WJEC Further Unit 1 (Further Unit 1) 2018 June

1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are such that \(\mathbf { A } = \left[ \begin{array} { c c } 4 & 2
   - 1 & - 3 \end{array} \right]\) and \(\mathbf { B } = \left[ \begin{array} { l l } 4 & 2
   2 & 1 \end{array} \right]\).
   1. Explain why \(\mathbf { B }\) has no inverse.
   2. 1. Find the inverse of \(\mathbf { A }\).
      2. Hence, find the matrix \(\mathbf { X }\), where \(\mathbf { A X } = \left[ \begin{array} { c } - 4
         1 \end{array} \right]\).
   3. Prove, by mathematical induction, that \(\sum _ { r = 1 } ^ { n } r ( r + 3 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 5 )\) for all positive integers \(n\).
   4. 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 \(\gamma\).
2. Find the cubic equation with roots \(3 \alpha , 3 \beta , 3 \gamma\). Give your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\), 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 \(r ( \cos \theta + \mathrm { i } \sin \theta )\). Another complex number is defined as \(w = \sqrt { 5 } ( \cos 2 \cdot 68 + \mathrm { i } \sin 2 \cdot 68 )\).
2. Express \(z w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).

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

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

7. 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 \(P\).
2. Give a geometric interpretation of the locus of \(P\).

8. 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 \(T\).
2. Find the equation of the line of fixed points of \(T\).
3. Find \(T ^ { 2 }\) and hence write down \(T ^ { - 1 }\).

9. 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. The vector equation of the line \(L _ { 2 }\) is given by \(\mathbf { r } = 2 \mathbf { i } - 4 \mathbf { j } + \mu ( 4 \mathbf { j } + 7 \mathbf { k } )\).
2. Show that \(L _ { 1 }\) and \(L _ { 2 }\) do not intersect.
3. Find a vector in the direction of the common perpendicular to \(L _ { 1 }\) and \(L _ { 2 }\).