First slide

Binomial theorem for positive integral Index

Question

The value/values of x in the expression ${(x + x^{\log_{10} x})}^{5}$ if the third term in the expansion is 10,00,000 is/are

Moderate

A

10
B

100
C

$10^{- 5 / 2}$
D

$10^{- 3 / 2}$

Solution

Inclusion of log x implies x > 0.
Now, 3^rd term in the expansion is
$T_{2 + 1} =^{5} C_{2} x^{5 - 2} {(x^{\log_{10} x})}^{2} = 1000000 (given)$
or $x^{3 + 2 \log_{10} x} = 10^{5}$
Taking logarithm of both sides, we get
$(3 + 2 \log_{10} x) \log_{10} x = 5$
or $2 y^{2} + 3 y - 5 = 0, where \log_{10} x = y$
or $(y - 1) (2 y + 5) = 0 or y = 1 or - 5 / 2$
or $\log_{10} x = 1 or - 5 / 2$
$∴ x = 10^{1} = 10 or 10^{- 5 / 2}$

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