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June 2021 p12 q4
1007
The coefficient of x in the expansion of \(\left(4x + \frac{10}{x}\right)^3\) is p. The coefficient of \(\frac{1}{x}\) in the expansion of \(\left(2x + \frac{k}{x^2}\right)^5\) is q.
\(Given that p = 6q, find the possible values of k.\)
Solution
First, find the coefficient of \(x\) in the expansion of \(\left(4x + \frac{10}{x}\right)^3\).
The term that gives \(x\) is \(\binom{3}{2} (4x)^2 \left(\frac{10}{x}\right)^1 = 3 \times 16x^2 \times \frac{10}{x} = 480x\).
Thus, \(p = 480\).
Next, find the coefficient of \(\frac{1}{x}\) in the expansion of \(\left(2x + \frac{k}{x^2}\right)^5\).
The term that gives \(\frac{1}{x}\) is \(\binom{5}{3} (2x)^3 \left(\frac{k}{x^2}\right)^2 = 10 \times 8x^3 \times \frac{k^2}{x^4} = 80k^2 \times \frac{1}{x}\).
