Calculus PYQ — Page 11

The number of critical points of the function $f(x)=(x-2)^{2 / 3}(2 x+1)$ is

A variable line $L$ passes through the point $(3,5)$ and intersects the positive coordinate axes at the points $\mathrm{A}$ and $\mathrm{B}$. The minimum area of the triangle $\mathrm{OAB}$, where $\mathrm{O}$ is the origin, is :

If ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\sqrt{1-\mathrm{sin}2x}dx=\alpha +\beta \sqrt{2}+\gamma \sqrt{3}$, where $\alpha ,\beta$ and $\gamma$ are rational numbers, then $3\alpha +4\beta -\gamma$ is equal to _____.

One of the points of intersection of the curves $y=1+3 x-2 x^2$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2}, 2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}(l \sqrt{5}+\mathrm{m})-\mathrm{n} \log _{\mathrm{e}}(1+\sqrt{5})$, where $l, \mathrm{~m}, \mathrm{n} \in \mathbf{N}$. Then $l+\mathrm{m}+\mathrm{n}$ is equal to

Let $\beta(\mathrm{m}, \mathrm{n})=\int_0^1 x^{\mathrm{m}-1}(1-x)^{\mathrm{n}-1} \mathrm{~d} x, \mathrm{~m}, \mathrm{n}>0$. If $\int_0^1\left(1-x^{10}\right)^{20} \mathrm{~d} x=\mathrm{a} \times \beta(\mathrm{b}, \mathrm{c})$, then $100(\mathrm{a}+\mathrm{b}+\mathrm{c})$ equals____

If the solution of the differential equation $(2x+3y-2)dx+(4x+6y-7)dy=0,y(0)=3$, is $\alpha x+\beta y+3{\mathrm{log}}_{e}|2x+3y-\gamma |=6$, then $\alpha +2\beta +3\gamma$ is equal to ______.

If $a=\underset{x\rightarrow 0}{\mathrm{lim}}\frac{\sqrt{1+\sqrt{1+{x}^{4}}}-\sqrt{2}}{{x}^{4}}$ and $b=\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\mathrm{sin}}^{2}x}{\sqrt{2}-\sqrt{1+\mathrm{cos}x}}$, then the value of $a{b}^{3}$ is :

Let the slope of the line $45x+5y+3=0$ be $27{r}_{1}+\frac{9{r}_{2}}{2}$ for some ${r}_{1},{r}_{2}\in R$. Then $\underset{x\rightarrow 3}{\mathrm{lim}}({\int }_{3}^{x}\frac{8{t}^{2}}{\frac{3{r}_{2}x}{2}-{r}_{2}{x}^{2}-{r}_{1}{x}^{3}-3x}dt)$ is equal to ______.

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}\frac{a-b\mathrm{cos}2x}{{x}^{2}};x<0 \\ {x}^{2}+cx+2;0\leq x\leq 1 \\ 2x+1;x>1\end{matrix}$ If $f$ is continuous everywhere in $R$ and $m$ is the number of points where $f$ is NOT differential then $m+a+b+c$ equals:

Let $f(x)={\begin{matrix}x-1,x\text{is even}, \\ 2x,x\text{is odd},\end{matrix}x\in N$. If for some $a\in N,f(f(f(a)))=21$, then $\underset{x\rightarrow {a}^{-}}{\mathrm{lim}}{\frac{{|x|}^{3}}{a}-[\frac{x}{a}]},$ where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to:

Let $f:(0, \pi) \rightarrow \mathbf{R}$ be a function given by $f(x)=$ \(\left\{\begin{array}{cc}\left(\frac{8}{7}\right)^{\frac{\tan 8 x}{\tan 7 x}}, & 0 < x < \frac{\pi}{2} \\ \mathrm{a}-8, & x=\frac{\pi}{2} \\ (1+|\cot x|)^{\frac{\mathrm{b}}{}|\tan x|}, & \frac{\pi}{2} < x < \pi\end{array}\right.\) where $\mathrm{a}, \mathrm{b} \in \mathbf{Z}$. If $f$ is continuous at $x=\frac{\pi}{2}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)={\begin{matrix}{x}^{2}+3x+a, & x\leq 1 \\ bx+2, & x>1\end{matrix}$ be differentiable on $R$. Then, the value of ${\int }_{-2}^{2}f(x)dx$ equals

The value of $k \in \mathrm{N}$ for which the integral $I_n=\int_0^1\left(1-x^k\right)^n d x, n \in \mathbb{N}$, satisfies $147 I_{20}=148 I_{21}$ is

The area (in sq. units) of the part of circle ${x}^{2}+{y}^{2}=169$ which is below the line $5x-y=13$ is $\frac{\pi \alpha }{2\beta }-\frac{65}{2}+\frac{\alpha }{\beta }{\mathrm{sin}}^{-1}(\frac{12}{13})$ where $\alpha ,\beta$ are coprime numbers. Then $\alpha +\beta$ is equal to

The value of $\lim _{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots \ldots \sqrt[10]{\cos 10 x}}{x^2}\right)$ is

If $y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}$, then at $\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y$ is equal to :

Let $f(x)=\sqrt{\underset{r\rightarrow x}{\mathrm{lim}}{\frac{2{r}^{2}[(f(r){)}^{2}-f(x)f(r)]}{{r}^{2}-{x}^{2}}-{r}^{3}{e}^{\frac{f(r)}{r}}}}$ be differentiable in $(-\infty ,0)\cup (0,\infty )$ and $f(1)=1$. Then the value of $ae$, such that $f(a)=0$, is equal to ______.

The value of $\int_{-\pi}^\pi \frac{2 y(1+\sin y)}{1+\cos ^2 y} d y$ is :

Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $\left(1+x\left(\lambda^2-x^2\right)\right)$ satisfies $\frac{x^2+x+2}{x^2+5 x+6} < 0$, be $(\alpha, \beta)$. Then $\alpha^2+\beta^2$ is equal to _________

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+4\right)^2 d y+\left(2 x^3 y+8 x y-2\right) d x=0$. If $y(0)=0$, then $y(2)$ is equal to

For $\mathrm{a}, \mathrm{b}>0$, let $f(x)=\left\{\begin{array}{cc}\frac{\tan ((\mathrm{a}+1) x)+\mathrm{b} \tan x}{x}, & x < 0 \\ 3, & x=0 \\ \frac{\sqrt{\mathrm{a} x+\mathrm{b}^2 x^2}-\sqrt{\mathrm{a} x}}{\mathrm{~b} \sqrt{\mathrm{a}} x \sqrt{x}}, & x>0\end{array}\right.$ be a continous function at $x=0$. Then $\frac{\mathrm{b}}{\mathrm{a}}$ is equal to :

If the integral $525\int _{0}^{\frac{\pi }{2}}\mathrm{sin}2x{\mathrm{cos}}^{\frac{11}{2}}x{(1+{\mathrm{cos}}^{\frac{5}{2}}x)}^{\frac{1}{2}}dx$ is equal to $(n\sqrt{2}-64)$, then $n$ is equal to ________

If the solution $y=y(x)$ of the differential equation $\left(x^4+2 x^3+3 x^2+2 x+2\right) \mathrm{d} y-\left(2 x^2+2 x+3\right) \mathrm{d} x=0$ satisfies $y(-1)=-\frac{\pi}{4}$, then $y(0)$ is equal to :

If $\log _e y=3 \sin ^{-1} x$, then $\left(1-x^2\right) y^{\prime \prime}-x y^{\prime}$ at $x=\frac{1}{2}$ is equal to