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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?

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? Correct Answer 5.2

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 x6-n(1/x)n = x0 ⇒ x6-n(x-1)n = x0 ⇒ x6-nx-n = x0 ⇒ 6 – n = n ⇒ 2n = 6 and n = 3. Thus, we have a constant term of 6C3 * 33 * b3 = 8000 20 * 27 * b3 = 80000 540 * b3 = 80000 b3 = 148.14 ⇒ b= 5.2.

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