Calculus PYQ — Page 32

The function $f(x)={x}^{3}-6{x}^{2}+ax+b$ is such that $f(2)=f(4)=0$. Consider two statements: $({S}_{1})$ there exists ${x}_{1},{x}_{2}\in (2,4),{x}_{1}<{x}_{2},$ such that ${f}^{'}({x}_{1})=-1$ and ${f}^{'}({x}_{2})=0.$ $({S}_{2})$ there exists ${x}_{3},{x}_{4}\in (2,4),{x}_{3}<{x}_{4}$, such that $f$ is decreasing in $(2,{x}_{4})$, increasing in $({x}_{4},4)$ and $2{f}^{'}({x}_{3})=\sqrt{3}f({x}_{4})$ then

The sum of all the local minimum values of the twice differentiable function $f:R\rightarrow R$ defined by $f(x)={x}^{3}-3{x}^{2}-\frac{3{f}^{"}(2)}{2}x+{f}^{"}(1)$ is:

Consider the integral $I={\int }_{0}^{10}\frac{[x]{e}^{[x]}}{{e}^{x-1}}dx$ where $[x]$ denotes the greatest integer less than or equal to $x$. Then the value of $I$ is equal to :

Let $f:R\rightarrow R$ be a function such that $f(2)=4$ and ${f}^{'}(2)=1.$ Then, the value of $\underset{x\rightarrow 2}{\mathrm{lim}}\frac{{x}^{2}f(2)-4f(x)}{x-2}$ is equal to:

The area of the region bounded by $y-x=2$ and ${x}^{2}=y$ is equal to :-

If ${\int }_{0}^{\pi }({\mathrm{sin}}^{3}x){e}^{-{\mathrm{sin}}^{2}x}dx=\alpha -\frac{\beta }{e}{\int }_{0}^{1}\sqrt{t}{e}^{t}dt,$ then $\alpha +\beta$ is equal to

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t=0.$ The number of bacteria is increased by $20%$ in $2$ hours. If the population of bacteria is $2000$ after $\frac{k}{{\mathrm{log}}_{e}(\frac{6}{5})}$ hours, then ${(\frac{k}{{\mathrm{log}}_{e}2})}^{2}$ is equal to:

Let $[t]$ denote the greatest integer $\leq t.$ The number of points where the function $f(x)=[x]|{x}^{2}-1|+\mathrm{sin}(\frac{\pi }{[x]+3})-[x+1],x\in (-2,2)$ is not continuous is _____ .

Let $a,b\in R,b\neq 0$. Defined a function, $f(x)={\begin{matrix}a\mathrm{sin}\frac{\pi }{2}(x-1),\mathrm{for}x\leq 0 \\ \frac{\mathrm{tan}2x-\mathrm{sin}2x}{b{x}^{3}},\mathrm{for}x>0\end{matrix}$ If $f$ is continuous at$x=0$, then $10-ab$ is equal to

Let $f:[0,\infty )\rightarrow [0,\infty )$ be defined as $f(x)={\int }_{0}^{x}[y]dy$ where $[x]$ is the greatest integer less than or equal to $x$. Which of the following is true?

Let $y=y(x)$ be the solution of the differential equation $\mathrm{cos}x(3\mathrm{sin}x+\mathrm{cos}x+3)dy=(1+y\mathrm{sin}x(3\mathrm{sin}x+\mathrm{cos}x+3))dx,0\leq x\leq \frac{\pi }{2},y(0)=0.$ Then, $y(\frac{\pi }{3})$ is equal to:

If $f(x)=\mathrm{sin}({\mathrm{cos}}^{-1}(\frac{1-{2}^{2x}}{1+{2}^{2x}}))$ and its first derivative with respect to $x$ is $-\frac{b}{a}{\mathrm{log}}_{e}2$ when $x=1,$ where $a$ and $b$ are integers, then the minimum value of $|{a}^{2}-{b}^{2}|$ is _______.

Let $f(x)=\mathrm{cos}(2{\mathrm{tan}}^{-1}\mathrm{sin}({\mathrm{cot}}^{-1}\sqrt{\frac{1-x}{x}})),0<x<1$. Then:

If $y=y(x)$ is an implicit function of $x$ such that ${\mathrm{log}}_{e}(x+y)=4xy$, then $\frac{{d}^{2}y}{d{x}^{2}}$ at $x=0$ is equal to

Let $f$ be a real valued function, defined on$R-{-1,1}$ and given by $f(x)=3{\mathrm{log}}_{e}|\frac{x-1}{x+1}|-\frac{2}{x-1}$. Then in which of the following intervals, function $f(x)$ is increasing?

Let $a$ be an integer such that all the real roots of the polynomial $2{x}^{5}+5{x}^{4}+10{x}^{3}+10{x}^{2}+10x+10$ lie in the interval $(a,a+1)$. Then, $|a|$ is equal to ______.

The minimum value of $\alpha$ for which the equation $\frac{4}{\mathrm{sin}x}+\frac{1}{1-\mathrm{sin}x}=\alpha$ has at least one solution in $(0,\frac{\pi }{2})$ is______.

If the value of $\underset{x\rightarrow 0}{\mathrm{lim}}{(2-\mathrm{cos}x\sqrt{\mathrm{cos}2x})}^{(\frac{x+2}{{x}^{2}})}$ is equal to ${e}^{a}$, then $a$ is equal to_____.

The number of points, at which the function $f(x)=|2x+1|-3|x+2|+|{x}^{2}+x-2|,x\in R$ is not differentiable, is

The integral $\int \frac{{e}^{3{\mathrm{log}}_{e}2x}+5{e}^{2{\mathrm{log}}_{e}2x}}{{e}^{4{\mathrm{log}}_{e}x}+5{e}^{3{\mathrm{log}}_{e}x}-7{e}^{2{\mathrm{log}}_{e}x}}dx,x>0$, is equal to (where $c$ is a constant of integration)

Let $f$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1).$ If ${\int }_{0}^{x}\sqrt{1-{({f}^{'}(t))}^{2}}\mathrm{dt}={\int }_{0}^{x}f(t)\mathrm{dt},0\leq x\leq 1$ and $f(0)=0,$ then $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{1}{{x}^{2}}{\int }_{0}^{x}f(t)\mathrm{dt}:$

The value of the integral ${\int }_{0}^{1}\frac{\sqrt{x}dx}{(1+x)(1+3x)(3+x)}$ is:

The value of the definite integral ${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{dx}{(1+{e}^{x\mathrm{cos}x})({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x)}$ is equal to :

The value of the definite integral ${\int }_{\pi /24}^{5\pi /24}\frac{dx}{1+\sqrt[3]{\mathrm{tan}2x}}$ is