If frac{3+2i sin theta}{1-2i sin theta} is a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $z = \frac{3+2i \sin 	heta}{1-2i \sin 	heta}$.To make $z$ a real number,the imaginary part must be zero.Multiply the numerator and denominat

Vedclass · Accepted Answer

$\pi$

Vedclass · Answer

(A) Let $z = \frac{3+2i \sin 	heta}{1-2i \sin 	heta}$.To make $z$ a real number,the imaginary part must be zero.Multiply the numerator and denominator by the conjugate of the denominator $(1+2i \sin 	heta)$:$z = \frac{(3+2i \sin 	heta)(1+2i \sin 	heta)}{(1-2i \sin 	heta)(1+2i \sin 	heta)}$$z = \frac{3 + 6i \sin 	heta + 2i \sin 	heta + 4i^2 \sin^2 	heta}{1 + 4 \sin^2 	heta}$Since $i^2 = -1$,we have:$z = \frac{(3 - 4 \sin^2 	heta) + i(8 \sin 	heta)}{1 + 4 \sin^2 	heta}$For $z$ to be real,the imaginary part must be $0$:$\frac{8 \sin 	heta}{1 + 4 \sin^2 	heta} = 0$This implies $\sin 	heta = 0$.Given $0 < 	heta < 2\pi$,the only solution is $	heta = \pi$.

If $\frac{3+2i \sin \theta}{1-2i \sin \theta}$ is a real number and $0 < \theta < 2\pi$,then $\theta$ is equal to

Similar Questions

If $\operatorname{cosec} \theta = \frac{p+q}{p-q}$,then $\cot \left(\frac{\pi}{4} + \frac{\theta}{2}\right)$ is equal to

The ratio of diameters of two wires of the same material is $n : 1$. The length of the wires is $4\, m$ each. On applying the same load,the increase in length of the thin wire will be

If the straight line $y=mx+c$ is parallel to the axis of the parabola $y^2=lx$ and intersects the parabola at $\left(\frac{c^2}{8}, c\right)$,then the length of the latus rectum is

$Fe$ is made passive by

During the depression in freezing point experiment,an equilibrium is established between the molecules of

Vedclass Test Series

Exam Paper Generator

Online Exam Module