Binomial Theorem
Grade None
Question:
<p>The value of x in the expression <span class="math-tex">\(\left[x+x^{\log _{10}(x)}\right]^{5}\)</span>, if the third term in the expansion is 10,00,000, is</p>
<p style="display:inline">10</p>
<p style="display:inline">11</p>
<p style="display:inline">13</p>
<p style="display:inline">12</p>
Step-by-Step Solution
Key Concept: Apply the general term formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ to set up an equation and solve for $x$ using logarithmic identities.
<p>T<sub>3</sub> = <sup>5</sup>C<sub>2</sub> x<sup>3</sup> <span class="math-tex">$\left(x^{\log _{10} x}\right)^{2}$</span> = 10<sup>6</sup> ...[given]<br />
If x = 10, then 10<span class="math-tex">$\cdot$</span>(10)<sup>3</sup><span class="math-tex">$\cdot$</span>(10)<sup>2</sup> = (10)<sup>6</sup> is satisfied ...[<span class="math-tex">$\because$</span> log<sub>10</sub>10 = 1]<br />
<span class="math-tex">$\therefore$</span> x = 10</p>
Correct Answer: A