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If in the expansion of \small (1+x)^m(1-x)^n, the coefficient of  x and \small x^2 are 3 and – 6 respectively, then m is

Option: 1

6


Option: 2

9


Option: 3

12


Option: 4

14


Answers (1)

best_answer

(1+x)^m(1-x)^n

\begin{aligned} = & \left(1+m x+\frac{m(m-1) x^2}{2 !}+\ldots\right)\left(1-n x+\frac{n(n-1)}{2 !} x^2-\ldots\right) \\ & =1+(m-n) x+\left[\frac{n^2-n}{2}-m n+\frac{\left(m^2-m\right)}{2}\right] x^2 \end{aligned}

Given, m – n = 3  or n = m – 3
Hence\ \frac{n^2-n}{2}-m n+\frac{m^2-m}{2}=-6\\ \\\ \begin{aligned} & \frac{(m-3)(m-4)}{2}-m(m-3)+\frac{m^2-m}{2}=-6 \\ \Rightarrow & m^2-7 m+12-2 m^2+6 m+m^2-m+12=0 \\ \Rightarrow & -2 m+24=0 \Rightarrow m=12 \end{aligned}

 

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SANGALDEEP SINGH

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