Calculus PYQ — Page 5

If $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{(\mathrm{a}-1) x}+2 \cos \mathrm{~b} x+(\mathrm{c}-2) \mathrm{e}^{-x}}{x \cos x-\log _{\mathrm{e}}(1+x)}=2$, then $\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}$ is equal to :

If $\alpha = \displaystyle\int_0^{2\sqrt{3}} \log_2(x^2 + 4)\,dx + \displaystyle\int_2^4 \sqrt{2^x - 4}\,dx$, then $\alpha^2$ is equal to _______.

If the solution curve $y=f(x)$ of the differential equation $\left(x^{2}-4\right) y^{\prime}-2 x y+2 x\left(4-x^{2}\right)^{2}=0, x>2$, passes through the point $(3,15)$, then the local maximum value of $f$ is $\_\_\_\_$.

If the function $f(x)=\frac{e^{x}\left(e^{\tan x-x}-1\right)+\log _{e}(\sec x+\tan x)-x}{\tan x-x}$ is continuous at $x=0$, then the value of $f(0)$ is equal to

If the curve $y = f(x)$ passes through the point $(1, e)$ and satisfies the differential equation $dy = y(2 + \log_e x)\,dx$, $x > 0$, then $f(e)$ is equal to :

If the area of the region $\left\{(x, y): 1-2 x \leqslant y \leqslant 4-x^{2}, x \geqslant 0, y \geqslant 0\right\}$ is $\frac{\alpha}{\beta}, \alpha, \beta \in \mathbf{N}, \operatorname{gcd}(\alpha, \beta)=1$, then the value of $(\alpha+\beta)$ is :

If the area of the region bounded by $16x^2 - 9y^2 = 144$ and $8x - 3y = 24$ is A, then $3(A + 6 \log_e(3))$ is equal to _______.

If $f(x)$ satisfies the relation $f(x)=e^{x}+\int_{0}^{1}\left(y+x e^{x}\right) f(y) d y$, then $e+f(0)$ is equal to $\_\_\_\_$.

If $y=y(x)$ satisfies the differential equation $16(\sqrt{x+9 \sqrt{x}})(4+\sqrt{9+\sqrt{x}}) \cos y \mathrm{~d} y=(1+2 \sin y) \mathrm{d} x, x>0$ and $y(256)=\frac{\pi}{2}, y(49)=\alpha$, then $2 \sin \alpha$ is equal to :

If $f(x)=\left\{\begin{array}{cc}\frac{a|x|+x^{2}-2(\sin |x|)(\cos |x|)}{x} &, x \neq 0 \\ b &, x=0\end{array}\right.$ is continuous at $x=0$, then $a+b$ is equal to

For the function $f(x) = e^{\sin|x|} - |x|$, $x \in \mathbb{R}$, consider the following statements: Statement I: $f$ is differentiable for all $x \in \mathbb{R}$. Statement II: $f$ is increasing in $\left(-\pi, -\dfrac{\pi}{2}\right)$. In the light of the above statements, choose the correct answer from the options given below:

Consider the following three statements for the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=\left|\log _{\mathrm{e}} x\right|-|x-1|$ : (I) $f$ is differentiable at all $x>0$. (II) $f$ is increasing in $(0,1)$. (III) $f$ is decreasing in $(1, \infty)$. Then.

$\operatorname{IfI}(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n\gt0$, then $I(9,14)+I(10,13)$ is

The value of $\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}}{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}+e^{\left(\left(6-\log _e x\right)^2+1\right)^{-1}}}\right) d x$ is

The value of $\int_{-1}^1 \frac{(1+\sqrt{|x|-x}) e^x+(\sqrt{|x|-x}) e^{-x}}{e^x+e^{-x}} d x$ is equal to

The sum of all local minimum values of the function $$ f(x)=\left\{\begin{array}{lr} 1-2 x, & x \lt -1 \\ \frac{1}{3}(7+2|x|), & -1 \leq x \leq 2 \\ \frac{11}{18}(x-4)(x-5), & x\gt2 \end{array}\right. $$ is

The shortest distance between the curves $y^2=8 \mathrm{x}$ and $x^2+y^2+12 y+35=0$ is :

The number of points of discontinuity of the function $f(\mathrm{x})=\left[\frac{\mathrm{x}^2}{2}\right]-[\sqrt{\mathrm{x}}], \mathrm{x} \in[0,4]$, where $[\cdot]$ denotes the greatest integer function is ________

The integral $\int_{-1}^{\frac{3}{2}}\left(\left|\pi^2 x \sin (\pi x)\right|\right) d x$ is equal to:

The integral $\int_0^\pi \frac{8 x d x}{4 \cos ^2 x+\sin ^2 x}$ is equal to

The integral $\int_0^\pi \frac{(x+3) \sin x}{1+3 \cos ^2 x} d x$ is equal to :

The integral \(80 \int_0^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta\) is equal to :

The area of the region $\{(x, y):|x-y| \leq y \leq 4 \sqrt{x}\}$ is

The area of the region $\left\{(x, y): x^2+4 x+2 \leq y \leq|x+2|\right\}$ is equal to