Real value of θ for which expression frac{{1 + i,cos θ }}{{1 - 2icos θ }} is a real number

English

Gujarati

Hindi

4-1.Complex numbers

normal

The real value of $\theta$ for which the expression $\frac{{1 + i\,\cos \theta }}{{1 - 2i\cos \theta }}$ is a real number is $\left( {n \in I} \right)$

A

$\left( {2n + 1} \right)\pi $

B

$\left( {2n + 1} \right)\pi /2$

C

$2n\,\,\pi $

D

None of these

Solution

$\frac{1+i \cos \theta}{1-2 i \cos \theta}=\frac{(1+i \cos \theta)(1+2 i \cos \theta)}{(1-2 i \cos \theta)(1+2 i \cos \theta)}$

$=\frac{1-2 \cos ^{2} \theta+3 i \cos \theta}{1+4 \cos ^{2} \theta}$

is a real number only if $\frac{3 \cos \theta}{1+4 \cos ^{2} \theta}=0$

i.e. if $\cos \theta=0$

i.e. if $\theta=(2 n+1) \pi / 2, n \in I$

So $(b)$ is correct alternative.

Standard 11

Mathematics

Share

Similar Questions

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2+3 z+4 z^2}{2-3 z+4 z^2}$ is a real number, then the value of $|z|^2$ is. . . . .

hard

(IIT-2022)

Let $z$ =${i^{2i}}$ , then $|z|$ is (where $i$ =$\sqrt { – 1}$ )

normal

Find the modulus and argument of the complex number $\frac{1+2 i}{1-3 i}$

medium

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are $TRUE$?

$(A)$ $\left|z+\frac{1}{2}\right| \leq \frac{1}{2}$ for all $z \in S$ $(B)$ $|z| \leq 2$ for all $z \in S$

$(C)$ $\left|z+\frac{1}{2}\right| \geq \frac{1}{2}$ for all $z \in S$ $(D)$ The set $S$ has exactly four elements

normal

(IIT-2020)

If ${z_1},{z_2},{z_3}$be three non-zero complex number, such that ${z_2} \ne {z_1},a = |{z_1}|,b = |{z_2}|$ and $c = |{z_3}|$ suppose that $\left| {\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}} \right| = 0$, then $arg\left( {\frac{{{z_3}}}{{{z_2}}}} \right)$ is equal to

hard