Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
The complete set of values of $x$ satisfying the inequality $\sin^{-1}(\sin 5) > x^2 - 4x$ is $(2 - \sqrt{\lambda - 2\pi}, 2 + \sqrt{\lambda - 2\pi})$, then $\lambda =$
Step-by-Step Solution
Key Concept: Solve the compound inequality to find the domain, then match the given interval endpoints.
From $5 - 2\pi > x^2 - 4x$ and $x^2 - 4x + (2\pi - 5) < 0$, we solve the quadratic to get $2 - \sqrt{9 - 2\pi} < x < 2 + \sqrt{9 - 2\pi}$. The length of this interval is $2\sqrt{9 - 2\pi}$, and setting $\lambda = 9$ satisfies the constraint.
Correct Answer: 9