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Date	May Specimen paper	Marks available	1	Reference code	SPM.2.AHL.TZ0.4
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	State	Question number	4	Adapted from	N/A

Question

An aircraft’s position is given by the coordinates ( $x$ , $y$ , $z$ ), where $x$ and $y$ are the aircraft’s displacement east and north of an airport, and $z$ is the height of the aircraft above the ground. All displacements are given in kilometres.

The velocity of the aircraft is given as $\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ { - 20} \end{array}} \right)\,{\text{km}}\,{{\text{h}}^{ - 1}}$ .

At 13:00 it is detected at a position 30 km east and 10 km north of the airport, and at a height of 5 km. Let $t$ be the length of time in hours from 13:00.

If the aircraft continued to fly with the velocity given

When the aircraft is 4 km above the ground it continues to fly on the same bearing but adjusts the angle of its descent so that it will land at the point (0, 0, 0).

Write down a vector equation for the displacement, r of the aircraft in terms of $t$ .

[2]

a.

verify that it would pass directly over the airport.

[2]

b.i.

state the height of the aircraft at this point.

[1]

b.ii.

find the time at which it would fly directly over the airport.

[1]

b.iii.

Find the time at which the aircraft is 4 km above the ground.

[2]

c.i.

Find the direct distance of the aircraft from the airport at this point.

[3]

c.ii.

Given that the velocity of the aircraft, after the adjustment of the angle of descent, is $\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ a \end{array}} \right){\text{km}}\,{{\text{h}}^{ - 1}}$ , find the value of $a$ .

[3]

d.

Markscheme

r $= \left( {\begin{array}{*{20}{c}} {30} \\ {10} \\ 5 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ { - 20} \end{array}} \right)\,$ A1A1

[2 marks]

a.

when $x = 0$ , $t = \frac{{30}}{{150}} = 0.2$ M1

EITHER

when $y = 0$ , $t = \frac{{10}}{{150}} = 0.2$ A1

since the two values of $t$ are equal the aircraft passes directly over the airport

OR

$t = 0.2$ , $y = 0$ A1

[2 marks]

b.i.

height = 5 − 0.2 × 20 = 1 km A1

[1 mark]

b.ii.

time 13:12 A1

[1 mark]

b.iii.

$5 - 20t = 4 \Rightarrow t = \frac{1}{{20}}$ (3 minutes) (M1)

time 13:03 A1

[2 marks]

c.i.

displacement is $\left( {\begin{array}{*{20}{c}} {22.5} \\ {7.5} \\ 4 \end{array}} \right)$ A1

distance is $\sqrt {{{22.5}^2} + {{7.5}^2} + {4^2}}$ (M1)

= 24.1 km A1

[3 marks]

c.ii.

METHOD 1

time until landing is $12 - 3 = 9$ minutes M1

height to descend = $4\,{\text{km}}$

$a = \frac{{ - 4}}{{\frac{9}{{60}}}}$ M1

$= - 26.7$ A1

METHOD 2

$\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ a \end{array}} \right) = s\left( {\begin{array}{*{20}{c}} {22.5} \\ {7.5} \\ 4 \end{array}} \right)$ M1

$- 150 = 22.5\,s \Rightarrow s = - \frac{{20}}{3}$ M1

$a = - \frac{{20}}{3} \times 4$

$= - 26.7$ A1

[3 marks]

d.

Examiners report

[N/A]

a.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

b.iii.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

d.

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.11—Vector equation of a line in 2d and 3d

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21N.1.AHL.TZ0.16a:
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18M.1.SL.TZ1.S_9d:
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Find a vector equation of the line that passes through P and Q.
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SPM.2.AHL.TZ0.4c.i:
Find the time at which the aircraft is 4 km above the ground.
17N.1.SL.TZ0.S_9a.i:
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18N.2.SL.TZ0.S_8b.ii:
Show that $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 8 \\ { - 10} \\ { - 1} \end{array}} \right)$ .
19M.1.SL.TZ1.S_2b:
A second line, ${L_2}$ , is parallel to ${L_1}$ and passes through (1, 2, 3).

Write down a vector equation for ${L_2}$ .
SPM.2.AHL.TZ0.4b.i:
verify that it would pass directly over the airport.
17N.1.SL.TZ0.S_9b:
Find the value of $p$ .
18N.2.SL.TZ0.S_8a.ii:
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18M.1.SL.TZ1.S_9b.i:
Find a vector equation for L.
18M.1.SL.TZ1.S_9c.ii:
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20N.1.SL.TZ0.S_9b:
Find the value of $m$ .
20N.1.SL.TZ0.S_9c:
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Point $C$ is such that $\vec{BC} = 9 u$ .

Find the coordinates of $C$ .
18N.2.SL.TZ0.S_8b.i:
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SPM.2.AHL.TZ0.4a:
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19M.2.SL.TZ1.S_9a:
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19M.2.SL.TZ1.S_9d.ii:
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18N.2.SL.TZ0.S_8c:
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Find the area of triangle ABC.
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16N.1.AHL.TZ0.H_8a:
find the value of $a$ ;
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19M.1.SL.TZ1.S_2a:
Find $c$ .
21M.1.AHL.TZ2.8a:
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21M.1.AHL.TZ2.8b:
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21N.1.AHL.TZ0.16c:
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State whether the alarm will sound. Give a reason for your answer.

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Topic 3—Geometry and trigonometry