Calculus PYQ — Page 40

The solution of the differential equation $\frac{dy}{dx}-\frac{y+3x}{{\mathrm{log}}_{e}(y+3x)}+3=0$ is (where $C$ is a constant of integration)

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of ${\int }_{1}^{2}|2x-[3x]|dx$ is

If ${y}^{2}+{\mathrm{log}}_{e}({\mathrm{cos}}^{2}x)=y,x\in (-\frac{\pi }{2},\frac{\pi }{2})$ then :

The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{1}{1+{e}^{\mathrm{sin}x}}dx$ is :

Let $y=y(x)$ be the solution of the differential equation $\mathrm{cos}x\frac{dy}{dx}+2y\mathrm{sin}x=\mathrm{sin}2x,x\in (0,\frac{\pi }{2})$ If $y(\pi /3)=0,$ then $y(\pi /4)$ is equal to :

The integral $\int \frac{dx}{{(x+4)}^{\frac{8}{7}}{(x-3)}^{\frac{6}{7}}}$ is equal to: (where $C$ is a constant of integration)

Let $y=y(x)$ be the solution of the differential equation, $\frac{2+\mathrm{sin}x}{y+1}.\frac{dy}{dx}=-\mathrm{cos}x$, $y>0,y(0)=1$. If $y(\pi )=a$ and $\frac{dy}{dx}$ at $x=\pi$ is $b$, then the ordered pair $(a,b)$ is equal to

If ${I}_{1}={\int }_{0}^{1}{(1-{x}^{50})}^{100}dx$ and ${I}_{2}={\int }_{0}^{1}{(1-{x}^{50})}^{101}dx$ such that ${I}_{2}=\alpha {I}_{1}$ then $\alpha$ equals to :

The general solution of the differential equation $\sqrt{1+{x}^{2}+{y}^{2}+{x}^{2}{y}^{2}}+xy\frac{dy}{dx}=0$ (where C is a constant of integration)

The function $f(x)={\begin{matrix}\frac{\pi }{4}+{\mathrm{tan}}^{-1}x, & |x|\leq 1 \\ \frac{1}{2}(|x|-1), & |x|>1\end{matrix}$ is :

If ${f}^{'}(x)={\mathrm{tan}}^{-1}⁡(\mathrm{sec}⁡x+\mathrm{tan}⁡x),-\frac{\pi }{2}<x<\frac{\pi }{2}$ and $f(0)=0$ , then $f(1)$ is equal to:

Let a function $f:[0,5]\rightarrow R$ be continuous, $f(1)=3$ and $F$ be defined as: $F(x)={\int }_{1}^{x}{t}^{2}g(t)dt,$ where $g(t)={\int }_{1}^{t}f(u)du.$ Then for the function $F(x),$ the point $x=1$ is:

The area (in sq. units) of the region $A={(x,y):(x-1)[x]\leq y\leq 2\sqrt{x},0\leq x\leq 2},$ where $[t]$ denotes the greatest integer function, is :

Let $f(x)=x{\mathrm{cos}}^{-1}⁡(-\mathrm{sin}⁡|x|),x\in [-\frac{\pi }{2},\frac{\pi }{2}],$ then which of the following is true?

${\int }_{-\pi }^{\pi }|\pi -|x||dx$ is equal to

The integral ${\int }_{1}^{2}{e}^{x}.{x}^{x}(2+{\mathrm{log}}_{e}x)\mathrm{dx}$ equals :

The value of $\alpha$ for which $4\alpha \int _{-1}^{2}{e}^{-\alpha |x|}dx=5$ , is

Let $S$ be the set of points where the function $,f(x)=|2-|x-3|,x\in R,$ is not differentiable. Then $\underset{x\in S}{\sum }f(f(x))$ is equal to

If a curve $y=f(x)$, passing through the point $(1,2)$, is the solution of the differential equation $2{x}^{2}dy=(2xy+{y}^{2})dx$, then $f(\frac{1}{2})$ is equal to

If $\frac{dy}{dx}=\frac{xy}{{x}^{2}+{y}^{2}};y(1)=1;$ then a value of $x$ satisfying $y(x)=e$ is:

The value of ${\int }_{0}^{2\pi }\frac{x{\mathrm{sin}}^{8}⁡x}{{\mathrm{sin}}^{8}⁡x+{\mathrm{cos}}^{8}⁡x}dx$ is equal to:

The function, $f(x)=(3x-7){x}^{\frac{2}{3}},x\in R$, is increasing for all $x$ lying in

Let $f$ be any function continuous on $[a,b]$ and twice differentiable on $(a,b)$ . If all $x\in (a,b),{f}^{'}(x)>0$ and ${f}^{''}(x)<0$ , then for any $c\in (a,b),\frac{f(c)-f(a)}{f(b)-f(c)}$

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}{x}^{5}\mathrm{sin}(\frac{1}{x})+5{x}^{2} & , & x<0 \\ 0 & , & x=0 \\ {x}^{5}\mathrm{cos}(\frac{1}{x})+\lambda {x}^{2} & , & x>0\end{matrix}$. The value of $\lambda$ for which ${f}^{"}(0)$ exists, is___.