Question

The independent term of x is 80000 in the expansion of (3x+b/x)^6, where b is a positive constant. What the value of b?

(a) 3.97

(b) 6.87

(c) 8.3

(d) 5.2

I have been asked this question by my school principal while I was bunking the class.

Enquiry is from Counting topic in division Counting of Discrete Mathematics

Answer 1

Correct choice is (d) 5.2

Explanation: By using the Binomial Theorem, the terms are of the form ^6Cn * (4x)^6-n * (b/x)^n.

For the term to be independent of x, we need x^6-n(1/x)^n = x^0  ⇒ x^6-n(x-1)^n = x^0  ⇒ x^6-nx^-n = x^0 ⇒ 6 – n = n ⇒ 2n = 6 and n = 3. Thus, we have a constant term of ^6C3 * 3^3 * b^3 = 8000

20 * 27 * b^3 = 80000

540 * b^3 = 80000

b^3 = 148.14 ⇒ b= 5.2.