WJEC Further Unit 1 (Further Unit 1) 2024 June

1. The complex numbers \(z , v\) and \(w\) are related by the equation

$$z = \frac { v } { w }$$ Given that \(v = - 16 + 11 \mathrm { i }\) and \(w = 5 + 2 \mathrm { i }\), find \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).

2. Given that \(x ^ { 2 } + 4 x + 5\) is a factor of \(x ^ { 3 } + x ^ { 2 } - 7 x - 15\), solve the equation \(x ^ { 3 } + x ^ { 2 } - 7 x - 15 = 0\).

3. The quadratic equation \(x ^ { 2 } + p x + q = 0\) has a repeated root \(\alpha\). A new quadratic equation has a repeated root \(\frac { 1 } { \alpha }\) and is of the form \(x ^ { 2 } + m x + m = 0\).
Find the values of \(p\) and \(q\) in the original equation.
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1. The complex numbers \(z\) and \(w\) are represented, respectively, by the points \(P ( x , y )\) and \(Q ( u , v )\) in Argand diagrams and

$$w = \frac { z } { 1 - z }$$

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

5. Given that $$\sum _ { r = k } ^ { 76 } ( r - 31 ) = 980$$ show that there are two possible values of \(k\).
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1. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram.

Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), are given by: $$\begin{aligned} & L _ { 1 } : | z - 2 + \mathrm { i } | = | z + 2 - 3 \mathrm { i } |
& L _ { 2 } : | z - 2 + \mathrm { i } | = \sqrt { 10 } \end{aligned}$$

1. Find the coordinates of the points of intersection of these loci.
2. On the same Argand diagram, sketch the loci \(L _ { 1 }\) and \(L _ { 2 }\). Clearly label the coordinates of the points of intersection.

7. Prove, by mathematical induction, that \(13 ^ { ( 2 n - 1 ) } + 8\) is a multiple of 7 for all positive integers \(n\).
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1. A point \(P\) is reflected in the line \(y = k x\), where \(k\) is a constant. It is then rotated anticlockwise about \(O\) through an acute angle \(\theta\), where \(\cos \theta = 0 \cdot 8\). The resulting transformation matrix is given by \(T\), where

$$T = \frac { 1 } { 85 } \left[ \begin{array} { r r } - 84 & - 13
- 13 & 84 \end{array} \right]$$

1. Determine the value of \(k\).
   Find the invariant points of \(T\).

9. Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), are defined by $$\begin{aligned} & \Pi _ { 1 } : 4 x - 3 y + 2 z = 5
& \Pi _ { 2 } : 6 x + y + z = 9 \end{aligned}$$

1. Find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Give your answer correct to three significant figures.
2. Find the perpendicular distance from the point \(A ( 5 , - 2 , - 6 )\) to the plane \(\Pi _ { 1 }\).
3. 1. Show that the point \(B ( 5,5,0 )\) lies on \(\Pi _ { 1 }\) and that the point \(C ( 1,3,0 )\) lies on \(\Pi _ { 2 }\).
   2. State an equation of a plane that contains the points \(B\) and \(C\).
      Additional page, if required. Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}