If a + ib = \(\frac { ({ x }^{ 2 }+1) }{ 2{ x }^{ 2 }+1 } \) , prove that \({ a }^{ 2 }+{ b }^{ 2 }=\frac { ({ x }^{ 2 }+1)^{ 2 } }{ (2{ x }+1)^{ 2 } } \)

4

(a)

Convert the complex numbers in polar form -1 + i.

4

(a)

Convert the complex numbers in polar form -3.

4

(a)

If \(|z+1|=z+2(1+i)\) then find z.

4

(a)

If z = x + iy, w = \(\frac { 1-iz }{ z-i } \) and \(|w|=1\) then show that z is purely real.

4

(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)

Show that a real value of x will satisfy the equation \(\frac{1-ix}{1+ix}=a-ib\) if a + b² = 1where a and b are real.

5

(a)

Find the square root of -5+12i.

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)

Find the square root of -8i.

5

(a)

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

5

(a)