First slide

Binomial theorem for positive integral Index

Question

If in the binomial expansion of $(1 - x)^{m} (1 + x)^{n}$ , the coefficients of $x and x^{2}$ are respectively 3 and -4, then the ratio $m : n$ is equal to

Moderate

A

10 : 7
B

8 : 11
C

10 : 13
D

7 : 10

Solution

We have
$\begin{array}{l} (1 - x)^{m} (1 + x)^{n} \\ = (1 - m x +^{m} C_{2} x^{2} - \dots) (1 + n x +^{n} C_{2} x^{2} + \dots) \\ = 1 + (n - m) x + (^{m} C_{2} - m n +^{n} C_{2}) x^{2} + \dots \end{array}$
we are given
$\begin{array}{l} n - m = 3,^{m} C_{2} - m n +^{n} C_{2} = - 4 \\ \Rightarrow n - m = 3, \frac{1}{2} (m^{2} - 2 m n + n^{2} - (m + n)) = - 4 \\ \Rightarrow n - m = 3, (m - n)^{2} - (m + n) = - 8 \end{array}$
$\begin{array}{l} \Rightarrow n - m = 3, m + n = 17 \\ ∴ n = 10, m = 7 \\ \Rightarrow m : n = 7 : 10 \end{array}$

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