Question

If $x^2+px+1$ is a factor of 2 $co{s}^2\theta x^3+2x+sin2\theta$, then

• A. $\theta =n\pi ,n\epsilon I$
• B. $\theta =n\pi +\frac{\pi }{2},n\epsilon I$
• C. $\theta =2n\pi ,n\epsilon I$
• D. $\theta =\frac{n\pi }{2},n\epsilon I$

Answer 1

The correct option is A $\theta =n\pi ,n\epsilon I$

$2co{s}^2\theta x^3+2x+sin2\theta =(x^2+px+1)(2co{s}^2\theta x+sin2\theta )$
On comparing coefficients of x and $x^2,$ then
$2=psin2\theta +2co{s}^2\theta \Rightarrow 2si{n}^2\theta =psin2\theta orp=tan\theta ...\left(i\right)and0=sin2\theta +2pco{s}^2\theta \therefore p=-tan\theta ...\left(ii\right)$
From Eqs. (i) and (ii),
$tan\theta =-tan\theta 2tan\theta =0tan\theta =0\theta =n\pi ,n\epsilon I.$