Calculus PYQ — Page 41

The derivative of ${\mathrm{tan}}^{-1}(\frac{\sqrt{1+{x}^{2}}-1}{x})$ with respect to ${\mathrm{tan}}^{-1}(\frac{2x\sqrt{1-{x}^{2}}}{1-2{x}^{2}})$ at $x=\frac{1}{2}$ is :

$\underset{x\rightarrow 0}{\mathrm{lim}}{(\frac{3{x}^{2}+2}{7{x}^{2}+2})}^{\frac{1}{{x}^{2}}}$ is equal to

If the value of the integral ${\int }_{0}^{\frac{1}{2}}\frac{{x}^{2}}{{(1-{x}^{2})}^{\frac{3}{2}}}dx$ is $\frac{k}{6}$, then $k$ is equal to:

$\underset{x\rightarrow 1}{\mathrm{lim}}(\frac{{\int }_{0}^{(x-1{)}^{2}}t\mathrm{cos}{t}^{2}dt}{(x-1)\mathrm{sin}(x-1)})$

Let $f(x)=|x-2|$ and $g(x)=f(f(x)),x\in [0,4]$. Then ${\int }_{0}^{3}(g(x)-f(x))dx$ is equal to

$\underset{x\rightarrow 2}{lim}\frac{{3}^{x}+{3}^{3-x}-12}{{3}^{-\frac{x}{2}}-{3}^{1-x}}$ is equal to

The area (in sq. units) of the region ${(x, y)\in {R}^{2}:{x}^{2}\leq y\leq 3-2x}$, is.

If ${x}^{3}dy+xy\cdot dx={x}^{2}dy+2ydx;y(2)=e$ and $x>1$, then $y(4)$ is equal to :

If $x=2\mathrm{sin}\theta -\mathrm{sin}2\theta$ and $y=2\mathrm{cos}\theta -\mathrm{cos}2\theta ,$$\theta \in [0,2\pi ],$ then $\frac{{d}^{2}y}{d{x}^{2}}$ at $\theta =\pi$ is:

The integral ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}{\mathrm{tan}}^{3}x\cdot {\mathrm{sin}}^{2}3x(2{\mathrm{sec}}^{2}x\cdot {\mathrm{sin}}^{2}3x+3\mathrm{tan}x\cdot \mathrm{sin}6x)dx$ is equal to:

Let $f:(-1,\infty )\rightarrow R$ be defined by $f(0)=1$ and $f(x)=\frac{1}{x}{\mathrm{log}}_{e}(1+x),x\neq 0$. Then the function $f$

If $\underset{x\rightarrow 1}{\mathrm{lim}}\frac{x+{x}^{2}+{x}^{3}+...+{x}^{n}-n}{x-1}=820,(n\in N)$ then the value of $n$ is equal to....

The area (in sq. units) of the largest rectangle $ABCD$ whose vertices $A$ and $B$ lie on the $x$-axis and vertices $C$ and $D$ lie on the parabola, $y={x}^{2}-1$ below the $x$-axis, is :

A spherical iron ball of $10cm$ radius is coated with a layer of ice of uniform thickness that melts at a rate of $50{cm}^{3}/min$ . When the thickness of ice is $5cm$ , then the rate (in $cm/min$ .) at which of the thickness of ice decreases, is:

The area (in sq. units) of the region ${(x,y)\in {R}^{2}|4{x}^{2}\leq y\leq 8x+12}$ is

Let $f(x)$ be a polynomial of degree $5$ such that $x=\pm 1$ are its critical points. If $\underset{x\rightarrow 0}{lim}(2+\frac{f(x)}{{x}^{3}})=4,$ then which one of the following is not true?

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{x({e}^{(\sqrt{1+{x}^{2}+{x}^{4}}-1)/x}-1)}{\sqrt{1+{x}^{2}+{x}^{4}}-1}$

Let $AD$ and $BC$ be two vertical poles at $A\text{and}B$ respectively on a horizontal ground. If $AD=8m$, $\mathrm{BC}=11m$, $\mathrm{AB}=10m;$ then the distance (in meters) of a point $M$ lying in between $\mathrm{AB}$ from the point $A$ such that ${\mathrm{MD}}^{2}+{\mathrm{MC}}^{2}$ is minimums, is__

If $\underset{x\rightarrow 0}{\mathrm{lim}}{\frac{1}{{x}^{8}}(1-\mathrm{cos}\frac{{x}^{2}}{2}-\mathrm{cos}\frac{{x}^{2}}{4}+\mathrm{cos}\frac{{x}^{2}}{2}\mathrm{cos}\frac{{x}^{2}}{4})}={2}^{-k}$ then the value of k is

If $\int \frac{\mathrm{cos}xdx}{{\mathrm{sin}}^{3}x{(1+{\mathrm{sin}}^{6}x)}^{\frac{2}{3}}}=f(x){(1+{\mathrm{sin}}^{6}x)}^{\frac{1}{\lambda }}+c$, where $c$ is a constant of integration, then $\lambda f(\frac{\pi }{3})$ is equal to

Let $f$ and $g$ be differentiable functions on $R$ such that $fog$ is the identity function. If for some $a,b\in R,{g}^{'}(a)=5$ and $g(a)=b,$ then ${f}^{'}(b)$ is equal to:

If $p(x)$ be a polynomial of degree three that has a local maximum value $8$ at $x=1$ and a local minimum value $4$ at $x=2$ then $p(0)$ is equal to

The integral ${\int }_{0}^{2}||x-1|-x|dx$ is equal to

$\underset{x\rightarrow 0}{lim}\frac{{\int }_{0}^{x}tsin(10t)dt}{x}$, is equal to