Complex roots with real coefficients

8 In this question you must show detailed reasoning. You are given that i is a root of the equation \(z ^ { 4 } - 2 z ^ { 3 } + 3 z ^ { 2 } + a z + b = 0\), where \(a\) and \(b\) are real constants.

1. Show that \(a = - 2\) and \(b = 2\).
2. Find the other roots of this equation.

2 The cubic equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ where \(p , q\) and \(r\) are real, has roots \(\alpha , \beta\) and \(\gamma\).

1. Given that $$\alpha + \beta + \gamma = 4 \quad \text { and } \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 20$$ find the values of \(p\) and \(q\).
2. Given further that one root is \(3 + \mathrm { i }\), find the value of \(r\).

3 The cubic equation $$z ^ { 3 } + 2 ( 1 - \mathrm { i } ) z ^ { 2 } + 32 ( 1 + \mathrm { i } ) = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. It is given that \(\alpha\) is of the form \(k \mathrm { i }\), where \(k\) is real. By substituting \(z = k \mathrm { i }\) into the equation, show that \(k = 4\).
2. Given that \(\beta = - 4\), find the value of \(\gamma\).

2 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + 6 z + q = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
2. Given that \(p\) and \(q\) are real and that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 12\) :
   1. explain why the cubic equation has two non-real roots and one real root;
   2. find the value of \(p\).
3. One root of the cubic equation is \(- 1 + 3 \mathrm { i }\). Find:
   1. the other two roots;
   2. the value of \(q\).

3 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + 25 z + q = 0$$ where \(p\) and \(q\) are real, has a root \(\alpha = 2 - 3 \mathrm { i }\).

1. Write down another non-real root, \(\beta\), of this equation.
2. Find:
   1. the value of \(\alpha \beta\);
   2. the third root, \(\gamma\), of the equation;
   3. the values of \(p\) and \(q\).

11 The equation \(x ^ { 3 } - 8 x ^ { 2 } + c x + d = 0\) where \(c\) and \(d\) are real numbers, has roots \(\alpha , \beta , \gamma\).
When plotted on an Argand diagram, the triangle with vertices at \(\alpha , \beta , \gamma\) has an area of 8 . Given \(\alpha = 2\), find the values of \(c\) and \(d\). Fully justify your solution.
[0pt] [5 marks]

12 Abel and Bonnie are trying to solve this mathematical problem: $$\begin{gathered} z = 2 - 3 \mathrm { i } \text { is a root of the equation } \\ 2 z ^ { 3 } + m z ^ { 2 } + p z + 91 = 0 \end{gathered}$$ Find the value of \(m\) and the value of \(p\). Abel says he has solved the problem.
Bonnie says there is not enough information to solve the problem.
12

1. Abel's solution begins as follows: Since \(z = 2 - 3 \mathrm { i }\) is a root of the equation, \(z = 2 + 3 \mathrm { i }\) is another root. State one extra piece of information about \(m\) and \(p\) which could be added to the problem to make the beginning of Abel's solution correct.

   12	Prove that Bonnie is right.

   13(a) Explain why \(\int _ { 3 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - 2 x } \mathrm {~d} x\) is an improper integral. [1 mark] 13(b) Evaluate \(\int _ { 3 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - 2 x } \mathrm {~d} x\) Show the limiting process. [9 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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4 The function f is a quartic function with real coefficients.
The complex number 5i is a root of the equation \(\mathrm { f } ( x ) = 0\)
Which one of the following must be a factor of \(\mathrm { f } ( x )\) ?
Circle your answer.
( \(x ^ { 2 } - 25\) )
\(\left( x ^ { 2 } - 5 \right)\)
\(\left( x ^ { 2 } + 5 \right)\)
\(\left( x ^ { 2 } + 25 \right)\)