Question

The quadratic equation $\left(1 - sin \theta \right)x^{2}+2\left(1 - sin ⁡ \theta \right)x-3sin ⁡ \theta =0$ has both roots complex for all $\theta $ lying in the interval

• A. $\left(- \frac{\pi }{2} , \frac{\pi }{2}\right)$
• B. $\left(0 , \frac{3 \pi }{2}\right)$
• C. $\left(\frac{\pi }{6} , \frac{7 \pi }{6}\right)$
• D. $\left(\frac{7 \pi }{6} , \frac{11 \pi }{6}\right)$

Answer 1

Answer: (D)

$D=4\left(1 - sin \theta \right)^{2}+4\left(1 - sin ⁡ \theta \right)\cdot 3sin ⁡ \theta < 0$
$\Rightarrow $ $4\left(1 - sin \theta \right)\left(1 - sin ⁡ \theta + 3 sin ⁡ \theta \right) < 0$
$\Rightarrow $ $4\left(1 - sin \theta \right)\left(1 + 2 sin ⁡ \theta \right) < 0$
$\Rightarrow $ $8\left(sin \theta - 1\right)\left(sin ⁡ \theta + \frac{1}{2}\right)>0$
$\Rightarrow $ $sin \theta < -\frac{1}{2 \, }or \, sin ⁡ \theta >1$
$sin \theta < -\frac{1}{2}\Rightarrow \theta \in \left(\frac{7 \pi }{6} , \frac{11 \pi }{6}\right)$