TOMATO Objective 106 | ISI Entrance Problem

This is a problem from Test of Mathematics, TOMATO Problem 106. It is useful for ISI and CMI Entrance Exam. Try out the problem.

Problem: TOMATO Objective 106

The term that is independent of x in the expansion of

$[\frac{3x^2}{2}-\frac{1}{3x}]^9$

a) ${{9} \choose {6}}(\frac{1}{3})^3(\frac{3}{2})^6$

b) ${{9} \choose {5}}(\frac{3}{2})^5(-\frac{1}{3})^4$

c) ${{9} \choose {3}}(\frac{1}{6})^3$

d) ${{9} \choose {4}}(\frac{3}{2})^4(-\frac{1}{3})^5$

Solution:

$[\frac{3x^2}{2}-\frac{1}{3}]^9$

=$(\frac{3x^2}{2})^9+{{9}\choose{1}}(\frac{3x^2}{2})^8(-\frac{1}{3x})+...{{9}\choose{6}}(\frac{3x^2}{2})^3(-\frac{1}{3x})^6+...-(\frac{1}{3x})^9$

Therefore the term independent of x is

=${{9}\choose{6}}(\frac{3}{2})^3(\frac{1}{3})^6$

=$\frac{9!}{6!3!}[\frac{3x3x3}{2x2x2x3x3x3x3x3x3}]$

=${{9}\choose{3}}(\frac{1}{6})^3$ (c)