<p><strong>170.</strong> Number of real solution(s) of the equation \(|x-3|^{3x^2-10x+3} = 1\) is:</p>
Step-by-Step Solution
Key Concept: For an equation of the form |a|^b = 1, we need either: (1) |a| = 1, (2) b = 0 (with a ≠ 0), or (3) a = -1 and b is an even integer. Systematically find all solutions from each case.
<p><strong>Step 1:</strong> For |x-3|^(3x²-10x+3) = 1, we need one of three conditions:</p><p><strong>Case 1:</strong> |x-3| = 1</p><p>This gives x - 3 = 1 or x - 3 = -1</p><p>So x = 4 or x = 2 ✓ (2 solutions)</p><p><strong>Case 2:</strong> Exponent = 0 (with base ≠ 0)</p><p>3x² - 10x + 3 = 0</p><p>(3x - 1)(x - 3) = 0</p><p>x = 1/3 or x = 3</p><p>Check base: |1/3 - 3| = 8/3 ≠ 0 ✓ and |3 - 3| = 0 ✗</p><p>So x = 1/3 is valid (1 solution)</p><p><strong>Case 3:</strong> Base = -1 and exponent is even</p><p>|x-3| = 1 already covered, and x - 3 = -1 is impossible</p><p><strong>Step 2:</strong> Verify no overlaps:</p><p>Solutions are: x ∈ {2, 4, 1/3}</p><p>∴ Answer: A (3 real solutions)</p>
Correct Answer: A