If the third term in the expansion of 1x+xlog10⁡x5, x>1, is 1000, then value of x is

A
$\sqrt{10}$
B
10
C
100
D
$10 \sqrt{10}$

Solution:

$\begin{array}{l} T_{3} =^{5} C_{2} {(\frac{1}{x})}^{5 - 2} {(x^{\log_{10} x})}^{2} = 1000 \\ \Rightarrow 10 x^{2 \log_{10} x - 3} = 1000 \\ \Rightarrow (2 t - 3) t = 2 where t = \log_{10} x \\ \Rightarrow 2 t^{2} - 3 t - 2 = 0 \\ \Rightarrow (2 t + 1) (t - 2) = 0 \\ \Rightarrow x = \frac{1}{\sqrt{10}} or x = 100 \end{array}$

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

Solution:

Related content