Question

If $cos\theta ,sin\varphi ,sin\theta$ are in $GP$, then the roots of $x^2+2cot\varphi x+1=0$ are always

• A. equal
• B. real
• C. imaginary
• D. greater than $1$

Answer 1

The correct option is B real

Since, $\cos \theta ,\sin \varphi ,\sin \theta$ are in G.P.
Therefore, ${\sin }^2\varphi =\cos \theta \sin \theta$ $...\left(1\right)$
$x^2+2\cot \varphi \cdot x+1=0$
Now, $D=4({\cot }^2\varphi -1)=4({\cot }^2\varphi +1-2)$
$\Rightarrow D=4(\frac{1}{{\sin }^2\varphi }-2)=4(\frac{2}{2\cos \theta \sin \theta }-2)=8(\frac{1}{\sin 2\theta }-1)$ [ from $\left(1\right)$ ]
Since, $\sin 2\theta \le 1$
Therefore, $D\ge 0$ and roots are real.
Ans: B