Q2.    Find a if the coefficients of x^2and x^3 in the expansion of (3 + ax)^9 are equal.

Answers (1)

As we know that the general  (r+1)^{th} term  T_{r+1} in the binomial expansion of  (a+b)^n  is given by 

T_{r+1}=^nC_ra^{n-r}b^r

So, the general  (r+1)^{th} term  T_{r+1} in the binomial expansion of  (3 + ax)^9  is

T_{r+1}=^nC_r3^{n-r}(ax)^r=^nC_r3^{n-r}a^rx^r

Now, x^2 will come when r=2 and  x^3 will come when r=3

So, the coefficient of x^2 is 

K_{x^2}=^nC_23^{9-2}a^2=^nC_23^7a^2

And the coefficient of x^3 is

K_{x^3}=^9C_33^{9-3}a^2=^9C_33^6a^3

Now, Given in the question,

K_{x^2}=K_{x^3}

^9C_23^7a^2=^9C_33^6a^3

\frac{9!}{2!7!}\times3=\frac{9!}{3!6!}\times a

a=\frac{18}{14}=\frac{9}{7}

Hence the value of a is 9/7.

Boost your Preparation for JEE Main 2021 with Personlized Coaching
 
Exams
Articles
Questions