Question

Consider the equation $a{z}^2+z+1=0$ having purely imaginary root where $a=\cos \theta +i\sin \theta ,i=\sqrt{-1}$ and function $f\left(x\right)=x^3-3x^2+3(1+\cos \theta )x+5,$ then answer the following questions.

Number of roots of the equation $\cos 2\theta =\cos \theta ,\theta \in [0,4\pi ]$ are

• A. $2$
• B. $3$
• C. $4$
• D. $6$

Answer 1

The correct option is C $4$

We have, $\cos 2\theta =\cos \theta$
$\Rightarrow 2{\cos }^2\theta -1=\cos \theta$
$\Rightarrow 2{\cos }^2\theta -\cos \theta -1=0$
$\Rightarrow (\cos \theta -1)(2\cos \theta +1)=0$
$\Rightarrow \cos \theta =1,-\frac{1}{2}$
Hence the solutions in the given interval are,
$\theta =0,2\pi ,\frac{2\pi }{3},\frac{4\pi }{3}\Rightarrow$ There are 4 solutions