Question

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

• A. $\frac{-1}{2}<\sin \theta <\frac{1}{2}$
• B. $\frac{-1}{2}<\sin \theta <0$
• C. $\frac{1}{2}<\sin \theta <1$
• D. None of these

Answer 1

Answer: (C)

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