Let $$f(x)={\int }_0^x\left(t+\sin \left(1-e^t\right)\right)dt,x\in \mathbb{R}$$. Then, \lim _\limits{x \rightarrow 0} \frac{f(x)}{x^3} is equal to

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

Let \int_\limits\alpha^{\log _e 4} \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=\frac{\pi}{6}. Then $${\mathrm{e}}^{\alpha }$$ and $${\mathrm{e}}^{-\alpha }$$ are the roots of the equation :

The value of $$k\in \mathbb{N}$$ for which the integral $$I_n={\int }_0^1{\left(1-x^k\right)}^{n}dx,n\in \mathbb{N}$$, satisfies $$147I_{20}=148I_{21}$$ is

The integral \int_\limits0^{\pi / 4} \frac{136 \sin x}{3 \sin x+5 \cos x} \mathrm{~d} x is equal to :

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

Let \beta(\mathrm{m}, \mathrm{n})=\int_\limits0^1 x^{\mathrm{m}-1}(1-x)^{\mathrm{n}-1} \mathrm{~d} x, \mathrm{~m}, \mathrm{n}>0. If \int_\limits0^1\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 $z$ is a complex number such that $|z|⩽1$, then the minimum value of $\left|z+\frac{1}{2}(3+4i)\right|$ is :

Let $\mathrm{S}=|\mathrm{z}\in \mathrm{C}:|z-1\mid =1$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2\sqrt{2}\mid$. Let $z_1,z_2\in \mathrm{S}$ be such that $\left|z_1\right|=\underset{z\in s}{\max }|z|$ and $\left|z_2\right|=\underset{z\in S}{\min }|z|$. Then ${\left|\sqrt{2}{z}_1-z_2\right|}^2$ equals :

If $S=\{z\in C:|z-i|=|z+i|=|z-1|\}$, then, $n(S)$ is :

Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$z_1+z_2=5$$ and $$z_1^3+z_2^3=20+15i$$ Then, $$\left|z_1^4+z_2^4\right|$$ equals -

If $$z=\frac{1}{2}-2i$$ is such that $$|z+1|=\alpha z+\beta (1+i),i=\sqrt{-1}$$ and $$\alpha ,\beta \in \mathbb{R}$$, then $$\alpha +\beta$$ is equal to

Let $$\mathrm{r}$$ and $$\theta$$ respectively be the modulus and amplitude of the complex number $$z=2-i\left(2\tan \frac{5\pi }{8}\right)$$, then $$(\mathrm{r},\theta )$$ is equal to

If $$z$$ is a complex number, then the number of common roots of the equations $$z^{1985}+z^{100}+1=0$$ and $$z^3+2z^2+2z+1=0$$, is equal to

If $$z=x+iy,xy\ne 0$$, satisfies the equation $$z^2+i\bar{z}=0$$, then $$\left|z^2\right|$$ is equal to :

Let $$z$$ be a complex number such that the real part of $$\frac{z-2i}{z+2i}$$ is zero. Then, the maximum value of $$|z-(6+8i)|$$ is equal to

Let $$\alpha$$ and $$\beta$$ be the sum and the product of all the non-zero solutions of the equation $$(\bar{z}{)}^2+|z|=0,z\in C$$. Then $$4({\alpha }^2+{\beta }^2)$$ is equal to :

The area (in sq. units) of the region $$S=\{z\in \mathbb{C}:|z-1|\le 2;(z+\bar{z})+i(z-\bar{z})\le 2,\mathrm{lm}(z)\ge 0\}$$ is

The sum of all possible values of $$\theta \in [-\pi ,2\pi ]$$, for which $$\frac{1+i\cos \theta }{1-2i\cos \theta }$$ is purely imaginary, is equal to :