Q.1 (b) If \(z=\frac1{\sqrt2}+\frac i{\sqrt2}\) and \(\bar z\) is the conjugate of z, then prove that \((z)^{10}+(\bar z)^{10}=0\).
Best answer
\(z=\cos\frac\pi4+i\sin\frac\pi4,\quad\bar z=\cos\frac\pi4-i\sin\frac\pi4\).
By De Moivre: \(z^{10}=\cos\frac{10\pi}4+i\sin\frac{10\pi}4\), \(\bar z^{10}=\cos\frac{10\pi}4-i\sin\frac{10\pi}4\).
\(z^{10}+\bar z^{10}=2\cos\frac{5\pi}2=2(0)\)
\(\therefore z^{10}+\bar z^{10}=0\) (proved). (Indeed \(z^{10}=i,\ \bar z^{10}=-i\).)
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