10
3
\end{array} \right) + \lambda \left( \begin{array} { c }
2
- 2
1
\end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }
5
2
4
\end{array} \right) + \mu \left( \begin{array} { c }
3
1
- 2
\end{array} \right)
\end{aligned}$$
\(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
- Find the position vector of \(P\).
- Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
[0pt]
[BLANK PAGE]
4. Solve the quadratic equation \(x ^ { 2 } - 4 x - 1 - 12 i = 0\) writing your solutions in the form \(a + b i\).
[0pt]
[8]
[0pt]
[BLANK PAGE]
5. \(\int _ { 1 } ^ { 2 } x ^ { 3 } \ln ( 2 x ) \mathrm { d } x\) can be written in the form \(p \ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\).
6.
(a) Use the binomial expansion, in ascending powers of \(x\), to show that
$$\sqrt { ( 4 - x ) } = 2 - \frac { 1 } { 4 } x + k x ^ { 2 } + \ldots$$
where \(k\) is a rational constant to be found.
A student attempts to substitute \(x = 1\) into both sides of this equation to find an approximate value for \(\sqrt { 3 }\).
(b) State, giving a reason, if the expansion is valid for this value of \(x\).
7. (a) Determine a sequence of transformations which maps the graph of \(y = \cos \theta\) onto the graph of \(y = 3 \cos \theta + 3 \sin \theta\)
Fully justify your answer.
(b) Hence or otherwise find the least value and greatest value of
$$4 + ( 3 \cos \theta + 3 \sin \theta ) ^ { 2 }$$
Fully justify your answer.
[0pt]
[BLANK PAGE]
8.
Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\)
$$f ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$$
is divisible by 7
[0pt]
[BLANK PAGE]
9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15b5fb91-3bc6-4167-afb9-91879ebbfc96-16_595_593_157_822}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where
$$\mathrm { f } ( x ) = 2 | 3 - x | + 5 , \quad x \geqslant 0$$
(a) State the range of f
(b) Solve the equation
$$f ( x ) = \frac { 1 } { 2 } x + 30$$
Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has two distinct roots, (c) state the set of possible values for \(k\).
[0pt]
[BLANK PAGE]
10.
A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram.
The shape of the cross-section of the sculpture can be modelled by the equation \(x ^ { 2 } + 2 x y + 2 y ^ { 2 } = 10\), where \(x\) and \(y\) are measured in metres.
The \(x\) and \(y\) axes are horizontal and vertical respectively.
\includegraphics[max width=\textwidth, alt={}, center]{15b5fb91-3bc6-4167-afb9-91879ebbfc96-18_224_478_667_804}
Find the maximum vertical height above the platform of the sculpture.
[0pt]
[BLANK PAGE]