Number Theory T...

The modulus of $$(1 + i) (3 + 4i) =$$

If $$|\mathrm{z}-4|<|\mathrm{z}-2|$$ then

If $$z_1$$, $$z_2$$, $$z_3$$ are complex numbers such that $$\left| { z }_{ 1 } \right| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =\left| \dfrac { 1 }{ { z }_{ 1 } } +\dfrac { 1 }{ { z }_{ 2 } } +\dfrac { 1 }{ { z }_{ 3 } }  \right| =1$$, then $$|z_1+z_2+z_3|$$ is:

If z is a complex number satisfying $$|z^2+1|=4|z|$$, then the minimum value of $$|z|$$ is

lf $$(x+iy)(2+cos\theta+isin\theta)=3$$ then $$x^{2}+y^{2}-4x+3$$ is

If $$|{z}|-z=1+2i$$ then $$z=$$

If $$z=2-i\sqrt{3}$$ then $$z^{4}-4z^{2}+8z+35$$ is :

For any two non-zero complex numbers $$Z_1$$ and $$Z_2$$, the value of $$\left(\left|{ Z }_{ 1 }\right|+\left|{ Z }_{ 2 }\right|\right)\left|\dfrac{{ Z }_{ 1 }}{\left|{Z}_{1}\right|}+\dfrac{{Z}_{2}}{\left|{Z}_{2}\right|}\right|$$ is:

lf $$z_{1},\ z_{2}$$ are roots of equation $$z^{2}-az+a^{2}=0$$, then $$|\displaystyle \frac{z_{1}}{z_{2}}|=$$

I. lf $$\left |z-\displaystyle \frac{2}{z}\right |=2$$ then the greatest value of $$|z|$$ is $$\sqrt{3}+1$$
II. $$\left | z+1 \right |-\left |  z-1\right |=\frac{3}{2}$$ then the least value of $$\left | z \right |=\frac{3}{4}$$