Calculus PYQ — Page 46

Let $f(x)=\int _{0}^{x}g(t)dt,$ where $g$ is a non-zero even function. If $f(x+5)=g(x),$ then $\int _{0}^{x}f(t)dt$ equals

Let $I={\int }_{a}^{b}({x}^{4}-2{x}^{2})dx.$ If $I$ is minimum then the ordered pair $(a, b)$ is

Let $f(x)=5-|x-2|$ and $g(x)=|x+1|,$ $x \in R.$ If $f(x)$ attains maximum value at $\alpha$ and $g(x)$ attains minimum value at $\beta ,$ then $\underset{x\rightarrow -\alpha \beta }{lim}\frac{(x-1)({x}^{2}-5x+6)}{{x}^{2}-6x+8}$ is equal to

Let $f:R\rightarrow R$ be a function defined as $f(x)={\begin{matrix} 5, if x\leq 1 \\ a+bx, if 1<x<3 \\ b+5x, if 3\leq x<5 \\ 30, if x\geq 5\end{matrix}$ Then $f$ is:

Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $\frac{x^{\mathrm{m}} y^{\mathrm{n}}}{\left(1+x^{2 \mathrm{~m}}\right)\left(1+y^{2 \mathrm{n}}\right)}$ is :

Let $f:R\rightarrow R$ be a differentiable function satisfying ${f}^{'}(3)+{f}^{'}(2)=0.$ Then $\underset{x\rightarrow 0}{lim}{(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)})}^{\frac{1}{x}}$ is equal to

Let $y=y(x)$ be the solution of the differential equation, $x\frac{dy}{dx}+y=x{\mathrm{log}}_{e}x, (x>1)$. If $2y(2)={\mathrm{log}}_{e}4-1$, then $y(e)$ is equal to

$\underset{x\rightarrow 0}{lim}\frac{x+2sinx}{\sqrt{{x}^{2}+2sinx+1} - \sqrt{{sin}^{2}x-x+1}}$ is

If $\int _{0}^{x}f(t)dt={x}^{2}+\int _{x}^{1}{t}^{2}f(t)dt,$ then ${f}^{'}(\frac{1}{2})$ is

The integral $\int cos(\mathrm{lnx})dx$, is equal to

The integral ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}{sec}^{\frac{2}{3}}x\cdot cose{c}^{\frac{4}{3}}xdx$ is equal to

The solution of the differential equation, $\frac{\mathrm{d} y}{\mathrm{~d} x}=(x-y)^{2}$, when $y(1)=1,$ is:

For each $x\in R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $\underset{x\rightarrow {0}^{-}}{lim}\frac{x([x]+|x|)\mathrm{sin}⁡[x]}{|x|}$ is equal to

Let $f:R\rightarrow R$ be differentiable at $c\in R$ and $f(c)=0$. If $g(x)=|f(x)|$, then at $x=c, g$ is:

The integral $\int \frac{3{x}^{13}+2{x}^{11}}{{(2{x}^{4}+3{x}^{2}+1)}^{4}}dx$, is equal to

If $\int \frac{dx}{{({x}^{2}-2x+10)}^{2}}=A({\mathrm{tan}}^{-1}(\frac{x-1}{3})+\frac{f(x)}{{x}^{2}-2x+10})+C$, then (where $C$ is a constant of integration)

The value of $\int _{0}^{\pi /2}\frac{{sin}^{3}x}{sinx+cosx}dx$ is:

If $\int {x}^{5}{e}^{-{x}^{2}}dx=g(x){e}^{-{x}^{2}}+c$, where $c$ is a constant of integration, then $g(-1)$ is equal to

The integral $\int \frac{2{x}^{3}-1}{{x}^{4}+x}dx$, is equal to

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

If ${x}^{2}+{y}^{2}+\mathrm{sin}y=4$, then the value of $\frac{{d}^{2}y}{d{x}^{2}}$ at the point $(-2,0)$ is :

If $f(x)={\int }_{0}^{x}t(\mathrm{sin}x-\mathrm{sin}t)dt$, then

If $I_1=\int_0^1 e^{-x} \cos ^2 x d x ; I_2=\int_0^1 e^{-x^2} \cos ^2 x d x$ and $I_3=\int_0^1 e^{-x^3} d x$; then

Let $g(x)=\mathrm{cos}{x}^{2}, f(x)=\sqrt{x},$ and $\alpha ,\beta (\alpha <\beta )$ be the roots of the quadratic equation $18{x}^{2}-9\pi x+{\pi }^{2}=0$. Then the area (in sq. units) bounded by the curve $y=(gof)(x)$ and the lines $x=\alpha ,x=\beta$ and $y=0,$ is