Find the locus of z if |3z - 5| = 3 |z + 1| where z = x + iy.

Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.

If \(\frac { (a+i)^{ 2 } }{ 2a-i } \) = p + iq, show that p²+q² = \(\frac { ({ a }^{ 2 }+i)^{ 2 } }{ 4a^{ 2 }+1 } \).

Find the value of \(\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1\)

Show that \(i^{n+100}+i^{n+50}+i^{n+48}+i^{n+46}=0, \forall n \in N\)

Prove that \((1+i)^{4}\left(1+\frac{1}{i}\right)^{4}=16\)

Evaluate : \(\left[i^{17}-\left(\frac{1}{i}\right)^{34}\right]^{2}\)

Find the real values of x and y if x + 4iy = ix + y + 3

Find the values of x and y if \(\left(\frac{3}{\sqrt{5}} x-5\right)+i 2 \sqrt{5} y=\sqrt{2}\)

If \((x+i y)^{\frac{1}{3}}=\mathrm{a}+\mathrm{i b}\), prove that \(\frac{x}{a}+\frac{y}{b}=4\left(a^{2}-b^{2}\right)\)

Find the Additive and Multiplicative inverse of 1 - i.

If \(x+i y=\frac{a+i}{a-i}\), prove that ay -1 = x.

Given z = 3 + 5i, Find the value of \(z^{3}+\bar z+198\)

If \(\frac{(a+i)^{2}}{2 a-i}=p+i q\), show that \(p^{2}+q^{2}=\frac{\left(a^{2}+1\right)^{2}}{4 a^{2}+1}\)

(i) If (a+ ib)² = x + iy, prove that \(x^{2}+y^{2}=\left(a^{2}+b^{2}\right)^{2}\)
(ii) If (cos \(\theta\) - isin \(\theta\))² = x - iy, prove that \(x^{2}+y^{2}=1\)