Binomial Theorem
Grade 11
Question:
<p>The coefficient of the term independent of x in the expansion of (1 + x + 2x<sup>3</sup>) <span class="math-tex">\(\left(\frac{3 x^{2}}{2}-\frac{1}{3 x}\right)^{9}\)</span> is</p>
<p style="display:inline"><span class="math-tex">\(\frac{19}{54}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{17}{54}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{4}\)</span></p>
Step-by-Step Solution
Key Concept: Identify the constant term by summing the products of each polynomial term with the specific binomial term whose power of x is the negative of that polynomial term's exponent.
<p>t<sub>r+1</sub> of <span class="math-tex">$\left(\frac{3 x^{2}}{2}-\frac{1}{3 x}\right)^{9}$</span> = <sup>9</sup>C<sub>r</sub> <span class="math-tex">$\left(\frac{3}{2} x^{2}\right)^{r}\left(-\frac{1}{3 x}\right)^{9-r}$</span><br />
= <sup>9</sup>C<sub>r</sub> <span class="math-tex">$\left(\frac{3}{2}\right)^{r}\left(-\frac{1}{3}\right)^{9-r}$</span> x<sup>3r-9</sup><br />
t<sub>r+1</sub> is independent of x, if 3r - 9 = 0 <span class="math-tex">$\Rightarrow$</span> r = 3<br />
For r = 3, <sup>9</sup>C<sub>r</sub> <span class="math-tex">$\left(\frac{3}{2}\right)^{r}\left(-\frac{1}{3}\right)^{9-r}$</span> = <sup>9</sup>C<sub>3</sub> <span class="math-tex">$\left(\frac{3}{2}\right)^{3}\left(-\frac{1}{3}\right)^{6}$</span><br />
= <span class="math-tex">$\frac{7}{18}$</span><br />
t<sub>r+1</sub> contains <span class="math-tex">$\frac{1}{x^{3}}$</span>, if 3r - 9 = -3 <span class="math-tex">$\Rightarrow$</span> r = 2<br />
For r = 2, <sup>9</sup>C<sub>r</sub> <span class="math-tex">$\left(\frac{3}{2}\right)^{r}\left(-\frac{1}{3}\right)^{9-r}$</span><br />
= <sup>9</sup>C<sub>2</sub> <span class="math-tex">$\left(\frac{3}{2}\right)^{2}\left(-\frac{1}{3}\right)^{7}=-\frac{1}{27}$</span><br />
<span class="math-tex">$\Rightarrow$</span> Coefficient of the term independent of x in the given expression = <span class="math-tex">$\frac{-2}{27}+\frac{7}{18}=\frac{-4+21}{54}=\frac{17}{54}$</span></p>
Correct Answer: B