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Complex Numbers and Quadratic Equations — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers and Quadratic Equations1 Mark
MCQ
Choose the correct answer.
The real value of $\theta$ for which the expression $\frac{1+\text{i}\cos\theta}{1-2\text{i}\cos\theta}$ is a real number is:
  • A
    $\text{n}\pi+\frac{\pi}{4}$
  • B
    $\text{n}\pi+(-1)^{\text{n}}\frac{\pi}{4}$
  • ✓
    $2\text{n}\pi\pm\frac{\pi}{2}$
  • D
    None of these.
✓

Answer

Correct option: C.
$2\text{n}\pi\pm\frac{\pi}{2}$
Let $\text{z}=\frac{1+\text{i}\cos\theta}{1-2\text{i}\cos\theta}=\frac{1+\text{i}\cos\theta}{1-2\text{i}\cos\theta}\times\frac{1+2\text{i}\cos\theta}{1+2\text{i}\cos\theta}$
$=\frac{1+2\text{i}\cos\theta+\text{i}\cos\theta+2\text{i}^2\cos^2\theta}{1-4\text{i}^2\cos^2\theta}$
$=\frac{1-3\text{i}\cos\theta-2\cos^2\theta}{1+4\cos^2\theta}$
$=\frac{1-2\cos^2\theta}{1+4\cos^2\theta}+\frac{3\cos\theta}{1+4\cos^2\theta}\text{i}$
If $z$ is real number, then
$\frac{3\cos\theta}{1+4\cos^2\theta}=0$
$\Rightarrow3\cos\theta=0$
$\Rightarrow\cos\theta=0$
$\therefore\ \theta=(2\text{n}+1)\frac{\pi}{2},\text{ n}\in\text{N}$

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