The distances of the roots of the equation tan θ0 zn + tan θ1 zn-1 + … + tan θn = 3 from z = 0, where θ0, θ1, θ2, …, θn ∈ [0 , π/4] satisfy

Q: The distances of the roots of the equation tan θ0 zn + tan θ1 zn-1 + … + tan θn = 3 from z = 0, where θ0, θ1, θ2, …, θn ∈ [0 , π/4] satisfy

(A) greater than 2/3

(B) less than 2/3

(C) greater than |cos θ1| + |cos θ2| + … + |cos θn|

(D) less than |cos θ1| + |cos θ2| + … + |cos θn|

Sol: 3 = |tan θ0 zn + tan θ1 zn-1 + .. + tan θn|

3 ≤ |tan θ0| |z|n + |tan θ1| |z|n-1 + … + |tan θn|

3 ≤ |z|n + |z|n-1 + … + 1

Since |tan θi| ≤ 1

3 < 1 + |z| + |z|2 + … ∞

$\large 3 < \frac{1}{1-|z|}$

3 – 3 |z| < 1

– 3 |z| < 1 – 3 = – 2

⇒ 3 |z| > 2

⇒ |z| > 2/3.

Hence (A) is the correct answer.