Question

Let \(S = \{θ ∈ (0, 2π) : 7 cos^2θ – 3 sin^2θ – 2 cos^22θ = 2\}\). 
Then, the sum of roots of all the equations \(x^2 – 2 (tan^2θ + cot^2θ) x + 6 sin^2θ = 0, θ ∈ S\), is _______.

Answer 1

Correct Answer: 16

7 cos²θ – 3 sin²θ – 2 cos²2θ = 2
\(\begin{array}{l} \Rightarrow 4\left(\frac{1+\cos2\theta}{2}\right)+3\cos2\theta-2\cos^22\theta=2 \end{array}\)
⇒ 2 + 5 cos²θ – 2 cos² 2θ = 2
⇒ cos 2θ = 0 or 5/2 (rejected)
\(\begin{array}{l} \Rightarrow \cos2\theta=0=\frac{1-\tan^2\theta}{1+\tan^2\theta}\Rightarrow \tan^2\theta =1\end{array}\)
∴ Sum of roots = 2 (tan²θ + cot²θ) = 2 × 2 = 4
But as tanθ = ±1 for \(π/4, 3π/4, 5π/4, 7π/4\) in the interval (\(0, 2π\))
∴ Four equations will be formed
Hence, sum of roots of all the equations = 4 × 4 = 16.