First slide

Binomial theorem for positive integral Index

Question

If the value of the third term in the expansion of ${(x + x^{\log_{10} x})}^{5}$ is $10^{6}$ , then x may have value(s)

Moderate

A

1, 10
B

$10, 10^{- 5 / 2}$
C

$10, 10^{- 3 / 2}$
D

$10^{2}, 10^{- 3 / 2}$

Solution

$T_{3} =^{5} C_{2} x^{5 - 2} {(x^{\log_{10} x})}^{2} = 10^{6} \Rightarrow x^{2 \log_{10} x + 3} = 10^{5}$
$\Rightarrow t (2 t + 3) = 5$ where $t = \log_{10} x \Rightarrow t = - 5 / 2, 1$

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