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Question

The values of $$x$$ in the expansion $$\displaystyle \left ( x+x^{log_{10}x} \right )^{5}$$ , if the third term in the expansion is $$10,00,000$$


A
10
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B
102
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C
103
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D
None of these
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Solution

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

Mathematics

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