Complex Numbers
Purely Imaginary Condition
nta_pyq_2024_apr
Grade 11
Question:
The sum of all possible values of $\theta\in[-\pi,2\pi]$, for which $\dfrac{1+i\cos\theta}{1-2i\cos\theta}$ is purely imaginary, is equal to:
$3\pi$
$2\pi$
$5\pi$
$4\pi$
Step-by-Step Solution
Key Concept: For purely imaginary, real part $=0$. Multiply numerator and denominator by conjugate of denominator; set real part $=0$. This gives $2-4\cos^2\theta=0\Rightarrow\cos^2\theta=1/2$.
$\theta=-\pi/4,-3\pi/4,\pi/4,3\pi/4,5\pi/4,7\pi/4$. Sum $=3\pi$.
Correct Answer: 1