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# 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) $\binom{9}{6}(\frac{1}{3})^3(\frac{3}{2})^6$

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

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

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

Solution:

$latex [\frac{3x^2}{2}-\frac{1}{3}]^9 =(\frac{3x^2}{2})^9+\binom{9}{1}(\frac{3x^2}{2})^8(-\frac{1}{3x})+…\binom{9}{6}(\frac{3x^2}{2})^3(-\frac{1}{3x})^6+…-(\frac{1}{3x})^9$

Therefore the term independent of x is

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

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

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

## Some Useful Links:

Our ISI CMI Entrance Program

Sequence Problem | ISI Entrance B.Math 2008 Obj 1 – Video

## By Dr. Ashani Dasgupta

Ph.D. in Mathematics, University of Wisconsin, Milwaukee, United States.

Research Interest: Geometric Group Theory, Relatively Hyperbolic Groups.

Founder, Cheenta

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