<p><strong>262.</strong> If \(\sec^{-1}(x) + \tan^{-1}\sqrt{9y^2 - 1} + \sin^{-1}(x^2 + y^2) = \lambda\) has no solution, then exhaustive set of values of \(\lambda\) is equal to:</p>
Step-by-Step Solution
Key Concept: Find the range of the left side by determining valid domains: sec⁻¹(x) requires |x|≥1, tan⁻¹(√(9y²-1)) requires 9y²-1≥0, and sin⁻¹(x²+y²) requires x²+y²≤1. The constraint x²+y²≤1 combined with |x|≥1 creates a contradiction, making the domain empty.
<p><strong>Step 1: Identify domain constraints</strong></p><p>For sec⁻¹(x): |x| ≥ 1 (so x ≤ -1 or x ≥ 1)</p><p>For tan⁻¹(√(9y²-1)): 9y² - 1 ≥ 0 ⟹ |y| ≥ 1/3</p><p>For sin⁻¹(x²+y²): x² + y² ≤ 1</p><p><strong>Step 2: Check compatibility of constraints</strong></p><p>From sec⁻¹(x): |x| ≥ 1 means x² ≥ 1</p><p>From sin⁻¹(x²+y²): x² + y² ≤ 1</p><p>If x² ≥ 1 and y² ≥ 0, then x² + y² ≥ 1</p><p>This contradicts x² + y² ≤ 1</p><p><strong>Step 3: Determine when equality holds</strong></p><p>For a solution to exist: x² = 1, y² = 0, and x² + y² = 1 simultaneously</p><p>But tan⁻¹(√(9y²-1)) requires |y| ≥ 1/3, so y = 0 is excluded</p><p><strong>Step 4: Conclusion</strong></p><p>The domain is empty for all values of λ, meaning the equation has no solution for ANY value of λ.</p><p>∴ Answer: C (Exhaustive set is ℝ or all real numbers)</p>
Correct Answer: C