First slide

Theory of equations

Question

If $\sin A, \sin B, \cos A$ are in G.P., then roots of $x^{2} + 2 x \cot B + 1 = 0$ are always

Moderate

A

Real
B

Imaginary
C

Greater than 1
D

Equal

Solution

Given $\sin^{2} B = \sin A \cos A$
$\Rightarrow \cos 2 B = 1 - \sin 2 A \geq 0$
Now for $x^{2} + 2 x \cot B + 1 = 0$
Consider $D = 4 (\cot^{2} B - 1) = 4 \cos 2 B {cosec}^{2} B \geq 0$
Hence, roots are always real.

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