Let α θ and β θ be the distinct roots of 2 x 2 +(cos θ) x-1=0, θ in(0,2 π) . If m and M are the minimum and the maximum values of α θ 4 +β θ 4 , then 16( M + m ) equals :

Mathematics · Quadratic Equations

JEE Main 2025 — 22 January, Evening Shift — Question 12

Let $\alpha_{\theta}$ 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 :

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

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

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

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

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

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

Answer key and solution verified before publishing.

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