CAIE P3 (Pure Mathematics 3) 2021 March

Solve the equation \(\ln(x^3 - 3) = 3 \ln x - \ln 3\). Give your answer correct to 3 significant figures. [3]

The polynomial \(ax^3 + 5x^2 - 4x + b\), where \(a\) and \(b\) are constants, is denoted by p\((x)\). It is given that \((x + 2)\) is a factor of p\((x)\) and that when p\((x)\) is divided by \((x + 1)\) the remainder is 2. Find the values of \(a\) and \(b\). [5]

By first expressing the equation \(\tan(x + 45°) = 2 \cot x + 1\) as a quadratic equation in \(\tan x\), solve the equation for \(0° < x < 180°\). [6]

The variables \(x\) and \(y\) satisfy the differential equation $$(1 - \cos x)\frac{dy}{dx} = y \sin x.$$ It is given that \(y = 4\) when \(x = \pi\).

1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\). [6]
2. Sketch the graph of \(y\) against \(x\) for \(0 < x < 2\pi\). [1]

1. Express \(\sqrt{7} \sin x + 2 \cos x\) in the form \(R \sin(x + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places. [3]
2. Hence solve the equation \(\sqrt{7} \sin 2\theta + 2 \cos 2\theta = 1\), for \(0° < \theta < 180°\). [5]

Let \(\text{f}(x) = \frac{5a}{(2x - a)(3a - x)}\), where \(a\) is a positive constant.

1. Express f\((x)\) in partial fractions. [3]
2. Hence show that \(\int_a^{2a} \text{f}(x) \, dx = \ln 6\). [4]

Two lines have equations \(\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}\).

1. Show that the lines are skew. [5]
2. Find the acute angle between the directions of the two lines. [3]

The complex numbers \(u\) and \(v\) are defined by \(u = -4 + 2\text{i}\) and \(v = 3 + \text{i}\).

1. Find \(\frac{u}{v}\) in the form \(x + \text{i}y\), where \(x\) and \(y\) are real. [3]
2. Hence express \(\frac{u}{v}\) in the form \(re^{\text{i}\theta}\), where \(r\) and \(\theta\) are exact. [2]

In an Argand diagram, with origin \(O\), the points \(A\), \(B\) and \(C\) represent the complex numbers \(u\), \(v\) and \(2u + v\) respectively.

1. State fully the geometrical relationship between \(OA\) and \(BC\). [2]
2. Prove that angle \(AOB = \frac{3}{4}\pi\). [2]

Let \(\text{f}(x) = \frac{e^{2x} + 1}{e^{2x} - 1}\), for \(x > 0\).

1. The equation \(x = \text{f}(x)\) has one root, denoted by \(a\). Verify by calculation that \(a\) lies between 1 and 1.5. [2]
2. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
3. Find f\('(x)\). Hence find the exact value of \(x\) for which f\('(x) = -8\). [6]

\includegraphics{figure_10} The diagram shows the curve \(y = \sin 2x \cos^2 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\).

1. Using the substitution \(u = \sin x\), find the exact area of the region bounded by the curve and the \(x\)-axis. [5]
2. Find the exact \(x\)-coordinate of \(M\). [6]