Question

The quadratic equation tan$\theta x^2+2(sec\theta +cos\theta )x+(tan\theta +3\sqrt{2}cot\theta )$ always has

• A. Equal roots for all θ
• B. Complex roots for all θ
• C. Real and distinct roots for all θ
• D. Real roots or complex roots depending upon θ

Answer 1

The correct option is B

Complex roots for all θ

Discriminant, $△=B^2-4ac$
$=\left(2\right(sec\theta +cos\theta ){)}^2-4tan\theta (tan\theta +3\sqrt{2}cot\theta )$
$=4(se{c}^2\theta +co{s}^2\theta +2)-4(ta{n}^2\theta +3\sqrt{2})]$
$=4\left[\right(se{c}^2\theta +co{s}^2\theta +2-(ta{n}^2\theta +3\sqrt{2})]$
$=4\left[\right(se{c}^2\theta +ta{n}^2+2+co{s}^2\theta +3\sqrt{2}\left)\right]$
$=4[1+2+co{s}^2\theta -3\sqrt{2})$
$=4[3+co{s}^2\theta -3\sqrt{2}]$
$△=4[co{s}^2+3-3\sqrt{2}]$
$co{s}^2\theta \le 1$
$△\le 4[1+3-3\sqrt{2}]$
$=4(4-3\sqrt{2})$
$\simeq 4(4-3$x$1.4$
$=4(-0.2)$
$\Rightarrow △is-ve$
$\Rightarrow$No real roots