SPS SPS SM Pure 2023 June — Question 10 5 marks

Exam BoardSPS
ModuleSPS SM Pure (SPS SM Pure)
Year2023
SessionJune
Marks5
TopicCurve Sketching
TypeQuadratic modelling problems
DifficultyModerate -0.3 This is a straightforward application of coordinate geometry requiring students to find a quadratic equation from three given points (symmetry simplifies this to vertex form), then substitute a value to check a height. The setup is clear, the algebra is routine, and no novel problem-solving insight is required—slightly easier than average due to the symmetric setup and standard techniques.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02n Sketch curves: simple equations including polynomials1.02z Models in context: use functions in modelling
\includegraphics{figure_5} \includegraphics{figure_6} A suspension bridge cable \(PQR\) hangs between the tops of two vertical towers, \(AP\) and \(BR\), as shown in Figure 5. A walkway \(AOB\) runs between the bases of the towers, directly under the cable. The towers are 100 m apart and each tower is 24 m high. At the point \(O\), midway between the towers, the cable is 4 m above the walkway. The points \(P\), \(Q\), \(R\), \(A\), \(O\) and \(B\) are assumed to lie in the same vertical plane and \(AOB\) is assumed to be horizontal. Figure 6 shows a symmetric quadratic curve \(PQR\) used to model this cable. Given that \(O\) is the origin,
  1. find an equation for the curve \(PQR\). [3] Lee can safely inspect the cable up to a height of 12 m above the walkway. A defect is reported on the cable at a location 19 m horizontally from one of the towers.
  2. Determine whether, according to the model, Lee can safely inspect this defect. [2]
\includegraphics{figure_5}

\includegraphics{figure_6}

A suspension bridge cable $PQR$ hangs between the tops of two vertical towers, $AP$ and $BR$, as shown in Figure 5.

A walkway $AOB$ runs between the bases of the towers, directly under the cable.

The towers are 100 m apart and each tower is 24 m high.

At the point $O$, midway between the towers, the cable is 4 m above the walkway.

The points $P$, $Q$, $R$, $A$, $O$ and $B$ are assumed to lie in the same vertical plane and $AOB$ is assumed to be horizontal.

Figure 6 shows a symmetric quadratic curve $PQR$ used to model this cable.

Given that $O$ is the origin,

(a) find an equation for the curve $PQR$. [3]

Lee can safely inspect the cable up to a height of 12 m above the walkway.

A defect is reported on the cable at a location 19 m horizontally from one of the towers.

(b) Determine whether, according to the model, Lee can safely inspect this defect. [2]

\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q10 [5]}}
This paper (18 questions)
View full paper
Q1 3 Q2 3 Q3 3 Q4 6 Q5 5 Q6 5 Q7 6 Q8 7 Q9 4 Q10 5 Q11 10 Q12 6 Q13 6 Q14 6 Q15 6 Q16 8 Q17 5 Q18 6