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
B
102
C
103
D
None of these

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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