Trigonometric Equations
Quadratic in x — Discriminant for Real Roots
nta_pyq_2024_apr
Grade 11
Question:
Let $S=\{\sin^22\theta:(\sin^4\theta+\cos^4\theta)x^2+(\sin2\theta)x+(\sin^6\theta+\cos^6\theta)=0$ has real roots$\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3\left((\alpha-2)^2+(\beta-1)^2\right)$ equals _____
Step-by-Step Solution
Key Concept: For real roots: $D\geq0$. $D=(\sin2\theta)^2-4(\sin^4\theta+\cos^4\theta)(\sin^6\theta+\cos^6\theta)\geq0$. Using $\sin^4\theta+\cos^4\theta=1-\frac{\sin^22\theta}{2}$ and $\sin^6\theta+\cos^6\theta=1-\frac{3\sin^22\theta}{4}$.
$\alpha=2-2/\sqrt{3}$, $\beta=1$. $3((\alpha-2)^2+(\beta-1)^2)=3\cdot4/3=4$.
Correct Answer: 4