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Date	May Specimen paper	Marks available	3	Reference code	SPM.2.SL.TZ0.5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Sketch	Question number	5	Adapted from	N/A

Question

The braking distance of a vehicle is defined as the distance travelled from where the brakes are applied to the point where the vehicle comes to a complete stop.

The speed, $s\,{\text{m}}\,{{\text{s}}^{ - 1}}$ , and braking distance, $d\,{\text{m}}$ , of a truck were recorded. This information is summarized in the following table.

This information was used to create Model A, where $d$ is a function of $s$ , $s$ ≥ 0.

Model A: $d\left( s \right) = p{s^2} + qs$ , where $p$ , $q \in \mathbb{Z}$

At a speed of $6\,{\text{m}}\,{{\text{s}}^{ - 1}}$ , Model A can be represented by the equation $6p + q = 2$ .

Additional data was used to create Model B, a revised model for the braking distance of a truck.

Model B: $d\left( s \right) = 0.95{s^2} - 3.92s$

The actual braking distance at $20\,{\text{m}}\,{{\text{s}}^{ - 1}}$ is $320\,{\text{m}}$ .

Write down a second equation to represent Model A, when the speed is $10\,{\text{m}}\,{{\text{s}}^{ - 1}}$ .

[2]

a.i.

Find the values of $p$ and $q$ .

[2]

a.ii.

Find the coordinates of the vertex of the graph of $y = d\left( s \right)$ .

[2]

Using the values in the table and your answer to part (b), sketch the graph of $y = d\left( s \right)$ for 0 ≤ $s$ ≤ 10 and −10 ≤ $d$ ≤ 60, clearly showing the vertex.

[3]

Hence, identify why Model A may not be appropriate at lower speeds.

[1]

Use Model B to calculate an estimate for the braking distance at a speed of $20\,{\text{m}}\,{{\text{s}}^{ - 1}}$ .

[2]

Calculate the percentage error in the estimate in part (e).

[2]

It is found that once a driver realizes the need to stop their vehicle, 1.6 seconds will elapse, on average, before the brakes are engaged. During this reaction time, the vehicle will continue to travel at its original speed.

A truck approaches an intersection with speed $s\,{\text{m}}\,{{\text{s}}^{ - 1}}$ . The driver notices the intersection’s traffic lights are red and they must stop the vehicle within a distance of $330\,{\text{m}}$ .

Using model B and taking reaction time into account, calculate the maximum possible speed of the truck if it is to stop before the intersection.

[3]

Markscheme

$p{\left( {10} \right)^2} + q\left( {10} \right) = 60$ M1

$10p + q = 6\left( {100p + 10q = 60} \right)$ A1

[2 marks]

a.i.

$p = 1$ , $q = - 4$ A1A1

Note: If $p$ and $q$ are both incorrect then award M1A0 for an attempt to solve simultaneous equations.

[2 marks]

a.ii.

(2, −4) A1A1

Note: Award A1 for each correct coordinate.
Award A0A1 if parentheses are missing.

[2 marks]

Note: Award A1 for smooth quadratic curve on labelled axes and within correct window.
Award A1 for the curve passing through (0, 0) and (10, 60). Award A1 for the curve passing through their vertex. Follow through from part (b).

[3 marks]

the graph indicates there are negative stopping distances (for low speeds) R1

Note: Award R1 for identifying that a feature of their graph results in negative stopping distances (vertex, range of stopping distances…).

[1 mark]

$0.95 \times {20^2} - 3.92 \times 20$ (M1)

$= 302\,\left( {\text{m}} \right)\,\,\left( {301.6 \ldots } \right)$ A1

[2 marks]

$\left| {\frac{{301.6 - 320}}{{320}}} \right| \times 100$ M1

$= 5.75$ (%) A1

[2 marks]

$330 = 1.6 \times s + 0.95 \times {s^2} - 3.92 \times s$ M1A1

Note: Award M1 for an attempt to find an expression including stopping distance (model B) and reaction distance, equated to 330. Award A1 for a completely correct equation.

$19.9\left( {{\text{m}}\,{{\text{s}}^{ - 1}}} \right)\,\,\left( {19.8988 \ldots } \right)$ A1