(a) Determine the ratio of the constant terms in the binomial theorem expanvion of \( \left(2 x^{2}+\frac{1}{x}\right)^{9} \) and \( \left(2 x^{3}+\frac{3}{x^{2}}\right)^{3} \) b) Solve the simultaneous equations. \[ \begin{array}{l} 3 \log _{5} x-2 \log _{4} y=2 \\ 2 \log _{5} x+3 \log _{4} y=10 \end{array} \]
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The general term in the expansion is given by: \[ T_k = \binom{9}{k} (2x^2)^{9-k} \left(\frac{1}{x}\right)^k \] Simplifying, we get: \[ T_k = \binom{9}{k} 2^{9-k} x^{18-2k-k} = \binom{9}{k} 2^{9-k} x^{18-3k} \] For the term to be constant, the power of \(x\) Show more…
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