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Date	May 2021	Marks available	4	Reference code	21M.2.SL.TZ1.8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Show that	Question number	8	Adapted from	N/A

Question

Two straight fences meet at point $A$ $A$ and a field lies between them.

A horse is tied to a post, $P$ $P$ , by a rope of length $r$ $r$ metres. Point $D$ $D$ is on one fence and point $E$ $E$ is on the other, such that $PD = PE = PA = r$ $PD = PE = PA = r$ and $D ˆ P E = θ$ $D \hat{P} E = θ$ radians. This is shown in the following diagram.

The length of the arc $DE$ $DE$ shown in the diagram is $28 m$ $28 m$ .

A new fence is to be constructed between points $B$ $B$ and $C$ $C$ which will enclose the field, as shown in the following diagram.

Point $C$ $C$ is due west of $B$ $B$ and $AC = 800 m$ $AC = 800 m$ . The bearing of $B$ $B$ from $A$ $A$ is $195 °$ $195 °$ .

Write down an expression for $r$ $r$ in terms of $θ$ $θ$ .

[1]

a.

Show that the area of the field that the horse can reach is $\frac{392}{θ^{2}} (θ + sin θ)$ $\frac{392}{θ^{2}} (θ + \sin θ)$ .

[4]

b.

The area of field that the horse can reach is $460 m^{2}$ $460 m^{2}$ . Find the value of $θ$ $θ$ .

[2]

c.

Hence, find the size of $D ˆ A E$ $D \hat{A} E$ .

[2]

d.

Find the size of $A ˆ B C$ $A \hat{B} C$ .

[2]

e.i.

Find the length of new fence required.

[3]

e.ii.

Markscheme

$r = \frac{28}{θ}$ $r = \frac{28}{θ}$ A1

[1 mark]

a.

recognising sum of area of sector and area of triangle required (M1)

$\frac{1}{2} r^{2} θ + \frac{1}{2} r \times r \times sin (π - θ) (= \frac{r^{2}}{2} (θ + sin (π - θ))$ $\frac{1}{2} r^{2} θ + \frac{1}{2} r \times r \times \sin (π - θ) (= \frac{r^{2}}{2} (θ + \sin (π - θ))$ A1

$sin (π - θ) = sin θ$ $\sin (π - θ) = \sin θ$ (substitution seen anywhere) A1

$\frac{1}{2} {(\frac{28}{θ})}^{2} θ + \frac{1}{2} {(\frac{28}{θ})}^{2} \sin θ$ OR $\frac{1}{2} {(\frac{28}{θ})}^{2} (θ + \sin θ)$ A1

area $= \frac{392}{θ^{2}} (θ + \sin θ)$ AG

[4 marks]

b.

$\frac{392}{θ^{2}} (θ + \sin θ) = 460$ (M1)

$θ = 1.43917 \dots$

$θ = 1.44$ A1

[2 marks]

c.

$\frac{π - (π - θ)}{2}$ OR $\frac{θ}{2}$ (M1)

$D \hat{A} E = 0.719588 \dots$

$D \hat{A} E = 0.720$ A1

[2 marks]

d.

$A \hat{B} C = 195 - 180 + 90$ (A1)

$= 105 °$ A1

[2 marks]

e.i.

choosing sine rule (M1)

$\frac{BC}{sin D \hat{A} E} = \frac{800}{\sin 105}$ OR $\frac{BC}{sin D \hat{A} E} = \frac{800}{\sin 1.83}$ A1

$BC = 546 (m)$ A1

[2 marks]

e.ii.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.

[N/A]

e.i.

[N/A]

e.ii.

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.2—2d and 3d trig, sine rule, cosine rule, area

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19M.1.SL.TZ1.T_8a:
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18N.2.AHL.TZ0.H_11c:
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19M.2.SL.TZ2.S_3b:
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16N.1.SL.TZ0.T_11b:
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17M.1.SL.TZ2.T_3b:
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19N.2.SL.TZ0.S_6:
The diagram below shows a triangular-based pyramid with base ${\text{ADC}}$ .
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19N.1.SL.TZ0.S_9a.ii:
$\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}}$ .
22M.2.SL.TZ1.7b:
Given that $A \hat{V} B = 40 °$ , find $AB$ .
22M.2.SL.TZ2.7b:
Find the area of this piece of land.
22M.2.SL.TZ2.7c:
Find $CÂB$ .
22M.2.SL.TZ2.7d:
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20N.1.SL.TZ0.S_2a:
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20N.1.SL.TZ0.S_2b:
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21M.2.SL.TZ1.8e.ii:
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SPM.1.SL.TZ0.1b:
Find the area of triangle ABC.
18N.2.SL.TZ0.S_7a:
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16N.2.SL.TZ0.T_5d:
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21M.2.SL.TZ1.8a:
Write down an expression for $r$ in terms of $θ$ .
21M.2.SL.TZ1.8e.i:
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21M.3.AHL.TZ1.2c:
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21M.3.AHL.TZ1.2d:
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17M.2.AHL.TZ2.H_4a:
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18M.1.SL.TZ1.T_8b:
Find the size of angle APB.
19N.1.SL.TZ0.S_9b:
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19M.1.AHL.TZ1.H_4a:
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19N.1.AHL.TZ0.H_9c.i:
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19N.1.AHL.TZ0.H_9d:
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EXN.2.SL.TZ0.2b:
Show that $L = \sqrt{18 - 18 \cos θ}$ .
17N.2.SL.TZ0.S_1a:
Find BC.
SPM.2.SL.TZ0.7c:
Find ${\text{B}}\mathop {\text{C}}\limits^ \wedge {\text{A}}$ .
21M.2.SL.TZ1.8d:
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SPM.2.SL.TZ0.7d:
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EXN.1.AHL.TZ0.7:
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In $PQRS$ , $PQ = x$ , $SR = y$ , $R \hat{S} P = α$ and $Q \hat{R} S = β$ .

