Number Theory T...

If $$\mathrm{{z} _ { 1 }} = 10 + 6\mathrm{i} ,  \mathrm{{ z } _ { 2 }}= 4 + 6 \mathrm { i }$$ and $$\mathrm{ z}$$ is a complex number such that $$\operatorname { amp } \left( \dfrac { \mathrm { z } - \mathrm { z } _ { 1 } } { \mathrm { z } - \mathrm { z } _ { 2 } } \right) = \dfrac { \pi } { 4 }$$ , then the value of $$\left| \mathrm{z} - 7 - 9 \mathrm { i } \right|$$ is equal to

The modulus of the complex number $$z$$ such that $$\left| z + 3 - i\right | = 1$$ and $$\arg{z} = \pi$$ is equal to

The roots of the equation $$(3b+c-4a)x^2+(3c+a-4b)x+(3a+b-4c)= 0$$ are 

$$\frac { { z }_{ 2 }-{ 2z }_{ 2 } }{ { z }_{ 2 }-{ z }_{ 1 }{ z }_{ 2 } } $$ is unimodular then

If z is a complex number of unit modules and argument $$\theta $$, then the real part of $$\dfrac { z(1-\bar { z } ) }{ z(1+z) } $$ is :

This equation $$(x-5)^{11}+(x-5^{2})^{11}+....+(x-5^{11})^{11}=0$$ has 

If z= $$\dfrac { 1 }{ { \left( 2+3i \right)  }^{ 2 } } ,\quad then\left| z \right| $$=

The complex numbers $${ z }_{ 1 },{ z }_{ 2 },{ z }_{ 2 }$$ satisfying $$\dfrac { { z }_{ 1 }+{ z }_{ 3 } }{ { z }_{ 2 }-{ z }_{ 3 } } =\dfrac { 1-i\sqrt { 3 }  }{ 2 } $$

If $$z$$ is a complex number such that $$| z | = 1 , z \neq 1 ,$$ then the real part of $$\frac { z - 1 } { z + 1 }$$ is

If z be any complex number such that $$|3z-2|+|3z+2|=4$$, then locus of z is