Binomial Theorem
Grade None
Question:
<p>The absolute difference of the coefficients of x<sup>10</sup> and x<sup>7</sup> in the expansion of <span class="math-tex">\(\left(2 x^2+\frac{1}{2 x}\right)^{11}\)</span> is equal to</p>
<p style="display:inline">13<sup>3</sup> - 13</p>
<p style="display:inline">11<sup>3</sup> - 11</p>
<p style="display:inline">10<sup>3</sup> - 10</p>
<p style="display:inline">12<sup>3</sup> - 12</p>
Step-by-Step Solution
Key Concept: Use the general term formula $T_{r+1} = {}^{n}C_{r} a^{n-r} b^{r}$ to equate the resultant exponent of $x$ to the desired powers and solve for $r$.
<p>T<sub>r+1</sub> = <span class="math-tex">${ }^{11} \mathrm{C}_{\mathrm{r}}\left(2 \mathrm{x}^2\right)^{11-\mathrm{r}}\left(\frac{1}{2 \mathrm{x}}\right)^{\mathrm{r}}$</span> = <sup>11</sup>C<sub>r</sub>2<sup>11-2r</sup> x<sup>22-3r</sup><br />
For coefficients of x<sup>10</sup> & x<sup>7</sup><br />
<span class="math-tex">$\Rightarrow$</span> 22 - 3r = 10 and 22 - 3r = 7<br />
r = 4 and r = 5; So, coefficient of x<sup>10</sup> = <sup>11</sup>C<sub>4</sub><span class="math-tex">$\cdot$</span>2<sup>3</sup><br />
and coefficient of x<sup>7</sup> = <sup>11</sup>C<sub>5</sub><span class="math-tex">$\cdot$</span>2<sup>1</sup><br />
difference = <sup>11</sup>C<sub>4</sub><span class="math-tex">$\cdot$</span>2<sup>3</sup> - <sup>11</sup>C<sub>5</sub><span class="math-tex">$\cdot$</span>2<br />
= <span class="math-tex">$\frac{11 \times 10 \times 9 \times 8}{24} \times 8-\frac{11 \times 10 \times 9 \times 8 \times 7}{120} \times 2$</span><br />
= 11 <span class="math-tex">$\times$</span> 10 <span class="math-tex">$\times$</span> 3 <span class="math-tex">$\times$</span> 8 - 11 <span class="math-tex">$\times$</span> 3 <span class="math-tex">$\times$</span> 4 <span class="math-tex">$\times$</span> 7<br />
= 11 <span class="math-tex">$\times$</span> 3 <span class="math-tex">$\times$</span> 4 <span class="math-tex">$\times$</span> (20 - 7) = 11 <span class="math-tex">$\times$</span> 12 <span class="math-tex">$\times$</span> 13 = 12(12 - 1)(12 + 1)<br />
= 12<sup>3</sup> - 12</p>
Correct Answer: D