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Question: If \(\left( \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} \right)\)then \(n\)...

If (cos⁡π2+isin⁡π2)\left( \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} \right)then nn

A

1

B

0

C

(1+cos⁡θ+isin⁡θi+sin⁡θ+icos⁡θ)4=cos⁡nθ+isin⁡nθ\left( \frac{1 + \cos\theta + i\sin\theta}{i + \sin\theta + i\cos\theta} \right)^{4} = \cos n\theta + i\sin n\theta

D

None of these

Answer

0

Explanation

Solution

If zz ......(i)

We know that if two complex numbers are equal, their moduli must also be equal, therefore from (i), we have

∣z2∣=∣z∣2∣z2∣=∣zˉ∣2z=zˉ|z^{2}| = |z|^{2}|z^{2}| = |\bar{z}|^{2}z = \bar{z}, zˉ2=z2‾{\bar{z}}^{2} = \overline{z^{2}}

⇒∣z∣|z|⇒ ∣z+2z∣=2\left| z + \frac{2}{z} \right| = 2

⇒ 3−1\sqrt{3} - 1⇒ 3+1\sqrt{3} + 1⇒3\sqrt{3}

Trick : By inspection, 2+3\sqrt{2} + \sqrt{3}