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Find a if the coefficients of $x^2$ and $x^3$ in the expansion of $(3+ax)^9$ are equal.

$\begin{array}{1 1}(A)\;\large\frac{9}{4}\\(B)\;\large\frac{9}{5}\\(C)\;\large\frac{9}{7}\\(D)\;\large\frac{9}{8}\end{array} $

1 Answer

Toolbox:
  • General term =$T_{r+1}=nC_ra^{n-r}b^r$
  • $C(n,r)=\large\frac{n!}{r!(n-r)!}$
General term =$T_{r+1}=9C_r.3^{9-r}a^rx^r$
Putting $r=2$
Coefficient of $x^2=9C_23^{9-2}a^2$
$\Rightarrow \large\frac{9\times 8}{2}$$3^7a^2$
$\Rightarrow 4.3^9a^2$----(1)
Putting $r=3$
Coefficient of $x^3=9C_33^{9-3}a^3$
$\Rightarrow \large\frac{9\times 8\times 7\times 6!}{3\times 2\times 6!}$$\times 3^6....a^3$
$\Rightarrow 4\times 7\times 3^7.a^3$------(2)
Equating (1) and (2)
$4.3^9a^2=4\times 7\times 3^7a^3$
(Or) $3^2=7a$
$a=\large\frac{9}{7}$
Hence (C) is the correct answer.
answered Jun 23, 2014 by sreemathi.v
 

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