<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: The equation |x-3|^(3x²-10x+3) = 1 holds when: (1) base equals 1, (2) base equals -1 with even exponent, or (3) exponent equals 0 with non-zero base. Systematically check each case.
<p><strong>Step 1: Identify when |x-3|^(3x²-10x+3) = 1</strong></p><p>This equation holds when:</p><ul><li>(i) |x-3| = 1</li><li>(ii) |x-3| = -1 (impossible, absolute value ≥ 0)</li><li>(iii) 3x² - 10x + 3 = 0 and |x-3| ≠ 0</li></ul><p><strong>Step 2: Solve Case (i): |x-3| = 1</strong></p><p>x - 3 = 1 ⟹ x = 4</p><p>x - 3 = -1 ⟹ x = 2</p><p>Both solutions are valid (base = 1 for any real exponent).</p><p><strong>Step 3: Solve Case (iii): 3x² - 10x + 3 = 0</strong></p><p>Using the quadratic formula: x = (10 ± √(100-36))/6 = (10 ± 8)/6</p><p>x = 3 or x = 1/3</p><p><strong>Step 4: Check validity for Case (iii)</strong></p><p>For x = 3: |3-3| = 0, so 0^0 is undefined. ✗ Invalid</p><p>For x = 1/3: |1/3 - 3| = 8/3 ≠ 0, so (8/3)^0 = 1 ✓ Valid</p><p><strong>Step 5: Compile all solutions</strong></p><p>Valid solutions: x = 4, x = 2, x = 1/3</p><p><strong>∴ Number of real solutions = 3 (Answer: A)</strong></p>
Correct Answer: A