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  4. If 1 lies between the roots of 3 x2-3 sin θ-2 cos 2 θ=0 then

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Q. If 1 lies between the roots of $3 x^{2}-3 \sin \theta-2 \cos ^{2} \theta=0$ then

Complex Numbers and Quadratic Equations

Solution:

Since coefficient of $x^{2}>0$ and $1$ lies between the roots of
$3 x^{2}-3 \sin \theta-2 \cos ^{2} \theta=0$
$\therefore f(1)<0$
$\Rightarrow 3-3 \sin \theta-2 \cos ^{2} \theta<0$
$\Rightarrow 1+2\left(1-\cos ^{2} \theta\right)-3 \sin \theta<0 $
$\Rightarrow 2 \sin ^{2} \theta-3 \sin \theta+1<0 $
$\Rightarrow (2 \sin \theta-1)(\sin \theta-1)<0$
$\Rightarrow \frac{1}{2}<\sin \theta<1$