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
- 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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