Binomial Theorem
Grade 11
Question:
<p>If the term independent of x in the expansion of <span class="math-tex">\(\left(\sqrt{a} x^{2}+\frac{1}{2 x^{3}}\right)^{10}\)</span> is 105, then <span class="math-tex">\(a^{2}\)</span> is equal to:</p>
<p style="display:inline">6</p>
<p style="display:inline">4</p>
<p style="display:inline">9</p>
<p style="display:inline">2</p>
Step-by-Step Solution
Key Concept: Identify the term independent of x by setting the sum of exponents of x in the general binomial expansion term to zero and solving for r.
<p>Given term to expand is<br />
<span class="math-tex">$\left(\sqrt{a} x^{2}+\frac{1}{2 x^{3}}\right)^{10}$</span><br />
General term <span class="math-tex">$={ }^{10} C_{r}\left(\sqrt{a} x^{2}\right)^{10-r}\left(\frac{1}{2 x^{3}}\right)^{r}$</span><br />
<span class="math-tex">$\Rightarrow 20-2 r-3 r=0$</span><br />
<span class="math-tex">$\Rightarrow r=4$</span><br />
<span class="math-tex">$\therefore{ }^{10} C_{4} a^{3} \cdot \frac{1}{16}=105$</span><br />
<span class="math-tex">$\Rightarrow a^{3}=8 \Rightarrow a^{2}=4$</span></p>
Correct Answer: B