<p>If for some \(x \in R\), the frequency distribution of the marks obtained by 20 students in a test is</p><table border='1'><tr><th>Marks</th><td>2</td><td>3</td><td>5</td><td>7</td></tr><tr><th>Frequency</th><td>\((x+1)^2\)</td><td>\(2x-5\)</td><td>\(x^2-3x\)</td><td>\(x\)</td></tr></table><p>then the mean of the marks is ______.</p>
Step-by-Step Solution
Key Concept: The sum of all frequencies must equal 20 (total students). Use this constraint to find x, then calculate the mean using the formula: mean = Σ(marks × frequency)/total frequency.
<p><strong>Step 1:</strong> Set up the constraint equation. Since there are 20 students total:</p><p>(x+1)² + (2x-5) + (x²-3x) + x = 20</p><p><strong>Step 2:</strong> Expand and simplify:</p><p>x² + 2x + 1 + 2x - 5 + x² - 3x + x = 20</p><p>2x² + 2x - 4 = 20</p><p>2x² + 2x - 24 = 0</p><p>x² + x - 12 = 0</p><p>(x+4)(x-3) = 0</p><p>∴ x = 3 or x = -4</p><p><strong>Step 3:</strong> Check validity. For x = -4: frequencies are 9, -13, 28, -4 (negative values invalid). For x = 3: frequencies are 16, 1, 0, 3 (all non-negative, sum = 20) ✓</p><p><strong>Step 4:</strong> Calculate the mean with x = 3:</p><p>Mean = (2×16 + 3×1 + 5×0 + 7×3)/20</p><p>Mean = (32 + 3 + 0 + 21)/20</p><p>Mean = 56/20 = 2.8</p><p>∴ Answer: <strong>2.8</strong></p>
Correct Answer: 2.8