CAIE P2 (Pure Mathematics 2) 2024 November

The variables \(x\) and \(y\) satisfy the equation \(a^{2y} = e^{3x+k}\), where \(a\) and \(k\) are constants. The graph of \(y\) against \(x\) is a straight line.

1. Use logarithms to show that the gradient of the straight line is \(\frac{3}{2\ln a}\). [1]
2. Given that the straight line passes through the points \((0.4, 0.95)\) and \((3.3, 3.80)\), find the values of \(a\) and \(k\). [4]

Solve the inequality \(|x - 7| > 4x + 3\). [4]

The function \(\text{f}\) is defined by \(\text{f}(x) = \tan^2\left(\frac{1}{2}x\right)\) for \(0 \leqslant x < \pi\).

1. Find the exact value of \(\text{f}'\left(\frac{\pi}{3}\right)\). [3]

1. Find the exact value of \(\int_0^{\frac{\pi}{4}} \left(\text{f}(x) + \sin x\right) dx\). [4]

The polynomial \(\text{p}(x)\) is defined by $$\text{p}(x) = ax^3 - ax^2 - 15x + 18,$$ where \(a\) is a constant. It is given that \((x + 2)\) is a factor of \(\text{p}(x)\).

1. Find the value of \(a\). [2]
2. Hence factorise \(\text{p}(x)\) completely. [3]

1. Solve the equation \(\text{p}(\cos ec^2 \theta) = 0\) for \(-90° < \theta < 90°\). [3]

It is given that \(\int_a^{a^2} \frac{10}{2x+1} dx = 7\), where \(a\) is a constant greater than \(1\).

1. Show that \(a = \sqrt[9]{0.5e^{1.4}(2a+1) - 0.5}\). [5]
2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to \(3\) significant figures. Use an initial value of \(2\) and give the result of each iteration to \(5\) significant figures. [3]

A curve has parametric equations $$x = \frac{e^{2t} - 2}{e^{2t} + 1}, \quad y = e^{3t} + 1.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [4]
2. Find the exact gradient of the curve at the point where the curve crosses the \(y\)-axis. [3]

1. Prove that \(\cos(\theta + 30°)\cos(\theta + 60°) = \frac{1}{4}\sqrt{3} - \frac{1}{2}\sin 2\theta\). [4]
2. Solve the equation \(5\cos(2\alpha + 30°)\cos(2\alpha + 60°) = 1\) for \(0° < \alpha < 90°\). [4]
3. Show that the exact value of \(\cos 20° \cos 50° + \cos 40° \cos 70°\) is \(\frac{1}{2}\sqrt{3}\). [3]