WJEC Further Unit 1 (Further Unit 1) 2018 June

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that \(\mathbf{A} = \begin{bmatrix} 4 & 2 \\ -1 & -3 \end{bmatrix}\) and \(\mathbf{B} = \begin{bmatrix} 4 & 2 \\ 2 & 1 \end{bmatrix}\).

1. Explain why \(\mathbf{B}\) has no inverse. [1]
2. 1. Find the inverse of \(\mathbf{A}\). [3]
   2. Hence, find the matrix \(\mathbf{X}\), where \(\mathbf{AX} = \begin{bmatrix} -4 \\ 1 \end{bmatrix}\) [2]

Prove, by mathematical induction, that \(\sum_{r=1}^{n} r(r+3) = \frac{1}{3}n(n+1)(n+5)\) for all positive integers \(n\). [6]

A cubic equation has roots \(\alpha\), \(\beta\), \(\gamma\) such that $$\alpha + \beta + \gamma = -9, \quad \alpha\beta + \beta\gamma + \gamma\alpha = 20, \quad \alpha\beta\gamma = 0.$$

1. Find the values of \(\alpha\), \(\beta\) and \(\gamma\). [4]
2. Find the cubic equation with roots \(3\alpha\), \(3\beta\), \(3\gamma\). Give your answer in the form \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), \(d\) are constants to be determined. [4]

A complex number is defined by \(z = -3 + 4\mathrm{i}\).

1. 1. Express \(z\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\), where \(-\pi \leqslant \theta \leqslant \pi\).
   2. Express \(\bar{z}\), the complex conjugate of \(z\), in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\). [4]

Another complex number is defined as \(w = \sqrt{5}(\cos 2.68 + \mathrm{i}\sin 2.68)\).

1. Express \(zw\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\). [3]

1. Show that \(\frac{2}{n-1} - \frac{2}{n+1}\) can be expressed as \(\frac{4}{(n^2-1)}\). [1]
2. Hence, find an expression for \(\sum_{r=2}^{n} \frac{4}{(r^2-1)}\) in the form \(\frac{(an+b)(n+c)}{n(n+1)}\), where \(a\), \(b\), \(c\) are integers whose values are to be determined. [6]
3. Explain why \(\sum_{r=1}^{100} \frac{4}{(r^2-1)}\) cannot be calculated. [1]

1. Show that \(1 - 2\mathrm{i}\) is a root of the cubic equation \(x^3 + 5x^2 - 9x + 35 = 0\). [3]
2. Find the other two roots of the equation. [4]

The complex number \(z\) is represented by the point \(P(x, y)\) in the Argand diagram and $$|z - 4 - \mathrm{i}| = |z + 2|.$$

1. Find the equation of the locus of \(P\). [4]
2. Give a geometric interpretation of the locus of \(P\). [1]

The transformation \(T\) in the plane consists of a translation in which the point \((x, y)\) is transformed to the point \((x - 1, y + 1)\), followed by a reflection in the line \(y = x\).

1. Determine the \(3 \times 3\) matrix which represents \(T\). [4]
2. Find the equation of the line of fixed points of \(T\). [2]
3. Find \(T^2\) and hence write down \(T^{-1}\). [3]

The line \(L_1\) passes through the points \(A(1, 2, -3)\) and \(B(-2, 1, 0)\).

1. 1. Show that the vector equation of \(L_1\) can be written as $$\mathbf{r} = (1 - 3\lambda)\mathbf{i} + (2 - \lambda)\mathbf{j} + (-3 + 3\lambda)\mathbf{k}.$$
   2. Write down the equation of \(L_1\) in Cartesian form. [4]

The vector equation of the line \(L_2\) is given by \(\mathbf{r} = 2\mathbf{i} - 4\mathbf{j} + \mu(4\mathbf{j} + 7\mathbf{k})\).

1. Show that \(L_1\) and \(L_2\) do not intersect. [5]
2. Find a vector in the direction of the common perpendicular to \(L_1\) and \(L_2\). [5]