The range of the function f(\theta) = \sqrt{8\sin^2 \theta + 4\cos^2 \theta - 8\sin \theta\cos \theta} is -
[\sqrt{5} - 1, \sqrt{5} + 1]
[0, \sqrt{5} + 1]
[\sqrt{6} - \sqrt{20}, \sqrt{6} + \sqrt{20}]
none of these
Step-by-Step Solution
<div class="solution"><p>(A) ln{x}: {x}>0 excludes integers → 0 integers → (P),(S)</p><p>(B) Domain has exactly 3 admissible integers → (R)</p><p>(C) (x-1)²+1 on [0,2]: range [1,2], integers={1,2}: 2 → (Q),(S)</p><p>(D) [x]∈{-5,...,5}: range values √(25-n²) integers when n²=0,9,16,25 → 4+ integers → (T)</p><p><strong>Answer: A→P,S; B→R; C→Q,S; D→T</strong></p><div class="trap-box"><strong>Trap:</strong> Don't count real values — count only integers.</div><div class="key-concept"><strong>Key Concept:</strong> Integer counting in domain/range — direct integer testing is fastest</div></div>
Correct Answer: A,C