Question

If sin α and β and cosα are in GP, then roots of the equation x² + 2x cot β + 1 = 0 are always 

(A)  Equal 

(B)  Real 

(C)  Imaginary 

(D)  Greater than 1

Answer 1

Correct option (B) Real

sinα, sinβ, cosα are in GP. Therefore,

sin²β = sinα cosα ⇒ cos2β = 1 − sin2β ≥ 0

Now, the discriminant of the given equation 

4cot²β − 4 = 4 cos2β ⋅cosec²β ≥ 0 ⇒ roots are always real