If (1 + i)z = (1 - i)\(\overline { z }\) , then show that z = \(-i\overline { z } \).

2

(a)

We have, (1 + i)z = (1 - i)\(\overline { z } \)
\(\Rightarrow \frac { z }{ \overline { z } } =\frac { 1-i }{ 1+i } x \frac { 1-i }{ 1-i } =\frac { 1+1-2i }{ 1+1 }\)

Find the conjugate of the complex number \(\frac { 1-i }{ 1+i }\)

2

(a)

Evaluate \({ \left( 1+i \right) }^{ 6 }+{ \left( 1-i \right) }^{ 3 }\) . 

2

(a)

If \({ \left( x+iy \right) }^{ 1/3 }\) = a + ib, where x, y, a, b \(\epsilon \) R, then show that\(\frac { x }{ a } -\frac { y }{ b } =-2\left( { a }^{ 2 }+{ b }^{ 2 } \right) \).Firstly, use identity \({ \left( a+b \right) }^{ 3 }={ a }^{ 3 }+{ 3a }^{ 2 }b+{ 3ab }^{ 2 }+{ b }^{ 3 }\)and then equate the coefficients of real and imaginary parts.

2

(a)

Find the real value of \(\theta \) for which the expression \(\frac { 1+icos\theta }{ 1-2icos\theta } \) is a real number.

2

(a)

Evaluate \(\frac { { \left( 1-i \right) }^{ 3 } }{ 1-{ i }^{ 3 } } \)

2

(a)