Calculus PYQ — Page 24

Let the function $f(x)={\begin{matrix}\frac{{\mathrm{log}}_{e}(1+5x)-{\mathrm{log}}_{e}(1+\alpha x)}{x} & \mathrm{if}x\neq 0 \\ 10 & \mathrm{if}x=0\end{matrix}$ be continuous at $x=0$. Then $\alpha$ is equal to

If the maximum value of $a$, for which the function ${f}_{a}(x)={\mathrm{tan}}^{-1}2x-3ax+7$ is non-decreasing in $(-\frac{\pi }{6},\frac{\pi }{6})$, is $\bar{a}$, then ${f}_{\bar{a}}(\frac{\pi }{8})$ is equal to

If the solution curve $y=y(x)$ of the differential equation ${y}^{2}dx+({x}^{2}-xy+{y}^{2})dy=0$, which passes through the point $(1,1)$ and intersects the line $y=\sqrt{3}x$ at the point $(\alpha ,\sqrt{3}\alpha )$, then value of ${\mathrm{log}}_{e}(\sqrt{3}\alpha )$ is equal to

If $\int \frac{1}{x}\sqrt{\frac{1-x}{1+x}}dx=g(x)+c,g(1)=0$, then $g(\frac{1}{2})$ is equal to

Let $[t]$ denote the greatest integer $\leq t$ and ${t}$ denote the fractional part of $t$. Then integral value of $\alpha$ for which the left hand limit of the function $f(x)=[1+x]+\frac{{\alpha }^{2[x]+{x}}+[x]-1}{2[x]+{x}}$ at $x=0$ is equal to $\alpha -\frac{4}{3}$ is _____

If $f(\theta )=\mathrm{sin}\theta +{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(\mathrm{sin}\theta +t\mathrm{cos}\theta )\cdot f(t)dt$, then $|{\int }_{0}^{\frac{\pi }{2}}f(\theta )d\theta |$ is

If $\underset{x\rightarrow 1}{\mathrm{lim}}(\frac{\mathrm{sin}(3{x}^{2}-4x+1)-{x}^{2}+1}{2{x}^{3}-7{x}^{2}+ax+b})=-2$, then the value of $(a-b)$ is equal to

Let $a$ be an integer such that $\underset{x\rightarrow 7}{\mathrm{lim}}\frac{18-[1-x]}{[x-3a]}$ exists, where $[t]$ is greatest integer $\leq t$. Then $a$ is equal to

Let $\underset{0\leqslant x\leqslant 2}{Max}{\frac{9-{x}^{2}}{5-x}}=\alpha$ and $\underset{0\leqslant x\leqslant 2}{\mathrm{Min}}{\frac{9-{x}^{2}}{5-x}}=\beta$. If ${\int }_{\beta -\frac{8}{3}}^{2\alpha -1}Max{\frac{9-{x}^{2}}{5-x},x}dx={\alpha }_{1}+{\alpha }_{2}{\mathrm{log}}_{e}(\frac{8}{15})$, then ${\alpha }_{1}+{\alpha }_{2}$ is equal to ______

Let the slope of the tangent to a curve $y=f(x)$ at $(x,y)$ be given by $2\mathrm{tan}x(\mathrm{cos}x-y)$. if the curve passes through the point $(\frac{\pi }{4},0)$, then the value of ${\int }_{0}^{\frac{\pi }{2}}ydx$ is equal to

The slope of the tangent to a curve $C:y=y(x)$ at any point $[x,y)$ on it is $\frac{2{e}^{2x}-6{e}^{-x}+9}{2+9{e}^{-2x}}$. If $C$ passes through the points $(0,\frac{1}{2}+\frac{\pi }{2\sqrt{2}})$ and $(\alpha ,\frac{1}{2}{e}^{2\alpha })$ then ${e}^{\alpha }$ is equal to

The area of the region enclosed between the parabolas ${y}^{2}=2x-1$ and ${y}^{2}=4x-3$ is.

If $f(\alpha )={\int }_{1}^{\alpha }\frac{{\mathrm{log}}_{10}t}{1+t}dt,\alpha >0$, then $f({e}^{3})+f({e}^{-3})$ is equal to

Let the solution curve $y=y(x)$ of the differential equation $(1+{e}^{2x})(\frac{dy}{dx}+y)=1$ pass through the point $(0,\frac{\pi }{2})$. Then, $\underset{x\rightarrow \infty }{\mathrm{lim}}{e}^{x}y(x)$ is equal to

The number of points, where the function $f:R\rightarrow R,f(x)=|x-1|\mathrm{cos}|x-2|\mathrm{sin}|x-1|+(x-3)|{x}^{2}-5x+4|$, is NOT differentiable, is

Let $f:R\rightarrow R$ be a function defined by : $f(x)={\begin{matrix}\underset{t\leq x}{\mathrm{max}}{{t}^{3}-3t}; & x\leq 2 \\ {x}^{2}+2x-6; & 2<x<3 \\ [x-3]+9; & 3\leq x\leq 5 \\ 2x+1; & x>5\end{matrix}$ Where $[t]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I={\int }_{-2}^{2}f(x)dx$. Then the ordered pair $(m,I)$ is equal to

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined as $f(x)=[\begin{matrix}[{e}^{x}], & x<0 \\ a{e}^{x}+[x-1], & 0\leq x<1 \\ b+[\mathrm{sin}(\pi x)], & 1\leq x<2 \\ [{e}^{-x}]-c, & x\geq 2\end{matrix}$ where $a,b,c\in \mathbb{R}$ and $[t]$ denotes greatest integer less than or equal to $t$. Then, which of the following statements is true?

If $y={\mathrm{tan}}^{-1}(\mathrm{sec}{x}^{3}-\mathrm{tan}{x}^{3}),\frac{\pi }{2}<{x}^{3}<\frac{3\pi }{2}$, then

Let the function $f(x)=2{x}^{2}-{\mathrm{log}}_{e}x,x>0$, be decreasing in $(0,a)$ and increasing in $(a,4)$. A tangent to the parabola ${y}^{2}=4ax$ at a point $P$ on it passes through the point $(8a,8a-1)$ but does not pass through the point $(-\frac{1}{a},0)$. If the equation of the normal at $P$ is $\frac{x}{\alpha }+\frac{y}{\beta }=1$, then $\alpha +\beta$ is equal to

Let $f:R\rightarrow R$ be a function defined by $f(x)={(x-3)}^{{n}_{1}}{(x-5)}^{{n}_{2}},{n}_{1},{n}_{2}\in N$. The, which of the following is NOT true?

The curve $y(x)=a{x}^{3}+b{x}^{2}+cx+5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point $\mathrm{Q}$, where ${y}^{'}$ is equal to $3$. Then the local maximum value of $y(x)$ is

If the sum of all the roots of the equation ${e}^{2x}-11{e}^{x}-45{e}^{-x}+\frac{81}{2}=0$ is ${\mathrm{log}}_{e}P$, then $P$ is equal to _____.

Let $f:R\rightarrow R$ and $g:R\rightarrow R$ be two functions defined by $f(x)={\mathrm{log}}_{e}({x}^{2}+1)-{e}^{-x}+1$ and $g(x)=\frac{1-2{e}^{2x}}{{e}^{x}}\cdot$ Then, for which of the following range of $\alpha$, the inequality $f(g(\frac{{(\alpha -1)}^{2}}{3}))>f(g(\alpha -\frac{5}{3}))$ holds?

For the function $f(x)=4{\mathrm{log}}_{e}(x-1)-2{x}^{2}+4x+5,x>1$, which one of the following is NOT correct?