5
- 3
- 4
\end{array} \right) \quad \text { and } \quad \mathbf { n } = \left( \begin{array} { r }
3
- 1
2
\end{array} \right)$$
where \(O\) is the origin,
- find a Cartesian equation of \(\Pi\).
With respect to the fixed origin \(O\), the line \(l\) is given by the equation
$$\mathbf { r } = \left( \begin{array} { r }
7
3
- 2
\end{array} \right) + \lambda \left( \begin{array} { r }
- 1
- 5
3
\end{array} \right)$$
The line \(l\) intersects the plane \(\Pi\) at the point \(X\). - Show that the acute angle between the plane \(\Pi\) and the line \(l\) is \(21.2 ^ { \circ }\) correct to one decimal place.
- Find the coordinates of the point \(X\).
- Tyler invested a total of \(\pounds 5000\) across three different accounts; a savings account, a property bond account and a share dealing account.
Tyler invested \(\pounds 400\) more in the property bond account than in the savings account.
After one year
- the savings account had increased in value by \(1.5 \%\)
- the property bond account had increased in value by \(3.5 \%\)
- the share dealing account had decreased in value by \(2.5 \%\)
- the total value across Tyler's three accounts had increased by \(\pounds 79\)
Form and solve a matrix equation to find out how much money was invested by Tyler in each account.
- The cubic equation
$$x ^ { 3 } + 3 x ^ { 2 } - 8 x + 6 = 0$$
has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are \(( \alpha - 1 ) , ( \beta - 1 )\) and \(( \gamma - 1 )\), giving your answer in the form \(w ^ { 3 } + p w ^ { 2 } + q w + r = 0\), where \(p , q\) and \(r\) are integers to be found.
(5)
5.
$$\mathbf { M } = \left( \begin{array} { c c }
1 & - \sqrt { 3 }
\sqrt { 3 } & 1
\end{array} \right)$$