Question

If $-\pi < \theta < \pi ,$ the equation $\left(cos 3 \theta + 1\right)x^{2}+\left(2 cos ⁡ 2 \theta - 1\right)x+\left(1 - 2 cos ⁡ \theta \right)=0$ has more than two roots for

• A. no value of $\theta $
• B. one value of $\theta $
• C. two value of $\theta $
• D. all values of $\theta $

Answer 1

Answer: (A)

Given equation has more than two roots if it is an identity
$\Rightarrow cos 3 \theta +1=0;2cos ⁡ 2 \theta -1=0and1-2cos ⁡ \theta =0$
$\Rightarrow cos 3 \theta =-1\Rightarrow \theta =\pm\frac{\pi }{3}$ which does not satisfy $2cos 2 \theta -1=0$
Hence, no value possible