2025 Presbyterian High School Prelims A Math Paper 2
By using a suitable substitution, or otherwise, solve the equation \(3\log_6 x = \log_x 3 + \log_x 2 + \log_5 25.\)[4]
Sketch the graph of \(y = \log_6 x\) on the given axes. Label any axial intercepts.[1]
It is given that the equation of a curve is \(y = e^{2x+1} + 2e^{x+1} - 3e\). Explain why the curve has no stationary point.[3]
Find the \(x\)-intercept of the curve \(y = e^{2x+1} + 2e^{x+1} - 3e\).[3]
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Answer:(a) \(x=6\) or \(x=6^{-1/3}\). (b) Increasing logarithmic curve with vertical asymptote \(x=0\), \(x\)-intercept \((1,0)\), and no \(y\)-intercept. (c)(i) No stationary point. (c)(ii) \((0,0)\).