Parametric polynomials with root conditions

5

1. Solve the equation \(z ^ { 2 } - 2 p \mathrm { i } z - q = 0\), where \(p\) and \(q\) are real constants.
   In an Argand diagram with origin \(O\), the roots of this equation are represented by the distinct points \(A\) and \(B\).
2. Given that \(A\) and \(B\) lie on the imaginary axis, find a relation between \(p\) and \(q\).
3. Given instead that triangle \(O A B\) is equilateral, express \(q\) in terms of \(p\).

7

1. Solve the equation \(x ^ { 2 } + 4 x + 20 = 0\), giving your answers in the form \(c + d \mathrm { i }\), where \(c\) and \(d\) are integers.
2. The roots of the quadratic equation $$z ^ { 2 } + ( 4 + i + q i ) z + 20 = 0$$ are \(w\) and \(w ^ { * }\).
   1. In the case where \(q\) is real, explain why \(q\) must be - 1 .
   2. In the case where \(w = p + 2 \mathrm { i }\), where \(p\) is real, find the possible values of \(q\).
      [0pt] [5 marks] \(8 \quad\) The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { l l } 2 & 0
      0 & 1 \end{array} \right]\).
3. 1. Find the matrix \(\mathbf { A } ^ { 2 }\).
   2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
4. Given that the matrix \(\mathbf { B }\) represents a reflection in the line \(x + \sqrt { 3 } y = 0\), find the matrix \(\mathbf { B }\), giving the exact values of any trigonometric expressions.
5. Hence find the coordinates of the point \(P\) which is mapped onto \(( 0 , - 4 )\) under the transformation represented by \(\mathbf { A } ^ { 2 }\) followed by a reflection in the line \(x + \sqrt { 3 } y = 0\).
   [0pt] [6 marks] \(9 \quad\) A curve \(C\) has equation \(y = \frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) }\).
   The line \(L\) has equation \(y = \frac { 1 } { 2 } ( x - 1 )\).
6. Write down the equations of the asymptotes of \(C\).
7. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\).
8. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes.
9. Hence solve the inequality \(\frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) } \geqslant \frac { 1 } { 2 } ( x - 1 )\).

9 In this question you must show detailed reasoning. You are given that \(a\) is a real root of the equation \(x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } - 5 x = 0\).
You are also given that \(a + 2 + 3 \mathrm { i }\) is one root of the equation
\(z ^ { 4 } - 2 ( 1 + a ) z ^ { 3 } + ( 21 a - 10 ) z ^ { 2 } + ( 86 - 80 a ) z + ( 285 a - 195 ) = 0\). Determine all possible values of \(z\).

6 You are given that the cubic equation \(2 x ^ { 3 } + p x ^ { 2 } + q x - 3 = 0\), where \(p\) and \(q\) are real numbers, has a complex root \(\alpha = 1 + i \sqrt { 2 }\).

1. Write down a second complex root, \(\beta\).
2. Determine the third root, \(\gamma\).
3. Find the value of \(p\) and the value of \(q\).
4. Show that if \(n\) is an integer then \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 \times 3 ^ { \frac { 1 } { 2 } n } \times \cos n \theta + \frac { 1 } { 2 ^ { n } }\) where \(\tan \theta = \sqrt { 2 }\).