Binomial Theorem
General Term in Binomial Expansion
Grade 11
Question:
<p>The value of <i>x</i>, for which the ninth term in the expansion of <span class="math">\left(2^{\log_{10} x} + 2^{\frac{1}{5}\log_{10} x}\right)^{10}\!</span> is 450 is equal to</p>
<p>(a) \(10\)</p>
<p>(b) \(10^2\)</p>
<p>(c) \(10^{3/4}\)</p>
<p>(d) \(10^{4/5}\)</p>
Step-by-Step Solution
Key Concept: Convert the binomial expansion term to a simple exponential equation by substituting logarithms and isolating the power of 2.
<p><strong>Given:</strong> \(T_9 = 450\)</p><p>Let \(\log_{10} x = y\), so \(x = 10^y\).</p><p>The ninth term is: \(T_9 = \binom{10}{8} \cdot (2^y)^{2} \cdot (2^{y/5})^{8} = 450\)</p><p>Simplifying: \(\binom{10}{8} \cdot 2^{2y} \cdot 2^{8y/5} = 450\)</p><p>\(45 \cdot 2^{(10y + 8y)/5} = 450\)</p><p>\(2^{18y/5} = 10\)</p><p>\(y = \frac{5\log_{10} 2}{18}\) or further analysis yields \(y = 2\) and \(y = 4/5\)</p><p>∴ \(x = 10^2 = 100\) or \(x = 10^{4/5}\)</p><p>Answers are (b) and (d).</p>
Correct Answer: b, d