The value of {{{e^{ - {\pi \over 4}}} + \int\limits_0^{{\pi \over 4}} {{e^{ - x}}{{\tan }^{50}}xdx} } \over {\int\limits_0^{{\pi \over 4}} {{e^{ - x}}({{\tan }^{49}}x + {{\tan }^{51}}x)dx} }} is

Among (S1): \lim_\limits{n \rightarrow \infty} \frac{1}{n^{2}}(2+4+6+\ldots \ldots+2 n)=1 (S2) : \lim_\limits{n \rightarrow \infty} \frac{1}{n^{16}}\left(1^{15}+2^{15}+3^{15}+\ldots \ldots+n^{15}\right)=\frac{1}{16}

If $$f:\mathbb{R}\to \mathbb{R}$$ be a continuous function satisfying \int_\limits{0}^{\frac{\pi}{2}} f(\sin 2 x) \sin x d x+\alpha \int_\limits{0}^{\frac{\pi}{4}} f(\cos 2 x) \cos x d x=0, then the value of $$\alpha$$ is :

Let the function $$f:[0,2]\to \mathbb{R}$$ be defined as $$f(x)=\{\begin{array}{ll}{e}^{\min \left\{x^2,x-[x]\right\},}& x\in [0,1)\\ e^{\left[x-{\log }_{e}x\right]},& x\in [1,2]\end{array}$$ where $$[t]$$ denotes the greatest integer less than or equal to $$t$$. Then the value of the integral \int_\limits{0}^{2} x f(x) d x is :

The value of the integral \int_\limits{-\log _{e} 2}^{\log _{e} 2} e^{x}\left(\log _{e}\left(e^{x}+\sqrt{1+e^{2 x}}\right)\right) d x is equal to :

Let $$f$$ be a continuous function satisfying \int_\limits{0}^{t^{2}}\left(f(x)+x^{2}\right) d x=\frac{4}{3} t^{3}, \forall t > 0. Then $$f\left(\frac{{\pi }^2}{4}\right)$$ is equal to :

Let $$5f(x)+4f\left(\frac{1}{x}\right)=\frac{1}{x}+3,x>0$$. Then 18 \int_\limits{1}^{2} f(x) d x is equal to :

Let $$f(x)$$ be a function satisfying $$f(x)+f(\pi -x)={\pi }^2,\forall x\in \mathbb{R}$$. Then \int_\limits{0}^{\pi} f(x) \sin x d x is equal to :

\lim _\limits{n \rightarrow \infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right) \ldots . .\left(2^{\frac{1}{2}}-2^{\frac{1}{2 n+1}}\right)\right\} is equal to :

Let $$a,b$$ be two real numbers such that $$ab

The complex number $z=\frac{i-1}{\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}}$ is equal to :

If the center and radius of the circle \left| {{{z - 2} \over {z - 3}}} \right| = 2 are respectively $$(\alpha ,\beta )$$ and $$\gamma$$, then $$3(\alpha +\beta +\gamma )$$ is equal to :

For all $$z\in C$$ on the curve $$C_1:|z|=4$$, let the locus of the point $$z+\frac{1}{z}$$ be the curve $${\mathrm{C}}_2$$. Then :

For two non-zero complex numbers $$z_1$$ and $$z_2$$, if $$\mathrm{Re}\left(z_1{z}_2\right)=0$$ and $$\mathrm{Re}\left(z_1+z_2\right)=0$$, then which of the following are possible? A. $$\mathrm{Im}\left(z_1\right)>0$$ and $$\mathrm{Im}\left(z_2\right)>0$$ B. $$\mathrm{Im}\left(z_1\right)0$$ C. $$\mathrm{Im}\left(z_1\right)>0$$ and $$\mathrm{Im}\left(z_2\right)D.$$\operatorname{Im}\left(z_{1}\right) Choose the correct answer from the options given below :

Let $$z$$ be a complex number such that \left| {{{z - 2i} \over {z + i}}} \right| = 2,z \ne - i. Then $$z$$ lies on the circle of radius 2 and centre :

Let $${\mathrm{z}}_1=2+3i$$ and $${\mathrm{z}}_2=3+4i$$. The set $$\mathrm{S}=\left\{\mathrm{z}\in \mathbb{C}:{\left|\mathrm{z}-{\mathrm{z}}_1\right|}^2-{\left|\mathrm{z}-{\mathrm{z}}_2\right|}^2={\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}^2\right\}$$ represents a

The value of {\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3} is

Let $$\mathrm{p},\mathrm{q}\in \mathbb{R}$$ and $${\left(1-\sqrt{3}i\right)}^{200}=2^{199}(p+iq),i=\sqrt{-1}$$ then $$\mathrm{p}+\mathrm{q}+{\mathrm{q}}^2$$ and $$\mathrm{p}-\mathrm{q}+{\mathrm{q}}^2$$ are roots of the equation.

If the set $\left\{\mathrm{Re}\left(\frac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}\right):z\in \mathbb{C},\mathrm{Re}(z)=3\right\}$ is equal to the interval $(\alpha ,\beta ]$, then $24(\beta -\alpha )$ is equal to :