Calculus PYQ — Page 43

If $m$ is the minimum value of $k$ for which the function $f(x)=x\sqrt{kx-{x}^{2}}$ is increasing in the interval $[0,3]$ and $M$ is the maximum value of $f$ in $[0,3]$ when $k=m,$ then the ordered pair $(m, M)$ is equal to:

The region represented by $|x-y|\leq 2$ and $|x+y|\leq 2$ is bounded by a

Let $f:(-1, 1)\rightarrow R$ be a function defined by $f(x)=max{-|x|, -\sqrt{1-{x}^{2}}}.$ If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly

If $\underset{x\rightarrow 1}{\mathrm{lim}}\frac{{x}^{2}-ax+b}{x-1}=5,$ then $a+b$ is equal to:

If $\underset{x\rightarrow 1}{lim}\frac{{x}^{4}-1}{x-1}=\underset{x\rightarrow k}{lim}\frac{{x}^{3}-{k}^{3}}{{x}^{2}-{k}^{2}}$ , then $k$ is

The value of the integral $\int_{-2}^{2} \frac{\sin ^{2} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}} d x$ (where $[x]$ denotes the greatest integer less than or equal to x) is

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}=(tanx-y){sec}^{2}x$ , $x\in (-\frac{\pi }{2}, \frac{\pi }{2})$ , such that $y(0)=0$, then $y(-\frac{\pi }{4})$ is equal to:

The value of ${\int }_{0}^{\pi }{|\mathrm{cos}x|}^{3}dx$ is

If $f(x)={\begin{matrix}\frac{sin(p+1)x+sinx}{x} & , & x<0 \\ q & , & x=0 \\ \frac{\sqrt{x+{x}^{2}}-\sqrt{x}}{{x}^{3/2}} & , & x>0\end{matrix}$ is continuous at $x=0$ , then the ordered pair $(p, q)$ is equal to:

If the function $f$ defined on $(\frac{\pi }{6},\frac{\pi }{3})$ by $f(x)={\begin{matrix}\frac{\sqrt{2}cosx-1}{cotx-1},x\neq \frac{\pi }{4} \\ k, x=\frac{\pi }{4}\end{matrix}$ is continuous, then $k$ is equal to

Let $f(x)={\begin{matrix}max(|x|,{x}^{2}), & |x|\leq 2 \\ 8-2|x|, & 2<|x|\leq 4\end{matrix}.$ Let $S$ be the set of points in the interval $(-4,4)$ at which $f$ is not differentiable. Then $S$

The value of $\underset{y\rightarrow 0}{lim}\frac{\sqrt{1+\sqrt{1+{y}^{4}}}-\sqrt{2}}{{y}^{4}}$

Let $f(x)={e}^{x}-x$ and $g(x)={x}^{2}-x, \forall x \epsilon R$ . Then the set of all $x \epsilon R$ , where the function $h(x)=(fog)(x)$ is increasing, is:

If $x \log _{e}\left(\log _{e} x\right)-x^{2}+y^{2}=4(y>0),$ then $\frac{d y}{d x}$ at $x=e$ is equal to :

If $f(x)$ is a non-zero polynomial of degree four, having local extreme points at $x= –1, 0, 1;$ then the set $S={x\in R :f(x)=f(0)}$ contains exactly

Let $\sum _{k=1}^{10}f(a+k)=16({2}^{10}-1),$ where the function $f$ satisfies $f(x+y)=f(x)f(y)$ for all natural numbers $x, y$ and $f(1)=2.$ Then the natural number $'a'$ is:

Let $f:R\rightarrow R$ be a continuous and differentiable function such that $f(2)=6$ and ${f}^{'}(2)=\frac{1}{48}$. If ${\int }_{6}^{f(x)}4{t}^{3}dt=(x-2)g(x),$ then $\underset{x\rightarrow 2}{lim}g(x)$ is equal to

The value of the integral ${\int }_{0}^{1}x{cot}^{-1}(1-{x}^{2}+{x}^{4})dx$ is

Let $f(x)=\left\{\begin{array}{cl}-1, & -2 \leq x < 0 \\ x^{2}-1, & 0 \leq x \leq 2\end{array}\right.$ and $g(x)=|\eta(x)|+f(x \mid) .$ Then, in the interval $(-2,2), g$ is:

If the area (in sq. units) bounded by the parabola ${y}^{2}=4\lambda x$ and the line $y=\lambda x, \lambda >0,$ is $\frac{1}{9}$, then $\lambda$ is equal to

The area (in sq. units) of the region $A={(x,y):\frac{{y}^{2}}{2}\leq x\leq y+4}$ is:

If the function $f$ given by $f(x)={x}^{3}-3(a-2){x}^{2}+3ax+7$, for some $a\in R$ is increasing in $(0, 1]$ and decreasing in $[1, 5)$, then a root of the equation, $\frac{f(x)-14}{{(x-1)}^{2}}=0, (x\neq 1)$ is :

If $cosx\frac{dy}{dx}-ysinx=6x$, $(0<x<\frac{\pi }{2})$ and $y(\frac{\pi }{3})=0,$ then $y(\frac{\pi }{6})$ is equal to

If a curve passes through the point $(1,-2)$ and has slope of the tangent at any point $(x, y)$ on it as $\frac{{x}^{2}-2y}{x}$, then the curve also passes through the point