Number Theory T...

Which of the following is a pair of twin-prime number ? 

If $$ \displaystyle z_{0}=\frac{1-i}{2}$$,  then $$ \displaystyle \left (1+z_{0}  \right )\left (1+z_{0}^{{2}^{1}}  \right )\left (1+z_{0}^{{2}^{2}}  \right ).......... \left (1+z_{0}^{2^n}  \right )$$  must be

The number of solutions of $$log _{\frac{1}{5}}log_{\frac{1}{2}}(\left | z \right |^{2}+4\left | z \right |+3)< 0$$ is/are?

Let $$z$$ be a complex number and $$c$$ be a real number $$\geq $$ 1 such that z + $$c\left | z+1 \right |+i=0 ,$$ then $$c$$ belongs to 

If $$x = 2 + 5i($$where $$1 i = \sqrt{-1})$$ and $$2(\displaystyle \frac{1}{1! 9!}  + \frac{1}{3! 7!}) + \frac{1}{5! 5!} = \frac{2^{a}}{b!}$$ then $$ x^{3}-5x^{2}+33x-10 = $$

lf $$\displaystyle \log_{\tan 30^{\circ}}\left(\frac{2|Z|^{2}+2|Z|-3}{|z|+1}\right) <-2$$ then

Given $$z$$ is a complex number with modulus $$1$$. Then the equation in $$a$$, $$\left(\dfrac{1+ia}{1-ia}\right )^4=z$$ has

Which of the following is not a composite

number?

Let $$x_1, x_2,$$ are the roots of quadratic equation $$x^2 + ax + b = 0$$, Where $$ a, b$$ are complex numbers and $$y_1, y_2$$ are the roots of the quadratic equation $$y^2 + \left|a\right| y + \left|b\right| =0$$. If $$\left|x_1\right| = \left|x_2\right| = 1$$, then 

Dividing f(z) by $$z-i$$, we obtain the remainder $$i$$ and dividing it by $$z+i$$, we get the remainder $$1+i$$, then remainder upon the division of f(z) by $$z^2+1$$ is