Let αθ and βθ be the distinct roots of 2x^2+(cosθ)x-=0, θ∈(0, 2π).

Question

Let \(\alpha_{\theta}\ \text{and}\ \beta_{\theta}\) be the distinct roots of \(2 x^{2}+(\cos \theta) x-1=0, \theta \in(0,2 \pi).\) If m and M are the minimum and the maximum values of \(\alpha_{\theta}^{4}+\beta_{\theta}^{4}, \)then \(16(\mathrm{M}+\mathrm{m})\) equals :

(1) 24

(2) 25

(3) 27

(4) 17

Answer 1

Correct option is (2) 25 

\(\left(\alpha^{2}+\beta^{2}\right)^{2}-2 \alpha^{2} \beta^{2}\)

\(\left[(\alpha+\beta)^{2}-2 \alpha \beta\right]^{2}-2(\alpha \beta)^{2}\)

\(\left[\frac{\cos ^{2} \theta}{4}+1\right]^{2}-2 \cdot \frac{1}{4}\)

\(\left(\frac{\cos ^{2} \theta}{4}+1\right)^{2}-\frac{1}{2}\)

\(M=\frac{25}{16}-\frac{1}{2}=\frac{17}{16}\)

\(\mathrm{m}=\frac{1}{2}, 16(\mathrm{M}+\mathrm{m})=25\)