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Trigonometrical Equations
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If both roots of quadratic equation ${x^2} + \left( {\sin \,\theta + \cos \,\theta } \right)x + \frac{3}{8} = 0$ are positive and distinct then complete set of values of $\theta $ in $\left[ {0,2\pi } \right]$ is
A
$\left( {\frac{\pi }{{12}},\frac{{5\pi }}{{12}}} \right)$
B
$\left( {\frac{{13\pi }}{{12}},\frac{{17\pi }}{{12}}} \right)$
C
$\left( {\frac{{7\pi }}{{12}},\frac{{11\pi }}{{12}}} \right)$
D
$\left( {\frac{{19\pi }}{{12}},\frac{{23\pi }}{{12}}} \right)$
Solution
$\sin \theta+\cos \theta<0$
$(\sin \theta+\cos \theta)^{2}>\frac{3}{2}$
$\sin 2 \theta>\frac{1}{2}$
$2 n \pi+\frac{\pi}{6}<2 \theta<2 n \pi+\frac{5 \pi}{6}$
$n\pi + \frac{\pi }{{12}} < \theta < n\pi + \frac{{5\pi }}{{12}}$
Standard 11
Mathematics
