4.
Relative to a fixed origin \(O\),
- the point \(A\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\)
- the point \(B\) has position vector \(7 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\)
- the point \(C\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } - 3 \mathbf { k }\)
- Find \(| \overrightarrow { A B } |\) giving your answer as a simplified surd.
Given that \(A B C D\) is a parallelogram,
find the position vector of the point \(D\).
The point \(E\) is positioned such that
- \(A C E\) is a straight line
- \(A C : C E = 2 : 1\)
- Find the coordinates of the point \(E\).
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\section*{Solutions relying entirely on calculator technology are not acceptable.}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-12_855_1104_340_589}
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\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve \(C\) with equation
$$y = x ^ { 3 } - 10 x ^ { 2 } + 27 x - 23$$
The point \(P ( 5 , - 13 )\) lies on \(C\)
The line \(l\) is the tangent to \(C\) at \(P\)Use differentiation to find the equation of \(l\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.Hence verify that \(l\) meets \(C\) again on the \(y\)-axis.
The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).Use algebraic integration to find the exact area of \(R\).
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[TURN OVER FOR QUESTION 6]