CAIE P2 (Pure Mathematics 2) 2016 November

The sequence of values given by the iterative formula $$x_{n+1} = \frac{4}{x_n^2} + \frac{2x_n}{3},$$ with initial value \(x_1 = 2\), converges to \(\alpha\).

1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
2. State an equation that is satisfied by \(\alpha\), and hence find the exact value of \(\alpha\). [2]

\includegraphics{figure_2} The variables \(x\) and \(y\) satisfy the equation \(y = Kx^p\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \((1.28, 3.69)\) and \((2.11, 4.81)\), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places. [5]

1. Find \(\int \tan^2 4x \, dx\). [2]
2. Without using a calculator, find the exact value of \(\int_0^{\frac{\pi}{2}} (4 \cos 2x + 6 \sin 3x) \, dx\). [3]

The polynomial \(\mathrm{p}(x)\) is defined by $$\mathrm{p}(x) = ax^3 + 3x^2 + 4ax - 5,$$ where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(\mathrm{p}(x)\).

1. Use the factor theorem to find the value of \(a\). [2]
2. Factorise \(\mathrm{p}(x)\) and hence show that the equation \(\mathrm{p}(x) = 0\) has only one real root. [4]
3. Use logarithms to solve the equation \(\mathrm{p}(6^x) = 0\) correct to 3 significant figures. [2]

\includegraphics{figure_5} The diagram shows the curve \(y = \sqrt{1 + e^{4x}}\) for \(0 \leq x \leq 6\). The region bounded by the curve and the lines \(x = 0\), \(x = 6\) and \(y = 0\) is denoted by \(R\).

1. Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 2 decimal places. [3]
2. With reference to the diagram, explain why this estimate is greater than the exact area of \(R\). [1]
3. The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [4]

A curve has parametric equations $$x = \ln(t + 1), \quad y = t^2 \ln t.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [5]
2. Find the exact value of \(t\) at the stationary point. [2]
3. Find the gradient of the curve at the point where it crosses the \(x\)-axis. [2]

1. Express \(\sin 2\theta (3 \sec \theta + 4 \cosec \theta)\) in the form \(a \sin \theta + b \cos \theta\), where \(a\) and \(b\) are integers. [3]
2. Hence express \(\sin 2\theta (3 \sec \theta + 4 \cosec \theta)\) in the form \(R \sin(\theta + \alpha)\) where \(R > 0\) and \(0° < \alpha < 90°\). [3]
3. Using the result of part (ii), solve the equation \(\sin 2\theta (3 \sec \theta + 4 \cosec \theta) = 7\) for \(0° \leq \theta \leq 360°\). [4]