CAIE P3 (Pure Mathematics 3) 2010 June

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

The variables \(x\) and \(y\) satisfy the equation \(y^3 = Ae^{2x}\), where \(A\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line.

1. Find the gradient of this line. [2]
2. Given that the line intersects the axis of \(\ln y\) at the point where \(\ln y = 0.5\), find the value of \(A\) correct to 2 decimal places. [2]

Solve the equation $$\tan(45° - x) = 2\tan x,$$ giving all solutions in the interval \(0° < x < 180°\). [5]

Given that \(x = 1\) when \(t = 0\), solve the differential equation $$\frac{dx}{dt} = \frac{1}{x} - \frac{x}{4},$$ obtaining an expression for \(x^2\) in terms of \(t\). [7]

\includegraphics{figure_5} The diagram shows the curve \(y = e^{-x} - e^{-2x}\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(p\).

1. Find the exact value of \(p\). [4]
2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = p\) is equal to \(\frac{1}{4}\). [4]

The curve \(y = \frac{\ln x}{x + 1}\) has one stationary point.

1. Show that the \(x\)-coordinate of this point satisfies the equation $$x = \frac{x + 1}{\ln x},$$ and that this \(x\)-coordinate lies between 3 and 4. [5]
2. Use the iterative formula $$x_{n+1} = \frac{x_n + 1}{\ln x_n}$$ to determine the \(x\)-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

1. Prove the identity \(\cos 3\theta \equiv 4\cos^3 \theta - 3\cos \theta\). [4]
2. Using this result, find the exact value of $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cos^3 \theta \, d\theta.$$ [4]

1. The equation \(2x^3 - x^2 + 2x + 12 = 0\) has one real root and two complex roots. Showing your working, verify that \(1 + i\sqrt{3}\) is one of the complex roots. State the other complex root. [4]
2. On a sketch of an Argand diagram, show the point representing the complex number \(1 + i\sqrt{3}\). On the same diagram, shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(|z - 1 - i\sqrt{3}| \leq 1\) and \(\arg z \leq \frac{1}{4}\pi\). [5]

1. Express \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in partial fractions. [5]
2. Hence obtain the expansion of \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

The straight line \(l\) has equation \(\mathbf{r} = 2\mathbf{i} - \mathbf{j} - 4\mathbf{k} + \lambda(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})\). The plane \(p\) has equation \(3x - y + 2z = 9\). The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\). [3]
2. Find the acute angle between \(l\) and \(p\). [4]
3. Find an equation for the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(ax + by + cz = d\). [5]