Question

If $\sin \theta ,\sin \varphi$ and $\cos \theta$ are in G.P., then the roots of the equation $x^2+2x\cot \varphi +1=0$ are always

• A. Real
• B. Imaginary
• C. Equal
• D. Greater than $1$

Answer 1

Answer: (A)

Real

For G.P. ,
${\sin }^2\varphi =\sin \theta \cdot \cos \theta$

${\text{cosec}}^2\varphi =\frac{2}{\sin 2\theta }$

${\text{cosec}}^2\varphi -2=\frac{2}{\sin 2\theta }-2$

${\cot }^2\varphi -1=\frac{2}{\sin 2\theta }-2$

${\cot }^2\varphi -1>0$

$4({\cot }^2\varphi -1)>0$ or $D>0$
Hence, equation have always real roots .