CAIE FP1 (Further Pure Mathematics 1) 2005 November

Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z^5 = -16 + (16\sqrt{3})i,$$ giving each root in the form \(re^{i\theta}\). [4]

The sequence \(u_1, u_2, u_3, \ldots\) is such that \(u_1 = 1\) and $$u_{n+1} = -1 + \sqrt{(u_n + 7)}.$$

1. Prove by induction that \(u_n < 2\) for all \(n \geqslant 1\). [4]
2. Show that if \(u_n = 2 - \varepsilon\), where \(\varepsilon\) is small, then $$u_{n+1} \approx 2 - \frac{1}{6}\varepsilon.$$ [2]

The curve \(C\) has equation $$y = \frac{x^2}{x + \lambda},$$ where \(\lambda\) is a non-zero constant. Obtain the equations of the asymptotes of \(C\). [3] In separate diagrams, sketch \(C\) for the cases where

1. \(\lambda > 0\),
2. \(\lambda < 0\).

[4]

Solve the differential equation $$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 24e^{2x},$$ given that \(y = 1\) and \(\frac{dy}{dx} = 9\) when \(x = 0\). [7]

In the equation $$x^3 + ax^2 + bx + c = 0,$$ the coefficients \(a\), \(b\) and \(c\) are real. It is given that all the roots are real and greater than \(1\).

1. Prove that \(a < -3\). [1]
2. By considering the sum of the squares of the roots, prove that \(a^2 > 2b + 3\). [2]
3. By considering the sum of the cubes of the roots, prove that \(a^3 < -9b - 3c - 3\). [4]

Let $$I_n = \int_0^1 (1 + x^2)^{-n} dx,$$ where \(n \geqslant 1\). By considering \(\frac{d}{dx}(x(1 + x^2)^{-n})\), or otherwise, prove that $$2nI_{n+1} = (2n - 1)I_n + 2^{-n}.$$ [5] Deduce that \(I_3 = \frac{3}{32}\pi + \frac{1}{4}\). [3]

Write down an expression in terms of \(z\) and \(N\) for the sum of the series $$\sum_{n=1}^N 2^{-n}z^n.$$ [2] Use de Moivre's theorem to deduce that $$\sum_{n=1}^{10} 2^{-n}\sin\left(\frac{1}{10}n\pi\right) = \frac{1025\sin\left(\frac{1}{10}\pi\right)}{2560 - 2048\cos\left(\frac{1}{10}\pi\right)}.$$ [6]

Find the coordinates of the centroid of the finite region bounded by the \(x\)-axis and the curve whose equation is $$y = x^2(1 - x).$$ [7] Deduce the coordinates of the centroid of the finite region bounded by the \(x\)-axis and the curve whose equation is $$y = x(1 - x)^2.$$ [2]

The planes \(\Pi_1\) and \(\Pi_2\) have vector equations $$\mathbf{r} = \lambda_1(\mathbf{i} + \mathbf{j} - \mathbf{k}) + \mu_1(2\mathbf{i} - \mathbf{j} + \mathbf{k}) \quad \text{and} \quad \mathbf{r} = \lambda_2(\mathbf{i} + 2\mathbf{j} + \mathbf{k}) + \mu_2(3\mathbf{i} + \mathbf{j} - \mathbf{k})$$ respectively. The line \(l\) passes through the point with position vector \(4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\) and is parallel to both \(\Pi_1\) and \(\Pi_2\). Find a vector equation for \(l\). [6] Find also the shortest distance between \(l\) and the line of intersection of \(\Pi_1\) and \(\Pi_2\). [4]

It is given that the eigenvalues of the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 4 & 1 & -1 \\ -4 & -1 & 4 \\ 0 & -1 & 5 \end{pmatrix},$$ are \(1, 3, 4\). Find a set of corresponding eigenvectors. [4] Write down a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that $$\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1},$$ where \(n\) is a positive integer. [2] Find \(\mathbf{P}^{-1}\) and deduce that $$\lim_{n \to \infty} 4^{-n}\mathbf{M}^n = \begin{pmatrix} -\frac{1}{3} & 0 & -\frac{1}{3} \\ \frac{4}{3} & 0 & \frac{4}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \end{pmatrix}.$$ [5]

Find the rank of the matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 1 & 1 & 2 & 3 \\ 4 & 3 & 5 & 10 \\ 6 & 6 & 13 & 13 \\ 14 & 12 & 23 & 45 \end{pmatrix}.$$ [3] Find vectors \(\mathbf{x_0}\) and \(\mathbf{e}\) such that any solution of the equation $$\mathbf{A}\mathbf{x} = \begin{pmatrix} 0 \\ 2 \\ -1 \\ -3 \end{pmatrix} \quad (*)$$ can be expressed in the form \(\mathbf{x_0} + \lambda\mathbf{e}\), where \(\lambda \in \mathbb{R}\). [5] Hence show that there is no vector which satisfies \((*)\) and has all its elements positive. [3]

Answer only one of the following two alternatives. **EITHER** Show that \(\left(n + \frac{1}{2}\right)^3 - \left(n - \frac{1}{2}\right)^3 \equiv 3n^2 + \frac{1}{4}\). [1] Use this result to prove that \(\sum_{n=1}^N n^2 = \frac{1}{6}N(N + 1)(2N + 1)\). [2] The sums \(S\), \(T\) and \(U\) are defined as follows: \begin{align} S &= 1^2 + 2^2 + 3^2 + 4^2 + \ldots + (2N)^2 + (2N + 1)^2,
T &= 1^2 + 3^2 + 5^2 + 7^2 + \ldots + (2N - 1)^2 + (2N + 1)^2,
U &= 1^2 - 2^2 + 3^2 - 4^2 + \ldots - (2N)^2 + (2N + 1)^2. \end{align} Find and simplify expressions in terms of \(N\) for each of \(S\), \(T\) and \(U\). [5] Hence

1. describe the behaviour of \(\frac{S}{T}\) as \(N \to \infty\), [1]
2. prove that if \(\frac{S}{U}\) is an integer then \(\frac{T}{U}\) is an integer. [3]

**OR** The curves \(C_1\) and \(C_2\) have polar equations $$r = 4\cos\theta \quad \text{and} \quad r = 1 + \cos\theta$$ respectively, where \(-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Show that \(C_1\) and \(C_2\) meet at the points \(A\left(\frac{4}{3}, \alpha\right)\) and \(B\left(\frac{4}{3}, -\alpha\right)\), where \(\alpha\) is the acute angle such that \(\cos\alpha = \frac{1}{3}\). [2]
2. In a single diagram, draw sketch graphs of \(C_1\) and \(C_2\). [3]
3. Show that the area of the region bounded by the arcs \(OA\) and \(OB\) of \(C_1\), and the arc \(AB\) of \(C_2\), is $$4\pi - \frac{1}{3}\sqrt{2} - \frac{13}{2}\alpha.$$ [7]