CAIE FP1 (Further Pure Mathematics 1) 2018 November

The vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) in \(\mathbb{R}^3\) are given by $$\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 2 \\ 9 \\ 0 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 3 \\ 5 \\ 4 \end{pmatrix} \quad \text{and} \quad \mathbf{d} = \begin{pmatrix} 0 \\ -8 \\ -3 \end{pmatrix}.$$

1. Show that \(\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}\) is a basis for \(\mathbb{R}^3\). [3]
2. Express \(\mathbf{d}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\). [2]

The roots of the equation $$x^3 + px^2 + qx + r = 0$$ are \(\alpha\), \(2\alpha\), \(4\alpha\), where \(p\), \(q\), \(r\) and \(\alpha\) are non-zero real constants.

1. Show that $$2p\alpha + q = 0.$$ [4]
2. Show that $$p^3 r - q^3 = 0.$$ [2]

The sequence of positive numbers \(u_1\), \(u_2\), \(u_3\), \(\ldots\) is such that \(u_1 < 3\) and, for \(n \geqslant 1\), $$u_{n+1} = \frac{4u_n + 9}{u_n + 4}.$$

1. By considering \(3 - u_{n+1}\), or otherwise, prove by mathematical induction that \(u_n < 3\) for all positive integers \(n\). [5]
2. Show that \(u_{n+1} > u_n\) for \(n \geqslant 1\). [3]

A curve is defined parametrically by $$x = t - \frac{1}{2}\sin 2t \quad \text{and} \quad y = \sin^2 t.$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\).

1. Show that $$S = a\pi \int_0^\pi \sin^3 t \, dt,$$ where the constant \(a\) is to be found. [5]
2. Using the result \(\sin 3t = 3\sin t - 4\sin^3 t\), find the exact value of \(S\). [3]

It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf{A}\) with \(\mathbf{e}\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf{B}\) for which \(\mathbf{e}\) is also a corresponding eigenvector.

1. Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf{A} + \mathbf{B}\) with \(\mathbf{e}\) as a corresponding eigenvector. [2]

The matrix \(\mathbf{A}\), given by $$\mathbf{A} = \begin{pmatrix} 2 & 0 & 1 \\ -1 & 2 & 3 \\ 1 & 0 & 2 \end{pmatrix}$$ has \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\), \(\begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) as eigenvectors.

1. Find the corresponding eigenvalues. [3]

The matrix \(\mathbf{B}\) has eigenvalues \(4\), \(5\) and \(1\) with corresponding eigenvectors \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\), \(\begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) respectively.

1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A} + \mathbf{B})^3 = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}\). [3]

The curve \(C\) has equation $$y = \frac{x^2 + ax - 1}{x + 1},$$ where \(a\) is constant and \(a > 1\).

1. Find the equations of the asymptotes of \(C\). [3]
2. Show that \(C\) intersects the \(x\)-axis twice. [1]
3. Justifying your answer, find the number of stationary points on \(C\). [2]
4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis. [3]

1. Use de Moivre's theorem to show that $$\sin 8\theta = 8\sin \theta \cos \theta(1 - 10\sin^2 \theta + 24\sin^4 \theta - 16\sin^6 \theta).$$ [6]
2. Use the equation \(\frac{\sin 8\theta}{\sin 2\theta} = 0\) to find the roots of $$16x^6 - 24x^4 + 10x^2 - 1 = 0$$ in the form \(\sin k\pi\), where \(k\) is rational. [4]

The plane \(\Pi_1\) has equation $$\mathbf{r} = \begin{pmatrix} 5 \\ 1 \\ 0 \end{pmatrix} + s\begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix} + t\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}.$$

1. Find a cartesian equation of \(\Pi_1\). [3]

The plane \(\Pi_2\) has equation \(3x + y - z = 3\).

1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\), giving your answer in degrees. [2]
2. Find an equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\). [5]

The curve \(C\) has polar equation $$r = 5\sqrt{\cot \theta},$$ where \(0.01 \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Find the area of the finite region bounded by \(C\) and the line \(\theta = 0.01\), showing full working. Give your answer correct to \(1\) decimal place. [3]

Let \(P\) be the point on \(C\) where \(\theta = 0.01\).

1. Find the distance of \(P\) from the initial line, giving your answer correct to \(1\) decimal place. [2]
2. Find the maximum distance of \(C\) from the initial line. [3]
3. Sketch \(C\). [2]

1. Find the particular solution of the differential equation $$\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 10x = 37\sin 3t,$$ given that \(x = 3\) and \(\frac{dx}{dt} = 0\) when \(t = 0\). [10]
2. Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx \sqrt{(37)}\sin(3t - \phi),$$ where the constant \(\phi\) is such that \(\tan \phi = 6\). [3]

Answer only one of the following two alternatives. EITHER

1. By considering \((2r + 1)^2 - (2r - 1)^2\), use the method of differences to prove that $$\sum_{r=1}^n r = \frac{1}{2}n(n + 1).$$ [3]
2. By considering \((2r + 1)^4 - (2r - 1)^4\), use the method of differences and the result given in part (i) to prove that $$\sum_{r=1}^n r^3 = \frac{1}{4}n^2(n + 1)^2.$$ [5]

The sums \(S\) and \(T\) are defined as follows: $$S = 1^3 + 2^3 + 3^3 + 4^3 + \ldots + (2N)^3 + (2N + 1)^3,$$ $$T = 1^3 + 3^3 + 5^3 + 7^3 + \ldots + (2N - 1)^3 + (2N + 1)^3.$$

1. Use the result given in part (ii) to show that \(S = (2N + 1)^2(N + 1)^2\). [1]
2. Hence, or otherwise, find an expression in terms of \(N\) for \(T\), factorising your answer as far as possible. [2]
3. Deduce the value of \(\frac{S}{T}\) as \(N \to \infty\). [2]

OR The curve \(C\) has equation $$x^2 + 2xy = y^3 - 2.$$

1. Show that \(A(-1, 1)\) is the only point on \(C\) with \(x\)-coordinate equal to \(-1\). [2]

For \(n \geqslant 1\), let \(A_n\) denote the value of \(\frac{d^n y}{dx^n}\) at the point \(A(-1, 1)\).

1. Show that \(A_1 = 0\). [3]
2. Show that \(A_2 = \frac{2}{5}\). [3]

Let \(I_n = \int_{-1}^0 x^n \frac{d^n y}{dx^n} dx\).

1. Show that for \(n \geqslant 2\), $$I_n = (-1)^{n+1} A_{n-1} - nI_{n-1}.$$ [3]
2. Deduce the value of \(I_3\) in terms of \(I_1\). [2]