to ______.

The value $9{\int }_{0}^{9}[\sqrt{\frac{10x}{x+1}}]\mathrm{dx}$, where $t$ denotes the greatest integer less than or equal to $t$, is _____.

Let $f(x)={\int }_{0}^{x}g(t){\mathrm{log}}_{e}(\frac{1-t}{1+t})\mathrm{dt}$, where $g$ is a continuous odd function. If ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(f(x)+\frac{{x}^{2}\mathrm{cosx}}{1+{e}^{x}})\mathrm{dx}={(\frac{\pi }{\alpha })}^{2}-\alpha$, then $\alpha$ is equal to _____.

If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and ${I}_{1}={\int }_{a}^{b}\mathrm{xsin}(4x-{x}^{2})\mathrm{dx},{I}_{2}={\int }_{a}^{b}\mathrm{sin}(4x-{x}^{2})\mathrm{dx}$ , then $36\frac{{I}_{1}}{{I}_{2}}$ is equal to :

Let $f(x)={(x+3)}^{2}{(x-2)}^{3},x\in [-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4,4]$, then the value of $M-m$ is :

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1,f(1))$ and $(3,f(3))$ make angles $\frac{\pi }{6}$ and $\frac{\pi }{4}$, respectively with positive $x$-axis. If $27{\int }_{1}^{3}({({f}^{'}(t))}^{2}+1){f}^{"}(t)dt=\alpha +\beta \sqrt{3}$ where $\alpha ,\beta$ are integers, then the value of $\alpha +\beta$ equals

The area (in square units) of the region enclosed by the ellipse $x^2+3 y^2=18$ in the first quadrant below the line $y=x$ is

Let the area of the region enclosed by the curve $y=\min \{\sin x, \cos x\}$ and the $x$ axis between $x=-\pi$ to $x=\pi$ be $A$. Then $A^2$ is equal to ___________

The area enclosed between the curves $y=x|x|$ and $y=x-|x|$ is :

The area of the region enclosed by the parabola $(y-2{)}^{2}=x-1$, the line $x-2y+4=0$ and the positive coordinate axes is __________.

The solution of the differential equation $\left(x^2+y^2\right) \mathrm{d} x-5 x y \mathrm{~d} y=0, y(1)=0$, is :

If the solution $y(x)$ of the given differential equation $\left(\mathrm{e}^y+1\right) \cos x \mathrm{~d} x+\mathrm{e}^y \sin x \mathrm{~d} y=0$ passes through the point $\left(\frac{\pi}{2}, 0\right)$, then the value of $\mathrm{e}^{y\left(\frac{\pi}{6}\right)}$ is equal to_________

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2x{(x+y)}^{3}-x(x+y)-1,y(0)=1$. Then, ${(\frac{1}{\sqrt{2}}+y(\frac{1}{\sqrt{2}}))}^{2}$ equals:

If the solution curve $y=y(x)$ of the differential equation $(1+{y}^{2})(1+{\mathrm{log}}_{e}x)dx+xdy=0,x>0$ passes through the point $(1,1)$ and $y(e)=\frac{\alpha -\mathrm{tan}(\frac{3}{2})}{\beta +\mathrm{tan}(\frac{3}{2})}$, then $\alpha +2\beta$ is

If the solution curve, of the differential equation $\frac{dy}{dx}=\frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is ${\mathrm{tan}}^{-1}(\frac{y-1}{x-1})-\frac{1}{\beta }{\mathrm{log}}_{e}(\alpha +{(\frac{y-1}{x-1})}^{2})={\mathrm{log}}_{e}|x-1|$, then $5\beta +\alpha$ is equal to

Let $y=y(x)$ be the solution of the differential equation $\mathrm{sec}xdy+{2(1-x)\mathrm{tan}x+x(2-x)}dx=0$ such that $y(0)=2$. Then $y(2)$ is equal to :

One of the points of intersection of the curves $y=1+3 x-2 x^2$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2}, 2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}(l \sqrt{5}+\mathrm{m})-\mathrm{n} \log _{\mathrm{e}}(1+\sqrt{5})$, where $l, \mathrm{~m}, \mathrm{n} \in \mathbf{N}$. Then $l+\mathrm{m}+\mathrm{n}$ is equal to

Let $f(x)=a x^3+b x^2+c x+41$ be such that $f(1)=40, f^{\prime}(1)=2$ and $f^{\prime}(1)=4$. Then $\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2$ is equal to:

The area of the region ${(x,y):{y}^{2}\leq 4x,x<4,\frac{xy(x-1)(x-2)}{(x-3)(x-4)}>0,x\neq 3}$ is

If $y=\frac{(\sqrt{x}+1)({x}^{2}-\sqrt{x})}{x\sqrt{x}+x+\sqrt{x}}+\frac{1}{15}(3{\mathrm{cos}}^{2}x-5){\mathrm{cos}}^{3}x,$ then $96{y}^{'}(\frac{\pi }{6})$ is equal to:

Let $f:[-1,2] \rightarrow \mathbf{R}$ be given by $f(x)=2 x^2+x+\left[x^2\right]-[x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{3+\alpha \mathrm{sin}x+\beta \mathrm{cos}x+{\mathrm{log}}_{e}(1-x)}{3{\mathrm{tan}}^{2}x}=\frac{1}{3}$, then $2\alpha -\beta$ is equal to :

Let $g(x)=3f(\frac{x}{3})+f(3-x)$ and ${f}^{"}(x)>0$ for all $x\in (0,3)$. If $g$ is decreasing in $(0,\alpha )$ and increasing in $(\alpha ,3)$, then $8\alpha$ is

If $y=y(x)$ is the solution curve of the differential equation $({x}^{2}-4)\mathrm{dy}-({y}^{2}-3y)\mathrm{dx}=0$, $x>2,y(4)=\frac{3}{2}$ and the slope of the curve is never zero, then the value of $y(10)$ equals :