Calculus PYQ — Page 12

If the value of the integral $\int_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x$ is $\frac{2}{\pi}$. Then, a value of $\alpha$ is

If ${\int }_{0}^{1}\frac{1}{\sqrt{3+x}+\sqrt{1+x}}dx=a+b\sqrt{2}+c\sqrt{3}$, where $a,b,c$ are rational numbers, then $2a+3b-4c$ is equal to :

If $5f(x)+4f(\frac{1}{x})={x}^{2}-2,\forall x\neq 0$ and $y=9{x}^{2}f(x),$ then $y$ is strictly increasing in:

If $\int_0^{\frac{\pi}{4}} \frac{\sin ^2 x}{1+\sin x \cos x} \mathrm{~d} x=\frac{1}{\mathrm{a}} \log _{\mathrm{e}}\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi}{\mathrm{b} \sqrt{3}}$, where $\mathrm{a}, \mathrm{b} \in \mathbf{N}$, then $\mathrm{a}+\mathrm{b}$ is equal to_________

Let $g(x)$ be a linear function and $f(x)={\begin{matrix}g(x), & x\leq 0 \\ {(\frac{1+x}{2+x})}^{\frac{1}{x}}, & x>0\end{matrix}$, is continuous at $x=0$. If ${f}^{'}(1)=f(-1)$, then the value of $g(3)$ is

Let $[\mathrm{t}]$ denote the greatest integer less than or equal to $\mathrm{t}$. Let $f:[0, \infty) \rightarrow \mathbf{R}$ be a function defined by $f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]$. Let $S$ be the set of all points in the interval $[0,8]$ at which $f$ is not continuous. Then $\sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to _______

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function given by $f(x)=\left\{\begin{array}{ll} \frac{1-\cos 2 x}{x^2}, & x < 0 \\ \alpha, & x=0, \\ \frac{\beta \sqrt{1-\cos x}}{x}, & x>0 \end{array}\right.$ where $\alpha, \beta \in \mathbf{R}$. If $f$ is continuous at $x=0$, then $\alpha^2+\beta^2$ is equal to :

Let $f(x)={x}^{3}+{x}^{2}{f}^{'}(1)+x{f}^{"}(2)+{f}^{'''}(3),x\in R$. Then ${f}^{'}(10)$ is equal to

Suppose $f(x)=\frac{({2}^{x}+{2}^{-x})\mathrm{tan}x\sqrt{{\mathrm{tan}}^{-1}({x}^{2}-x+1)}}{{(7{x}^{2}+3x+1)}^{3}}$. Then the value of ${f}^{'}(0)$ is equal to

Let $f(x)=x^5+2 \mathrm{e}^{x / 4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8 g^{\prime}(2)$ is :

Let for a differentiable function $f:(0,\infty )\rightarrow R$, $f(x)-f(y)\geq {\mathrm{log}}_{e}(\frac{x}{y})+x-y,\forall x,y\in (0,\infty )$. Then $\sum _{n=1}^{20}{f}^{'}(\frac{1}{{n}^{2}})$ is equal to

Let the set of all values of $p$, for which $f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7$ does not have any critical point, be the interval $(a, b)$. Then $16 a b$ is equal to _______

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 :

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 _________

For the function $f(x)=(\cos x)-x+1, x \in \mathbb{R}$, between the following two statements (S1) $f(x)=0$ for only one value of $x$ in $[0, \pi]$. (S2) $f(x)$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.

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 ________

Consider the function $f:(0,2)\rightarrow R$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by $g(x)={\begin{matrix}\text{min}{f(t)}, & 0<t\leq x\text{ and }0<x\leq 1 \\ \frac{3}{2}+x, & 1<x<2\end{matrix}$. Then

Let $I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x$. If $I(0)=3$, then $I\left(\frac{\pi}{12}\right)$ is equal to

If $\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} \mathrm{~d} x=\mathrm{A}\left(\frac{\alpha x-1}{\beta x+3}\right)^B+\mathrm{C}$, where $\mathrm{C}$ is the constant of integration, then the value of $\alpha+\beta+20 \mathrm{AB}$ is__________

Let $\int \frac{2-\tan x}{3+\tan x} \mathrm{~d} x=\frac{1}{2}\left(\alpha x+\log _{\mathrm{e}}|\beta \sin x+\gamma \cos x|\right)+C$, where $C$ is the constant of integration. Then $\alpha+\frac{\gamma}{\beta}$ is equal to :

Let $r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in \mathbb{N}$. Then the value of $\sum_{k=1}^{10} \frac{1}{7\left(r_k-1\right)}$ is equal to________

Let $[t]$ denote the largest integer less than or equal to $t$. If $\int_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) \mathrm{d} x=\mathrm{a}+\mathrm{b} \sqrt{2}-\sqrt{3}-\sqrt{5}+\mathrm{c} \sqrt{6}-\sqrt{7}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbf{Z}$, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to_______

Let $f:(0,\infty )\rightarrow R$ and $F(x)=\int _{0}^{x}tf(t)dt$. If $F({x}^{2})={x}^{4}+{x}^{5},$ then $\sum _{r=1}^{12}f({r}^{2})$ is equal to:

Let $S=(-1,\infty )$ and $f:S\rightarrow \mathbb{R}$ be defined as $f(x)=\int _{-1}^{x}{({e}^{t}-1)}^{11}{(2t-1)}^{5}{(t-2)}^{7}{(t-3)}^{12}{(2t-10)}^{61}dt$. Let $p=$ Sum of square of the values of $x$, where $f(x)$ attains local maxima on $S$. and $q=$Sum of the values of $x$, where $f(x)$ attains local minima on $S$. Then, the value of ${p}^{2}+2q$ is ________