Question

If the equation $(cose{c}^2\theta -4)x^2+(cot\theta +\sqrt{3})x+co{s}^2\frac{3\pi }{2}=0$ holds true for all real x, then the general value of $\theta$ can given by $\left(nϵZ\right)$

• A. $2n\pi +\frac{11\pi }{6}$
• B. $2n\pi +\frac{5\pi }{6}$
• C. $2n\pi \pm \frac{3\pi }{6}$
• D. $n\pi \pm \frac{11\pi }{6}$

Answer 1

Answer: (A), (B)

The correct options are
A

$2n\pi +\frac{11\pi }{6}$

B

$2n\pi +\frac{5\pi }{6}$

Given that the quadratic equation is an identity
$\therefore cose{c}^2\theta =4$ and $cot\theta =-\sqrt{3}$
$\Rightarrow cosec\theta =2$ or $-2andtan\theta =\frac{-1}{\sqrt{3}}$
$\theta =\frac{5\pi }{6}$ and $\frac{11\pi }{6}$