Calculus PYQ — Page 44

The derivative of ${tan}^{-1}(\frac{sinx-cosx}{sinx+cosx})$ with respect to $\frac{x}{2},$ where $x\in (0,\frac{\pi }{2})$, is

Let $\alpha \in (0,\frac{\pi }{2})$, be constant.If the integral $\int \frac{tanx+tan\alpha }{tanx-\mathrm{tan}\alpha }dx=A(x)cos2\alpha +B(x)sin2\alpha +C$, where C is a constant of integration, then the functions $A(x)$ and $B(x)$ are respectively

If $y(x)$ is the solution of the differential equation $\frac{d y}{d x}+\left(\frac{2 x+1}{x}\right) y=e^{-2 x}, x>0,$ where $y(1)=\frac{1}{2} e^{-2},$ then:

$\underset{x\rightarrow \frac{\pi }{4}}{lim}\frac{co{t}^{3}x-tanx}{cos(x+\frac{\pi }{4})}$ is

If $f:R\rightarrow R$ is a differentiable function and $f(2)=6,$ then $\underset{x\rightarrow 2}{lim}{\int }_{6}^{f(x)}\frac{2tdt}{(x-2)}$ is:

If $\int {x}^{5}{e}^{-4{x}^{3}}dx=\frac{1}{48}{e}^{-4{x}^{3}}f(x)+C$, where $C$ is a constant of integration, then $f(x)$ is equal to

The area (in sq. units) of the region bounded by the parabola, $y={x}^{2}+2$ and the lines, $y=x+1, x=0$ and $x=3$, is

Let $[\mathrm{x}]$ denote the greatest integer less than or equal to $\mathrm{X}$. Then : $\lim _{x \rightarrow 0} \frac{\tan \left(\pi \sin ^{2} x\right)+(|\mathrm{x}|-\sin (x[x]))^{2}}{x^{2}}$

The integral $\int_{\pi / 6}^{\pi / 4} \frac{\mathrm{d} x}{\sin 2 x\left(\tan ^{5} x+\cot ^{5} x\right)}$ equals:

A $2$m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate $25cm/sec$ , then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is $1$ m above the ground is:

If the area (in sq. units) of the region ${(x,y):{y}^{2}\leq 4x,x+y\leq 1,x\geq 0,y\geq 0}$ is $a\sqrt{2}+b,$ then $a-b$ is equal to

If $f(1)=1,{f}^{'}(1)=3$, then the derivative of $f(f(f(x)))+{(f(x))}^{2}$ at $x=1$ is:

Let $\mathrm{K}$ be the set of all real values of $x$ where the function $f(x)=\sin |x|-|x|+2(x-\pi) \cos |x|$ is not differentiable. Then the set $K$ is equal to :

Consider the differential equation, ${y}^{2}dx+(x-\frac{1}{y})dy=0$. If value of $y$ is $1$ when $x=1$, then the value of $x$ for which $y=2,$is

A value of $\alpha$ such that $\int _{\alpha }^{\alpha +1}\frac{dx}{(x+\alpha )(x+\alpha +1)}=lo{g}_{e}(\frac{9}{8})$ is

The general solution of the differential equation $({y}^{2}-{x}^{3})dx-xydy=0,(x\neq 0)$ is (where c is a constant of integration)

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius $3$ is:

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is ${tan}^{-1}(\frac{1}{2}).$ Water is poured into it at a constant rate of $5cubicm/min.$ Then the rate $($in $m/min),$ at which the level of water is rising at the instant when the depth of water in the tank is $10 m;$ is:

If $\int _{0}^{\frac{\pi }{2}}\frac{cotx}{cotx+\mathrm{cosec}x}dx=m(\pi +n),$ then $mn$ is equal to

Themaximum value of the finction $f(x)=3 x^{3}-18 x^{2}+27 x-40$ on the set $\mathrm{S}=\left\{x \in R: x^{2}+30 \leq 11 x\right\}$ is :

If $\int {e}^{secx}(secx\mathrm{tan}xf(x)+(secx\mathrm{tan}x+se{c}^{2}x))dx={e}^{secx}f(x)+C,$ then a possible choice of $f(x)$ is:

The area (in sq. units) of the region bounded by the curves $y={2}^{x}$ and $y=|x+1|$, in the first quadrant is

Let $f$ be a differentiable function such that $f(1)=2$ and ${f}^{'}(x)=f(x)$ for all $x\in R$. If $h(x)=f(f(x))$, then ${h}^{'}(1)$ is equal to :

Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x)-f(y)|\leq 2{|x-y|}^{3/2},$ for all $x,y\in R\text{.}$ If $f(0)=1$ then $\int _{0}^{1}{f}^{2}(x)dx$ is equal to