The values of $x$ in the expansion ${(x + x^{l o g_{10} x})}^{5}$ , if the third term in the expansion is $10, 00, 000$

10

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10^{2}

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10^{3}

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None of these

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Solution

The correct option is A $10$
$T_{3}$ $=^{5} C_{2} x^{3 + 2 l o g_{10} (x)}$
$= 10^{6}$
$x^{3 + 2 l o g_{10} (x)} = 10^{5}$
Now x has to be in powers of $10$ .
Let $x = 10^{k}$
Therefore, $10^{k (3 + 2 k)} = 10^{6}$
$2 k^{2} + 3 k = 5$
$2 k^{2} + 3 k - 5 = 0$
$2 k^{2} + 5 k - 2 k - 5 = 0$
$(k - 1) (2 k + 5) = 0$
$k = 1$ and $k = \frac{- 5}{2}$
Therefore, $x = 10$ and $x = 10^{\frac{- 5}{2}}$

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