First slide

Binomial theorem for positive integral Index

Question

If the third term in the expansion of ${[\frac{1}{x} + x^{\log_{10} x}]}^{5}$
$x > 1,$ is $1000,$ then $x$ equals

Moderate

A

10
B

1
C

$\frac{1}{\sqrt{10}}$
D

100

Solution

$\begin{array}{l} T_{3} =^{5} C_{2} {(\frac{1}{x})}^{3} {(x^{\log_{10} x})}^{2} \\ = 10 x^{2 \log_{10} x - 3} \end{array}$
As $T_{3} = 1000$ we get
$\begin{array}{l} (2 a - 3) a = 2 where a = \log_{10} x \\ \Rightarrow 2 a^{2} - 3 a - 2 = 0 \\ \Rightarrow (2 a + 1) (a - 2) = 0 \\ As x > 1, a = \log_{10} x > 0, thus, \\ a = 2 \Rightarrow \log_{10} x = 2 \Rightarrow x = 100 \end{array}$

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