First slide

Binomial theorem for positive integral Index

Question

The value of $x$ in the expression ${(x + x^{1 o g_{10}^{x}})}^{5}$ 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 $T_{2 + 1} = ^{5} C_{2} x^{5 - 2} {(x^{1 o g_{10}^{x}})}^{2} = 1000000$ (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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