<p><strong>Ex. 62:</strong> If the sum of the base 2 logarithms of the roots of the cubic \(f(x) = 0\) is 5, then the value of \(a\) is</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to relate the product of roots to the constant term coefficient, combined with logarithm properties.
<p>Let the roots of \(f(x) = 8x^3 - 4ax^2 - 2x - a = 0\) be \(r_1, r_2, r_3\).</p><p>Given: \(\log_2 r_1 + \log_2 r_2 + \log_2 r_3 = 5\)</p><p>This means: \(\log_2(r_1 r_2 r_3) = 5\)</p><p>Therefore: \(r_1 r_2 r_3 = 2^5 = 32\)</p><p>By Vieta's formulas, for \(8x^3 - 4ax^2 - 2x - a = 0\):</p><p>\(r_1 r_2 r_3 = \frac{a}{8}\)</p><p>Thus: \(\frac{a}{8} = 32\)</p><p>\(a = 256\)</p><p>∴ Answer is (d).</p>
Correct Answer: d