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Date	May 2022	Marks available	2	Reference code	22M.1.AHL.TZ1.6
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Find	Question number	6	Adapted from	N/A

Question

A vertical pole stands on a sloped platform. The bottom of the pole is used as the origin, $O$ , of a coordinate system in which the top, $F$ , of the pole has coordinates $(0, 0, 5.8)$ . All units are in metres.

The pole is held in place by ropes attached at $F$ .

One of these ropes is attached to the platform at point $A (3.2, 4.5, - 0.3)$ . The rope forms a straight line from $A$ to $F$ .

Find $\vec{AF}$ .

[1]

Find the length of the rope.

[2]

Find $FÂO$ , the angle the rope makes with the platform.

[5]

Markscheme

$(\begin{matrix} - 3.2 \\ - 4.5 \\ 6.1 \end{matrix})$ A1

[1 mark]

$\sqrt{{(- 3.2)}^{2} + {(- 4.5)}^{2} + 6 . 1^{2}}$ (M1)

$8.22800 \dots \approx 8.23 m$ A1

[2 marks]

EITHER

$\vec{AO} = (\begin{matrix} - 3.2 \\ - 4.5 \\ 0.3 \end{matrix})$ A1

$\cos θ = \frac{\vec{AO} \cdot \vec{AF}}{|\vec{AO}| |\vec{AF}|}$

$\vec{AO} \cdot \vec{AF} = {(- 3.2)}^{2} + {(- 4.5)}^{2} + (0.3 \times 6.1) (= 32.32)$ (A1)

$\cos θ = \frac{32.32}{\sqrt{3 . 2^{2} + 4 . 5^{2} + 0 . 3^{2}} \times 8.22800 \dots}$ (M1)

$= 0.710326 \dots$ (A1)

Note: If $\vec{OA}$ is used in place of $\vec{AO}$ then $\cos θ$ will be negative.
Award A1(A1)(M1)(A1) as above. In order to award the final A1, some justification for changing the resulting obtuse angle to its supplementary angle must be seen.

$AO = \sqrt{3 . 2^{2} + 4 . 5^{2} + 0 . 3^{2}} (= 5.52991 \dots)$ (A1)

$\cos θ = \frac{8.22800 \dots^{2} + 5.52991 \dots^{2} - 5 . 8^{2}}{2 \times 8.22800 \dots \times 5.52991 \dots}$ (M1)(A1)

$= 0.710326 \dots$ (A1)

THEN

$θ = 0.780833 \dots \approx 0.781$ OR $44.7384 \dots ° \approx 44.7 °$ A1

[5 marks]

Examiners report

Many candidates appeared unfamiliar with the notation $\vec{AF}$ for a vector and interpreted it to mean the magnitude of the vector. This notation is defined in the notation list in the subject guide. Candidates are recommended to be aware of this list and to use it throughout the course; this list indicates notation that will be used in the examination questions without introduction and thus it is important that candidates are familiar. It was also common to see $\vec{FA}$ in place of $\vec{AF}$ . Often, candidates were able to find the magnitude correctly even though they failed to write the vector. Correct answers to part (c) were more elusive. The use of scalar product or cosine rule was common although some incorrectly assumed that $F \hat{A} O$ was a right angle. Other errors were to find one of the other angles of the triangle. Some candidates using the scalar product found the obtuse angle between $\vec{OA}$ and $\vec{AF}$ , but failed to find its supplementary angle. In a question like this, it is important that a suitable level of accuracy is maintained throughout so that early rounding will not affect subsequent parts of the question. Candidates must also learn to final round answers and not to truncate them.

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.1—3d space, volume, angles, midpoints

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Question

Markscheme

Examiners report

Syllabus sections

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