If the real part of the complex number $z=\frac{3+2 i \cos \theta}{1-3 i \cos \theta}, \theta \in\left(0, \frac{\pi}{2}\right)$ is zero, then the value of $\sin ^{2} 3 \theta+\cos ^{2} \theta$ is equal to ____

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Complex Numbers

If the real part of the complex number

z = \frac{3 + 2 i c o s θ}{1 - 3 i c o s θ}, θ \in (0, \frac{π}{2})

is zero, then the value of

sin^{2} 3 θ + cos^{2} θ

is equal to ____

Answer: 1

Multiply numerator and denominator by the conjugate of the denominator:

z = \frac{( 3 + 2 i c o s θ ) ( 1 + 3 i c o s θ )}{1 + 9 c o s ^{2} θ} = \frac{3 + 9 i c o s θ + 2 i c o s θ - 6 c o s ^{2} θ}{1 + 9 c o s ^{2} θ} = \frac{3 - 6 c o s ^{2} θ + 11 i c o s θ}{1 + 9 c o s ^{2} θ}

. Real part is zero:

3 - 6 cos^{2} θ = 0 \Rightarrow cos^{2} θ = 1/2 \Rightarrow cos θ = 1/ 2

(since

θ \in (0, π /2)

). Thus

θ = π /4

. Then

sin^{2} 3 θ + cos^{2} θ = sin^{2} (3 π /4) + (1/ 2)^{2} = (1/ 2)^{2} + 1/2 = 1/2 + 1/2 = 1

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Integer answer