CAIE P3 (Pure Mathematics 3) 2013 June

Solve the equation \(|x - 2| = |\frac{1}{3}x|\). [3]

The sequence of values given by the iterative formula $$x_{n+1} = \frac{x_n(x_n^2 + 100)}{2(x_n^2 + 25)},$$ with initial value \(x_1 = 3.5\), converges to \(\alpha\).

1. Use this formula to calculate \(\alpha\) correct to 4 decimal places, showing the result of each iteration to 6 decimal places. [3]
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\). [2]

\includegraphics{figure_3} The variables \(x\) and \(y\) satisfy the equation \(y = Ae^{-kx^2}\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(x^2\) is a straight line passing through the points \((0.64, 0.76)\) and \((1.69, 0.32)\), as shown in the diagram. Find the values of \(A\) and \(k\) correct to 2 decimal places. [5]

The polynomial \(ax^3 - 20x^2 + x + 3\), where \(a\) is a constant, is denoted by \(\text{p}(x)\). It is given that \((3x + 1)\) is a factor of \(\text{p}(x)\).

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

\includegraphics{figure_5} The diagram shows the curve with equation $$x^3 + xy^2 + ay^2 - 3ax^2 = 0,$$ where \(a\) is a positive constant. The maximum point on the curve is \(M\). Find the \(x\)-coordinate of \(M\) in terms of \(a\). [6]

1. By differentiating \(\frac{1}{\cos x}\), show that the derivative of \(\sec x\) is \(\sec x \tan x\). Hence show that if \(y = \ln(\sec x + \tan x)\) then \(\frac{dy}{dx} = \sec x\). [4]
2. Using the substitution \(x = (\sqrt{3}) \tan \theta\), find the exact value of $$\int_1^3 \frac{1}{\sqrt{(3 + x^2)}} dx,$$ expressing your answer as a single logarithm. [4]

1. By first expanding \(\cos(x + 45°)\), express \(\cos(x + 45°) - (\sqrt{2}) \sin x\) in the form \(R \cos(x + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places. [5]
2. Hence solve the equation $$\cos(x + 45°) - (\sqrt{2}) \sin x = 2,$$ for \(0° < x < 360°\). [4]

1. Express \(\frac{1}{x^2(2x + 1)}\) in the form \(\frac{A}{x^2} + \frac{B}{x} + \frac{C}{2x + 1}\). [4]
2. The variables \(x\) and \(y\) satisfy the differential equation $$y = x^2(2x + 1)\frac{dy}{dx},$$ and \(y = 1\) when \(x = 1\). Solve the differential equation and find the exact value of \(y\) when \(x = 2\). Give your value of \(y\) in a form not involving logarithms. [7]

1. The complex number \(w\) is such that \(\text{Re } w > 0\) and \(w + 3w^* = iw^2\), where \(w^*\) denotes the complex conjugate of \(w\). Find \(w\), giving your answer in the form \(x + iy\), where \(x\) and \(y\) are real. [5]
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(|z - 2i| \leq 2\) and \(0 \leq \arg(z + 2) \leq \frac{1}{4}\pi\). Calculate the greatest value of \(|z|\) for points in this region, giving your answer correct to 2 decimal places. [6]

The points \(A\) and \(B\) have position vectors \(\mathbf{2i - 3j + 2k}\) and \(\mathbf{5i - 2j + k}\) respectively. The plane \(p\) has equation \(x + y = 5\).

1. Find the position vector of the point of intersection of the line through \(A\) and \(B\) and the plane \(p\). [4]
2. A second plane \(q\) has an equation of the form \(x + by + cz = d\), where \(b\), \(c\) and \(d\) are constants. The plane \(q\) contains the line \(AB\), and the acute angle between the planes \(p\) and \(q\) is \(60°\). Find the equation of \(q\). [7]