Find the modulus and the arguments of complex number z= - 1 - i\(\sqrt { 3 } \)

5

(a)

Find the modulus and the arguments of complex number \(z=-\sqrt { 3 } +i\)

5

(a)

Convert the complex numbers in polar form \(\sqrt { 3 } +i\)

5

(a)

If \(x+iy=\frac{(a+i)^2}{2a-i}\) show that \(x^2+y^2=\frac{(a^2+1^2)}{4a^2+1}.\)

5

(a)

Convert the following in the polar form \(\frac { 1+3i }{ 1-2i } \)

5

(a)

If a+i=\(\frac { ({ x }^{ 2 }+i)^{ 2 } }{ { 2x }^{ 2 }+1 } \) ,prove that a²+b² =\(\frac { ({ x }^{ 2 }+i)^{ 2 } }{ ({ 2x }^{ 2 }+1)^{ 2 } } \)

5

(a)

Let z₁= 2-i ,z₂=-2+i Find
(i)  \(Re\left( \frac { { z }_{ 1 }{ z }_{ 2 } }{ \bar { { Z }_{ 1 } } } \right) \) (ii) \(Im\left( \frac { 1 }{ { z }_{ 1 }\bar { { z }_{ 1 } } } \right) \)

5

(a)

Find the square root of \(-2+2\sqrt {3}i.\)

5

(a)

If α and β are different complex numbers with \(\left| \beta \right| \)=1then find \(\left| \frac { \beta -\alpha }{ 1-\bar { \alpha } \beta } \right| \).

5

(a)

If a+ib =\(\frac { x+i }{ x-i } \) where x is real,prove that  a²+b²=1 and \(\frac { b }{ a } =\frac { 2x }{ { x }^{ 2 }-1 } \)

5

(a)