Question

If the expansion of \(\left(x^2 + \frac{2}{x} \right)^n\) has a term independent of \(x\), then \(n\) is

• A. 23
• B. 18
• C. 16
• D. 13

Hint:

Isobars have the same mass number but belong to different elements.

Answer 1

The Correct Option is B

General term: \[ T_{r+1} = \binom{n}{r} (x^2)^{n-r} \left(\frac{2}{x}\right)^r = \binom{n}{r} 2^r x^{2(n - r) - r} = \binom{n}{r} 2^r x^{2n - 3r} \] For independence of \(x\): \[ 2n - 3r = 0 \Rightarrow r = \frac{2n}{3} \] r must be integer ⇒ \(n\) must be divisible by 3 ⇒ smallest integer satisfying this: \(n = 18\)