Calculus PYQ — Page 37

If $\underset{x\rightarrow \infty }{\mathrm{lim}}(\sqrt{{x}^{2}-x+1}-ax)=b$, then the ordered pair $(a,b)$ is:

The value of $\underset{h\rightarrow 0}{\mathrm{lim}}{\frac{\sqrt{3}\mathrm{sin}(\frac{\pi }{6}+h)-\mathrm{cos}(\frac{\pi }{6}+h)}{\sqrt{3}h(\sqrt{3}\mathrm{cos}h-\mathrm{sin}h)}}$ is :

If $f:R\rightarrow R$ is a function defined by $f(x)=[x-1]\mathrm{cos}(\frac{2x-1}{2})\pi ,$ where $[\cdot ]$ denotes the greatest integer function, then $f$ is:

If ${I}_{n}={\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\mathrm{cot}}^{n}xdx$, then

Let $y=y(x)$ be the solution of the differential equation $((x+2){e}^{(\frac{y+1}{x+2})}+(y+1))dx=(x+2)dy,y(1)=1.$ If the domain of $y=y(x)$ is an open interval $(\alpha ,\beta ),$ then $|\alpha +\beta |$ is equal to ___________.

Let $f:[0,3]\rightarrow R$ be defined by $f(x)=\mathrm{min}{x-[x],1+[x]-x}$ where $[x]$ is the greatest integer less than or equal to $x.$ Let $P$ denote the set containing all $x\in [0,3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x\in (0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $P$ and $Q$ is equal to _____.

The number of distinct real roots of the equation $3{x}^{4}+4{x}^{3}-12{x}^{2}+4=0$ is _________.

Let $f(x)$ and $g(x)$ be two functions satisfying $f({x}^{2})+g(4-x)=4{x}^{3}$ and $g(4-x)+g(x)=0,$ then the value of ${\int }_{-4}^{4}f({x}^{2})dx$ is

If ${\int }_{0}^{100\pi }\frac{{\mathrm{sin}}^{2}x}{{e}^{(\frac{x}{\pi }-[\frac{x}{\pi }])}}dx=\frac{\alpha {\pi }^{3}}{1+4{\pi }^{2}},\alpha \in R$ where $[x]$ is the greatest integer less than or equal to $x,$ then the value of $\alpha$ is:

Consider the function $f:R\rightarrow R$ defined by $f(x)={\begin{matrix}(2-\mathrm{sin}(\frac{1}{x}))|x|, & x\neq 0 \\ 0, & x=0\end{matrix}.$ Then $f$ is:

The area, enclosed by the curves $y=\mathrm{sin}x+\mathrm{cos}x$ and $y=|\mathrm{cos}x-\mathrm{sin}x|$ and the lines $x=0,x=\frac{\pi }{2},$ is :

Consider the function $f(x)=\frac{P(x)}{\mathrm{sin}(x-2)},x\neq 2$, and $f(x)=7,x=2$where $P(x)$ is a polynomial such that ${P}^{"}(x)$ is always a constant and $P(3)=9.$ If $f(x)$ is continuous at $x=2,$ then $P(5)$ is equal to __________.

If $\underset{x\rightarrow 0}{\mathrm{lim}}[\frac{\alpha x{e}^{x}-\beta {\mathrm{log}}_{e}(1+x)+\gamma {x}^{2}{e}^{-x}}{x{\mathrm{sin}}^{2}x}]=10,\alpha ,\beta ,\gamma \in R$, then the value of $\alpha +\beta +\gamma$ is __________.

A function $f$ is defined on $[-3,3]$ as $f(x)={\begin{matrix}\mathrm{min}{|x|,2-{x}^{2}},-2\leq x\leq 2 \\ [|x|],2<|x|\leq 3\end{matrix}$ where $[x]$ denotes the greatest integer $\leq x$. The number of points, where $f$ is not differentiable in $(-3,3)$ is ___ .

If $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of the integral ${\int }_{-\pi /2}^{\pi /2}[[x]-\mathrm{sin}x]dx$ is equal to:

The value of $\underset{x\rightarrow {0}^{+}}{\mathrm{lim}}\frac{{\mathrm{cos}}^{-1}(x-[x{]}^{2})\cdot {\mathrm{sin}}^{-1}(x-[x{]}^{2})}{x-{x}^{3}},$ where $[x]$ denotes the greatest integer $\leq x$ is:

The integral $\int \frac{1}{\sqrt[4]{(x-1{)}^{3}(x+2{)}^{5}}}dx$ is equal to : (where $C$ is a constant of integration)

Let $\alpha \in R$ be such that the function $f(x)={\begin{matrix}\frac{{\mathrm{cos}}^{-1}(1-{{x}}^{2}){\mathrm{sin}}^{-1}(1-{x})}{{x}-{{x}}^{3}}, & x\neq 0 \\ \alpha , & x=0\end{matrix}$ is continuous at $x=0,$ where ${x}=x-[x],[x]$ is the greatest integer less than or equal to $x$. Then :

Let $f:(-\frac{\pi }{4},\frac{\pi }{4})\rightarrow R$ be defined as, $f(x)={\begin{matrix}{(1+|\mathrm{sin}x|)}^{\frac{3a}{|\mathrm{sin}x|}} & ,-\frac{\pi }{4}<x<0 \\ b & ,x=0 \\ {e}^{\mathrm{cot}4x/\mathrm{cot}2x} & ,0<x<\frac{\pi }{4}\end{matrix}$ If $f$ is continuous at $x=0$ then the value of $6a+{b}^{2}$ is equal to:

The area (in sq. units) of the part of the circle ${x}^{2}+{y}^{2}=36$, which is outside the parabola ${y}^{2}=9x$, is equal to

Let $f(x)={\int }_{0}^{x}{e}^{t}f(t)dt+{e}^{x}$ be a differentiable function for all $x\in R$. Then $f(x)$ equals :

Let $g(t)={\int }_{-\pi /2}^{\pi /2}(\mathrm{cos}\frac{\pi }{4}t+f(x))dx,$ where $f(x)={\mathrm{log}}_{e}(x+\sqrt{{x}^{2}+1}),x\in R.$ Then which one of the following is correct?

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}\frac{\lambda |{x}^{2}-5x+6|}{\mu (5x-{x}^{2}-6)} & x<2 \\ {e}^{\frac{\mathrm{tan}(x-2)}{x-[x]}} & x>2 \\ \mu & x=2\end{matrix}$ where $[x]$ is the greatest integer less than or equal to $x$. If $f$ is continuous at $x=2$, then $\lambda +\mu$ is equal to :

If $I=\int _{1}^{2}\frac{dx}{\sqrt{2{x}^{3}-9{x}^{2}+12x+4}}$, then