SPS SPS SM Pure 2024 February — Question 11 11 marks

Exam BoardSPS
ModuleSPS SM Pure (SPS SM Pure)
Year2024
SessionFebruary
Marks11
TopicComposite & Inverse Functions
TypeFind inverse function
DifficultyStandard +0.3 This is a multi-part question on inverse functions that is slightly easier than average. Part (a) requires routine algebraic manipulation to find the inverse of a cubic function. Parts (b)-(d) involve finding intercepts, perpendicular bisectors, and a circle equation—all standard techniques. While it has multiple parts (4 marks worth), each step follows predictable methods without requiring novel insight or particularly challenging problem-solving.
Spec1.02v Inverse and composite functions: graphs and conditions for existence1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle
11. The curves \(y = \mathrm { h } ( x )\) and \(y = \mathrm { h } ^ { - 1 } ( x )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(y = \mathrm { h } ( x )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(y = \mathrm { h } ^ { - 1 } ( x )\) crosses the \(x\)-axis at D and the \(y\)-axis at C . \includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-22_789_798_568_242}
  1. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
  2. Determine the coordinates of A, B, C and D.
  3. Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined.
  4. Points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D lie on a circle. Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a , b\) and \(r ^ { 2 }\) are constants to be determined.
11.

The curves $y = \mathrm { h } ( x )$ and $y = \mathrm { h } ^ { - 1 } ( x )$, where $\mathrm { h } ( x ) = x ^ { 3 } - 8$, are shown below.\\
The curve $y = \mathrm { h } ( x )$ crosses the $x$-axis at B and the $y$-axis at A.\\
The curve $y = \mathrm { h } ^ { - 1 } ( x )$ crosses the $x$-axis at D and the $y$-axis at C .\\
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-22_789_798_568_242}
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\mathrm { h } ^ { - 1 } ( x )$.
\item Determine the coordinates of A, B, C and D.
\item Determine the equation of the perpendicular bisector of AB . Give your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be determined.
\item Points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D lie on a circle.

Determine the equation of the circle. Give your answer in the form $( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$, where $a , b$ and $r ^ { 2 }$ are constants to be determined.
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q11 [11]}}
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