Question

The binomial expansion of ${(x^k+\frac{1}{x^{2k}})}^{3n}$ $nϵN$ contains a term independent of $x$.

• A. only if k is an integer
• B. only if k is a natural number
• C. only if k is rational
• D. for any real k

Answer 1

The correct option is D for any real k

Writing the general term, we get
$T_{r+1}={}^n{C}_r{x}^{3nk-kr-2kr}$
$={}^n{C}_r{x}^{3nk-3kr}$
For term independent of x.
$3nk-3kr=0$
$3k(n-r)=0$
Or
$r=n$
Hence
$T_{n+1}$ will always be independent of x, irrespective of the value of k.