CAIE FP1 (Further Pure Mathematics 1) 2018 November

The roots of the cubic equation $$x^3 - 5x^2 + 13x - 4 = 0$$ are \(\alpha, \beta, \gamma\).

1. Find the value of \(\alpha^2 + \beta^2 + \gamma^2\). [3]
2. Find the value of \(\alpha^3 + \beta^3 + \gamma^3\). [2]

It is given that $$\mathbf{A} = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -2 & 1 \\ 0 & 0 & 1 \end{pmatrix}$$

1. Find the eigenvalue of \(\mathbf{A}\) corresponding to the eigenvector \(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\). [1]
2. Write down the negative eigenvalue of \(\mathbf{A}\) and find a corresponding eigenvector. [3]
3. Find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf{A} + \mathbf{A}^6\). [2]

The curve \(C\) has polar equation \(r = a \cos 3\theta\), for \(-\frac{1}{6}\pi \leqslant \theta \leqslant \frac{1}{6}\pi\), where \(a\) is a positive constant.

1. Sketch \(C\). [2]
2. Find the area of the region enclosed by \(C\), showing full working. [3]
3. Using the identity \(\cos 3\theta \equiv 4\cos^3 \theta - 3\cos \theta\), find a cartesian equation of \(C\). [3]

1. Find the general solution of the differential equation $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 2\frac{\mathrm{d}x}{\mathrm{d}t} + x = 4\sin t.$$ [7]
2. State an approximate solution for large positive values of \(t\). [1]

The linear transformation \(\mathrm{T} : \mathbb{R}^4 \to \mathbb{R}^4\) is represented by the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 3 & 2 & 0 & 1 \\ 6 & 5 & -1 & 3 \\ 9 & 8 & -2 & 5 \\ -3 & -2 & 0 & -1 \end{pmatrix}.$$

1. Find the rank of \(\mathbf{M}\). [3]

Let \(K\) be the null space of \(\mathrm{T}\).

1. Find a basis for \(K\). [3]
2. Find the general solution of $$\mathbf{M}\mathbf{x} = \begin{pmatrix} 2 \\ 5 \\ 8 \\ -2 \end{pmatrix}.$$ [3]

It is given that \(y = e^x u\), where \(u\) is a function of \(x\). The \(r\)th derivatives \(\frac{\mathrm{d}^r y}{\mathrm{d}x^r}\) and \(\frac{\mathrm{d}^r u}{\mathrm{d}x^r}\) are denoted by \(y^{(r)}\) and \(u^{(r)}\) respectively. Prove by mathematical induction that, for all positive integers \(n\), $$y^{(n)} = e^x\left[\binom{n}{0}u + \binom{n}{1}u^{(1)} + \binom{n}{2}u^{(2)} + \ldots + \binom{n}{r}u^{(r)} + \ldots + \binom{n}{n}u^{(n)}\right].$$ [8] [You may use without proof the result \(\binom{k}{r} + \binom{k}{r-1} = \binom{k+1}{r}\).]

Let $$S_N = \sum_{r=1}^{N}(3r + 1)(3r + 4) \quad \text{and} \quad T_N = \sum_{r=1}^{N}\frac{1}{(3r + 1)(3r + 4)}.$$

1. Use standard results from the List of Formulae (MF10) to show that $$S_N = N(3N^2 + 12N + 13).$$ [3]
2. Use the method of differences to show that $$T_N = \frac{1}{12} - \frac{1}{3(3N + 4)}.$$ [3]
3. Deduce that \(\frac{S_N}{T_N}\) is an integer. [2]
4. Find \(\lim_{N \to \infty} \frac{S_N}{N^3 T_N}\). [2]

1. By considering the binomial expansion of \(\left(z + \frac{1}{z}\right)^6\), where \(z = \cos \theta + \mathrm{i} \sin \theta\), express \(\cos^6 \theta\) in the form $$\frac{1}{32}(p + q \cos 2\theta + r \cos 4\theta + s \cos 6\theta),$$ where \(p, q, r\) and \(s\) are integers to be determined. [6]
2. Hence find the exact value of $$\int_{-\frac{1}{4}\pi}^{\frac{1}{4}\pi} \cos^6\left(\frac{1}{2}x\right) \mathrm{d}x.$$ [4]

The curve \(C\) has equation $$y = \frac{5x^2 + 5x + 1}{x^2 + x + 1}.$$

1. Find the equation of the asymptote of \(C\). [2]
2. Show that, for all real values of \(x\), \(-\frac{1}{5} \leqslant y < 5\). [4]
3. Find the coordinates of any stationary points of \(C\). [2]
4. Sketch \(C\), stating the coordinates of any intersections with the \(y\)-axis. [2]

The position vectors of the points \(A, B, C, D\) are $$\mathbf{i} + \mathbf{j} + 3\mathbf{k}, \quad 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}, \quad -\mathbf{i} + 3\mathbf{k}, \quad m\mathbf{j} + 4\mathbf{k},$$ respectively, where \(m\) is a constant.

1. Show that the lines \(AB\) and \(CD\) are parallel when \(m = \frac{3}{2}\). [1]
2. Given that \(m \neq \frac{3}{2}\), find the shortest distance between the lines \(AB\) and \(CD\). [5]
3. When \(m = 2\), find the acute angle between the planes \(ABC\) and \(ABD\), giving your answer in degrees. [6]

Answer only one of the following two alternatives. EITHER The curve \(C\) is defined parametrically by $$x = 18t - t^2 \quad \text{and} \quad y = 8t^{\frac{1}{2}},$$ where \(0 < t \leqslant 4\).

1. Show that at all points of \(C\), $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = \frac{-3(9 + t)}{2t^2(9 - t)^3}.$$ [4]
2. Show that the mean value of \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) with respect to \(x\) over the interval \(0 < x \leqslant 56\) is \(\frac{3}{70}\). [4]
3. Find the area of the surface generated when \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis, showing full working. [6]

OR Let \(I_n = \int_1^{\sqrt{2}} (x^2 - 1)^n \mathrm{d}x\).

1. Show that, for \(n \geqslant 1\), $$(2n + 1)I_n = \sqrt{2} - 2nI_{n-1}.$$ [5]
2. Using the substitution \(x = \sec \theta\), show that $$I_n = \int_0^{\frac{1}{4}\pi} \tan^{2n+1} \theta \sec \theta \, \mathrm{d}\theta.$$ [4]
3. Deduce the exact value of $$\int_0^{\frac{1}{4}\pi} \frac{\sin^7 \theta}{\cos^8 \theta} \, \mathrm{d}\theta.$$ [5]