5
\end{array} \right) + \lambda \left( \begin{array} { l }
2
2
3
\end{array} \right)$$
- Find the acute angle between \(\Pi\) and \(l\).
- Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
- \(S\) is the point \(( 4,5 , - 5 )\). Find the shortest distance from \(S\) to \(\Pi\).
2 The complex number \(2 + \mathrm { i }\) is denoted by \(z\).
- Show that \(z ^ { 2 } = 3 + 4 \mathrm { i }\).
- Plot the following on the Argand diagram in the Printed Answer Booklet.
- \(z\)
- \(z ^ { 2 }\)
- State the relationship between \(\left| z ^ { 2 } \right|\) and \(| z |\).
- State the relationship between \(\arg \left( z ^ { 2 } \right)\) and \(\arg ( z )\).
3 In this question you must show detailed reasoning.
Use the formula \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to evaluate \(121 ^ { 2 } + 122 ^ { 2 } + 123 ^ { 2 } + \ldots + 300 ^ { 2 }\).
4 You are given that the cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + x + 4 = 0\) has three roots, \(\alpha , \beta\) and \(\gamma\).
By making a suitable substitution to obtain a related cubic equation, determine the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\).
5 In this question you must show detailed reasoning.
An ant starts from a fixed point \(O\) and walks in a straight line for 1.5 s . Its velocity, \(v \mathrm {~cm} \mathrm {~s} ^ { - 1 }\), can be modelled by \(v = \frac { 1 } { \sqrt { 9 - t ^ { 2 } } }\).
By finding the mean value of \(v\) in \(0 \leqslant t \leqslant 1.5\), deduce the average velocity of the ant.