Question

If in the expansion of $(1 + px)^n, n \, \in \, N$, the coefficient of $x$ and $x^2 $ are $8$ and $24$, then

• A. $ n = 3, p = 2 $
• B. $ n = 5, p = 3$
• C. $ n = 4, p = 3$
• D. $ n = 4, p = 2 $

Answer 1

Answer: (D)

We have, $(1 + px)^n$
$\therefore \, T_{r+1} = \,{}^{n}C_r (1)^{n -r} (px)^r$
$\Rightarrow \:\:\:T_{r+1} = \,{}^{n}C_r \, p^r \, x^r$
Coefficient of $x^r = \,{}^nC_r \, p^r$
Now coefficient of $x = \,{}^n C_1 \, p =8 $ [put $r$ = 1]
$\Rightarrow \:\: np = 8 $ ....(i)
Also, coefficient of $x^2 =\,{}^nC_2 p^2 = 24 $ [put $r$ = 2]
$\Rightarrow \:\: \frac{n(n -1)}{2} p^2 = 24$
$\Rightarrow \:\: \frac{n(n -1)}{2} \left(\frac{8}{n} \right)^2 = 24$ (from (i))
$\Rightarrow \:\: \frac{64 (n - 1)}{2n} = 2 4\, \Rightarrow \: 4(n - 1) = 3n \:\:\: \Rightarrow \:\: n = 4$
From (i), we get $p = 2$