Question

If the coefficient of $5^{\text {th }}$ and $3^{\text {rd }}$ terms in the expansion of $\left(\sqrt{\frac{15}{2}} x+\frac{1}{x^2}\right)^n$ where $n \in N$ are equal, then the term independent of $x$ is

• A. $5^{\text {th }}$ term
• B. $4^{\text {th }}$ term
• C. $6^{\text {th }}$ term
• D. $7^{\text {th }}$ term

Answer 1

Answer: (A)

$ T_{r+1}={ }^n C_r\left(\frac{15}{2}\right)^{\frac{n-r}{2}} \cdot(x)^{-2 r} \cdot(x)^{n-r}$
$\therefore{ }^{ n } C _4\left(\frac{15}{2}\right)^{\frac{ n -4}{2}}={ }^{ n } C _2\left(\frac{15}{2}\right)^{\frac{ n -2}{2}} \Rightarrow n =12$
$\therefore T _{ r +1}={ }^{12} C _{ r }\left(\frac{15}{2}\right)^{\frac{12- r }{2}} \cdot( x )^{12-3 r }$
Put $r =4 \Rightarrow T _5$ is independent of $x $