CAIE P3 (Pure Mathematics 3) 2018 November

Solve the inequality \(3|2x - 1| > |x + 4|\). [4]

Showing all necessary working, solve the equation \(\sin(\theta - 30°) + \cos \theta = 2 \sin \theta\), for \(0° < \theta < 180°\). [4]

1. Find \(\int \frac{\ln x}{x^3} \, dx\). [3]
2. Hence show that \(\int_1^2 \frac{\ln x}{x^3} \, dx = \frac{1}{16}(3 - \ln 4)\). [2]

Showing all necessary working, solve the equation $$\frac{e^x + e^{-x}}{e^x + 1} = 4,$$ giving your answer correct to 3 decimal places. [5]

The equation of a curve is \(y = x \ln(8 - x)\). The gradient of the curve is equal to 1 at only one point, when \(x = a\).

1. Show that \(a\) satisfies the equation \(x = 8 - \frac{8}{\ln(8 - x)}\). [3]
2. Verify by calculation that \(a\) lies between 2.9 and 3.1. [2]
3. Use an iterative formula based on the equation in part (i) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

A certain curve is such that its gradient at a general point with coordinates \((x, y)\) is proportional to \(\frac{y^2}{x}\). The curve passes through the points with coordinates \((1, 1)\) and \((e, 2)\). By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\). [8]

A curve has equation \(y = \frac{3 \cos x}{2 + \sin x}\), for \(-\frac{1}{2}\pi \leqslant x \leqslant \frac{1}{2}\pi\).

1. Find the exact coordinates of the stationary point of the curve. [6]
2. The constant \(a\) is such that \(\int_0^a \frac{3 \cos x}{2 + \sin x} \, dx = 1\). Find the value of \(a\), giving your answer correct to 3 significant figures. [4]

Let \(f(x) = \frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2}\).

1. Express \(f(x)\) in partial fractions. [5]
2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

1. 1. Without using a calculator, express the complex number \(\frac{2 + 6i}{1 - 2i}\) in the form \(x + iy\), where \(x\) and \(y\) are real. [2]
   2. Hence, without using a calculator, express \(\frac{2 + 6i}{1 - 2i}\) in the form \(r(\cos \theta + i \sin \theta)\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\). [3]
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - 3i| \leqslant 1\) and \(\text{Re } z \leqslant 0\), where \(\text{Re } z\) denotes the real part of \(z\). Find the greatest value of \(\arg z\) for points in this region, giving your answer in radians correct to 2 decimal places. [5]

The line \(l\) has equation \(\mathbf{r} = 5\mathbf{i} - 3\mathbf{j} - \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + \mathbf{k})\). The plane \(p\) has equation $$(\mathbf{r} - \mathbf{i} - 2\mathbf{j}) \cdot (3\mathbf{i} + \mathbf{j} + \mathbf{k}) = 0.$$ The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\). [3]
2. Calculate the acute angle between \(l\) and \(p\). [4]
3. Find the equation of the line which lies in \(p\) and intersects \(l\) at right angles. [4]