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Date	May 2021	Marks available	1	Reference code	21M.1.AHL.TZ1.5
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Find	Question number	5	Adapted from	N/A

Question

A garden has a triangular sunshade suspended from three points $A (2, 0, 2), B (8, 0, 2)$ $A (2, 0, 2), B (8, 0, 2)$ and $C (5, 4, 3)$ $C (5, 4, 3)$ , relative to an origin in the corner of the garden. All distances are measured in metres.

Find $- - \to CA$ $\vec{CA}$ .

[1]

a.i.

Find $- - \to CB$ $\vec{CB}$ .

[1]

a.ii.

Find $- - \to CA \times - - \to CB$ $\vec{CA} \times \vec{CB}$ .

[2]

Hence find the area of the triangle $ABC$ $ABC$ .

[2]

Markscheme

$- - \to CA = ⎛ ⎜ ⎝ \begin{matrix} - 3 - 4 - 1 \end{matrix} ⎞ ⎟ ⎠$ $\vec{CA} = (\begin{matrix} - 3 \\ - 4 \\ - 1 \end{matrix})$ A1

[1 mark]

a.i.

$- - \to CB = ⎛ ⎜ ⎝ \begin{matrix} 3 - 4 - 1 \end{matrix} ⎞ ⎟ ⎠$ $\vec{CB} = (\begin{matrix} 3 \\ - 4 \\ - 1 \end{matrix})$ A1

[1 mark]

a.ii.

$- - \to CA \times - - \to CB = ⎛ ⎜ ⎝ \begin{matrix} 0 - 6 24 \end{matrix} ⎞ ⎟ ⎠$ $\vec{CA} \times \vec{CB} = (\begin{matrix} 0 \\ - 6 \\ 24 \end{matrix})$ (M1)A1

Note: Do not award (M1) if less than 2 entries are correct.

[2 marks]

area is $\frac{1}{2} \sqrt{6^{2} + 24^{2}} = 12.4 m^{2} (12.3693 \dots, 3 \sqrt{17})$ $\frac{1}{2} \sqrt{6^{2} + 24^{2}} = 12.4 m^{2} (12.3693 \dots, 3 \sqrt{17})$ (M1)A1

[2 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.10—Vector definitions

17N.1.AHL.TZ0.H_9a.i:
Find, in terms of a and b $- - \to OF$ $\overrightarrow {{\text{OF}}}$ .
22M.1.AHL.TZ1.6a:
Find $- - \to AF$ $\vec{AF}$ .
22M.1.AHL.TZ1.6b:
Find the length of the rope.
22M.1.AHL.TZ1.6c:
Find $FÂO$ $FÂO$ , the angle the rope makes with the platform.
21N.1.AHL.TZ0.16a:
Find the position vector $- \to OS$ $\vec{OS}$ of the ship at time $t$ $t$ hours.
17N.1.AHL.TZ0.H_9c:
Show that $μ = \frac{1}{13}$ $\mu = \frac{1}{{13}}$ , and find the value of $λ$ $\lambda$ .
17N.1.AHL.TZ0.H_9b.ii:
Find an expression for $- - \to OD$ $\overrightarrow {{\text{OD}}}$ in terms of a, b and $μ$ $\mu$ .
20N.1.SL.TZ0.S_9b:
Find the value of $m$ $m$ .
20N.1.SL.TZ0.S_9c:
Consider a unit vector $u$ , such that $u = p i - \frac{2}{3} j + \frac{1}{3} k$ , where $p > 0$ .

Point $C$ is such that $\vec{BC} = 9 u$ .

Find the coordinates of $C$ .
17N.1.AHL.TZ0.H_9b.i:
Find an expression for $\overrightarrow {{\text{OD}}}$ in terms of a, b and $\lambda$ ;
20N.1.SL.TZ0.S_9a:
Express $\vec{AB}$ in terms of $m$ .
21M.1.AHL.TZ1.5b:
Find $\vec{CA} \times \vec{CB}$ .
21M.1.AHL.TZ1.5a.ii:
Find $\vec{CB}$ .
17N.1.AHL.TZ0.H_9d:
Deduce an expression for $\overrightarrow {{\text{CD}}}$ in terms of a and b only.
21M.1.AHL.TZ1.5c:
Hence find the area of the triangle $ABC$ .
17N.1.AHL.TZ0.H_9e:
Given that area $\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})$ , find the value of $k$ .
17N.1.AHL.TZ0.H_9a.ii:
Find, in terms of a and b $\overrightarrow {{\text{AF}}}$ .
21N.1.AHL.TZ0.16b:
Find the value of $t$ when the ship will be closest to the lighthouse.
21N.1.AHL.TZ0.16c:
An alarm will sound if the ship travels within $20$ kilometres of the lighthouse.

State whether the alarm will sound. Give a reason for your answer.

Topic 3—Geometry and trigonometry

Question

Markscheme

Examiners report

Syllabus sections

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