SPS SPS SM 2023 October — Question 2

Exam BoardSPS
ModuleSPS SM (SPS SM)
Year2023
SessionOctober
TopicIndices and Surds

2 In this question you must show detailed reasoning. Express \(\frac { 8 + \sqrt { 7 } } { 2 + \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{808e5492-febe-434f-91b8-9b2888b17fcb-04_704_912_178_694} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of a curve \(C\) and a straight line \(l\).
Given that
  • \(C\) has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic expression in \(x\)
  • \(C\) cuts the \(x\)-axis at 0 and 6
  • \(l\) cuts the \(y\)-axis at 60 and intersects \(C\) at the point \(( 10,80 )\)
    use inequalities to define the region \(R\) shown shaded in Figure 3.
    [0pt] [BLANK PAGE]