WJEC Further Unit 1 (Further Unit 1) 2019 June

1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 3 & 7
   - 2 & 0 \end{array} \right)\), \(\mathbf { B } = \left( \begin{array} { l l } 5 & 1
   0 & 4 \end{array} \right)\).

The matrix \(\mathbf { X }\) is such that \(\mathbf { A X } = \mathbf { B }\). Showing all your working, find the matrix \(\mathbf { X }\).

2. The position vectors of the points \(A , B , C , D\) are given by
\(\mathbf { a } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k }\),
\(\mathbf { b } = 4 \mathbf { j } + 5 \mathbf { k }\),
\(\mathbf { c } = 7 \mathbf { i } - 3 \mathbf { k }\),
\(\mathbf { d } = - 3 \mathbf { i } - \mathbf { j } - 5 \mathbf { k }\),
respectively.

1. Find the vector equations of the lines \(A B\) and \(C D\).
2. Determine whether or not the lines \(A B\) and \(C D\) are perpendicular.

3. The complex numbers \(z\) and \(w\) are represented by the points \(Z\) and \(W\) in an Argand diagram. The complex number \(z\) is such that \(| z | = 6\) and \(\arg z = \frac { \pi } { 3 }\).
The point \(W\) is a \(90 ^ { \circ }\) clockwise rotation, about the origin, of the point \(Z\) in the Argand diagram.

1. Express \(z\) and \(w\) in the form \(x + \mathrm { i } y\).
2. Find the complex number \(\frac { z } { w }\).

4. Prove, by mathematical induction, that \(9 ^ { n } + 15\) is a multiple of 8 for all positive integers \(n\).

5. Given that \(x = - \frac { 1 } { 2 }\) and \(x = - 3\) are two roots of the equation $$2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 23 x + 15 = 0$$ find the remaining roots.

6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 1 | = | z - 2 \mathrm { i } |$$ show that the locus of \(P\) is a straight line.

7. (a) Find an expression for \(\sum _ { r = 1 } ^ { 2 m } ( r + 2 ) ^ { 2 }\) in the form \(\frac { 1 } { 3 } m \left( a m ^ { 2 } + b m + c \right)\), where \(a , b , c\) are integers whose values are to be determined.
(b) Hence, calculate \(\sum _ { r = 1 } ^ { 20 } ( r + 2 ) ^ { 2 }\).

8. The plane \(\Pi\) contains the three points \(A ( 3,5,6 ) , B ( 5 , - 1,7 )\) and \(C ( - 1,7,0 )\). Find the vector equation of the plane \(\Pi\) in the form r.n \(= d\).
Express this equation in Cartesian form.

9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively in Argand diagrams and $$w = z ^ { 2 } - 1$$

1. Show that \(v = 2 x y\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 3 x\). Find the equation of the locus of \(Q\).

10. The quadratic equation \(p x ^ { 2 } + q x + r = 0\) has roots \(\alpha\) and \(\beta\), where \(p , q , r\) are non-zero constants.

1. A cubic equation is formed with roots \(\alpha , \beta , \alpha + \beta\). Find the cubic equation with coefficients expressed in terms of \(p , q , r\).
2. Another quadratic equation \(p x ^ { 2 } - q x - r = 0\) has roots \(2 \alpha\) and \(\gamma\). Show that \(\beta = - 2 \gamma\).