CAIE P2 (Pure Mathematics 2) 2021 November

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\includegraphics[max width=\textwidth, alt={}, center]{6294c4f4-70a9-4b81-87e0-20e2cc24dd27-05_606_933_258_605} The variables \(x\) and \(y\) satisfy the equation \(a ^ { y } = k x\), where \(a\) and \(k\) are constants. The graph of \(y\) against \(\ln x\) is a straight line passing through the points \(( 1.03,6.36 )\) and \(( 2.58,9.00 )\), as shown in the diagram. Find the values of \(a\) and \(k\), giving each value correct to 2 significant figures.

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\includegraphics[max width=\textwidth, alt={}, center]{6294c4f4-70a9-4b81-87e0-20e2cc24dd27-08_663_433_260_854} The diagram shows the curve with parametric equations $$x = \ln ( 2 t + 3 ) , \quad y = \frac { 2 t - 3 } { 2 t + 3 }$$ The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { 2 t + 3 }\).
2. Find the gradient of the curve at \(A\).
3. Find the gradient of the curve at \(B\).