Question : If $\left(z+\frac{1}{z}\right)=4$, then what will be the value of $\frac{1}{2}\left(z^2+\frac{1}{z^2}\right)$?
Option 1: 14
Option 2: 16
Option 3: 7
Option 4: 8
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Correct Answer: 7
Solution :
$(z+\frac{1}{z})=4$
Squaring both sides, we get:
$(z+\frac{1}{z})^2=4^2$
⇒ $z^2+\frac{1}{z^2}+2×z×\frac{1}{z}=16$
⇒ $z^2+\frac{1}{z^2}=16-2$
⇒ $z^2+\frac{1}{z^2}=14$
Thus, $\frac{1}{2}(z^2+\frac{1}{z^2})$
$=\frac{1}{2}×14 = 7$
Hence, the correct answer is 7.
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