First slide

Binomial theorem for positive integral Index

Question

If the fourth term in the binomial expansion of ${(\sqrt{x^{(\frac{1}{1 + \log_{10} x})}} + x^{\frac{1}{12}})}^{6}$ is equal to 200, and $x > 1$ then the value of x is

NA

A

100
B

10⁴
C

10
D

10³

Solution

Given binomial is ${(\sqrt{x^{(\frac{1}{1 + \log_{10} x})}} + x^{\frac{1}{12}})}^{6}$
Since, the fourth term in the given expansion is 200
$∴^{6} C_{3} {(x^{\frac{1}{1 + \log_{10} x}})}^{\frac{3}{2}} {(x^{\frac{1}{12}})}^{3} = 200$
$\begin{array}{lc} \Rightarrow & 20 \times x^{[\frac{3}{2 (1 + \log_{10} x)} + \frac{1}{4}]} = 200 \\ \Rightarrow & x^{\frac{3}{2 (1 + \log_{10} x)} + \frac{1}{4}} = 10 \end{array}$
$\Rightarrow [\frac{3}{2 (1 + \log_{10} x)} + \frac{1}{4}] \log_{10} x = 1$
[applying log₁₀both sides]
$\begin{aligned} \Rightarrow [6 + (1 + \log_{10} x)] \log_{10} x = 4 (1 + \log_{10} x) \\ \Rightarrow (7 + \log_{10} x) \log_{10} x = 4 + 4 \log_{10} x \\ \Rightarrow t^{2} + 7 t = 4 + 4 t t^{2} + 3 t - 4 = 0 \\ \Rightarrow t = 1, - 4 = \log_{10} x \Rightarrow x = 10, 10^{- 4} \\ \Rightarrow x > 1 \\ Since,, x = 10 \end{aligned}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App