Find an expression for $PS$ in terms of $x, y, \sin β$ and $\sin (α + β)$ .
22M.2.SL.TZ2.1a:
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21N.2.SL.TZ0.5b:
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22M.2.SL.TZ2.7a:
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18M.2.SL.TZ1.T_1c:
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19M.2.SL.TZ1.T_2c:
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17N.1.SL.TZ0.T_10b:
Write down the angle of elevation of B from E.
18M.2.SL.TZ2.T_5c:
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17N.1.SL.TZ0.T_10c:
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19M.1.SL.TZ2.T_9b:
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Find the maximum height on the wall that can be reached by the ladder on the second truck.
16N.2.SL.TZ0.T_5f:
the length of the fence, BP.
17M.2.SL.TZ1.T_2a.i:
Write down an equation for the area of ABCDE using the above information.
17N.1.SL.TZ0.T_10a:
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19M.1.SL.TZ1.T_10b:
Find the area, in km², of triangle ABC.
21N.2.SL.TZ0.5a:
Given that the areas of the two shaded regions are equal, show that $θ = 2 \sin θ$ .
19M.1.SL.TZ1.S_3a:
Given that ${\text{sin}}\,\theta = \frac{3}{5}$ , find the value of ${\text{cos}}\,\theta$ .
19M.1.SL.TZ1.S_3b:
Find the value of ${\text{cos}}\,2\theta$ .
19M.2.AHL.TZ2.H_1:
In triangle ABC, AB = 5, BC = 14 and AC = 11.

Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.
18N.2.SL.TZ0.T_5b:
Calculate the area of triangle ABC.
19N.1.AHL.TZ0.H_9c.ii:
By considering triangle ${\text{CBD}}$ , find another expression for ${\text{B}}{{\text{D}}^2}$ in terms of $r$ and $q$ .
17N.1.SL.TZ0.S_4a:
Show that ${\text{AC}} = 7{\text{ cm}}$ .
17M.1.SL.TZ2.T_3a:
Find the length of one of the straight pieces in the wireframe.
19M.1.SL.TZ1.S_3c:
Hence or otherwise, find ${\text{BC}}$ .
19N.1.AHL.TZ0.H_9a:
Given that ${\text{cos}}\,75^\circ = q$ , show that ${\text{cos}}\,105^\circ = - q$ .
19N.1.SL.TZ0.S_9c:
Calculate the area of triangle ${\text{AOC}}$ .
17M.1.SL.TZ1.S_3:
The following diagram shows triangle PQR.

Find PR.
19M.1.AHL.TZ1.H_4b:
Find the two possible values for the length of the third side.
17N.1.SL.TZ0.S_4b:
The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the triangle.

Find the exact perimeter of this shape.
18M.2.SL.TZ2.S_2a:
Find DB.
18M.2.SL.TZ2.S_2b:
Find DC.
EXN.2.SL.TZ0.2a:
Show that arc $APB$ has length $6 π - 3 θ$ .
18N.2.SL.TZ0.S_7b:
Hence or otherwise, find the total distance along the road where the signal from the tower can reach cellular phones.
21M.1.SL.TZ2.6:
The following diagram shows triangle $ABC$ , with $AB = 10$ , $BC = x$ and $AC = 2 x$ .

Given that $\cos \hat{C} = \frac{3}{4}$ , find the area of the triangle.

Give your answer in the form $\frac{p \sqrt{q}}{2}$ where $p, q \in ℤ^{+}$ .
21M.2.SL.TZ1.8c:
The area of field that the horse can reach is $460 m^{2}$ . Find the value of $θ$ .
21N.2.SL.TZ0.3:
Consider a triangle $ABC$ , where $AC = 12, CB = 7$ and $B \hat{A} C = 25 °$ .

Find the smallest possible perimeter of triangle $ABC$ .
17M.2.SL.TZ2.T_4e:
Calculate the total length of the track.
17M.1.SL.TZ1.T_13a:
Find the length of BC.
18N.2.SL.TZ0.T_5c:
Find the length of AC.
16N.2.SL.TZ0.T_5b:
the size of angle DAB;
19M.1.SL.TZ2.T_9a:
Find the angle of elevation from B to C.
17M.2.SL.TZ2.T_4b:
Show that angle DBC is 48.7°, correct to three significant figures.
17M.1.SL.TZ1.T_13b:
Find the area of the postage stamp.
20N.2.SL.TZ0.T_3d:
Pedro draws a circle, with centre at point $E$ , passing through point $C$ . Part of the circle is shown in the diagram.

Show that point $A$ lies outside this circle. Justify your reasoning.
17M.2.SL.TZ1.T_2c:
Show that angle ${\rm{B\hat AE}} = 67.4^\circ$ , correct to one decimal place.
17M.2.SL.TZ2.T_4d:
Find the length of the path CE.
16N.2.SL.TZ0.T_5a:
the length of BD;
19M.2.SL.TZ2.T_2d:
The length of path AD is 287 m.

Find the area of the region ABCD.
16N.1.SL.TZ0.T_11a:
Calculate CD, the height of the observation deck above the ground.
19M.1.SL.TZ1.T_10a:
Find the size, in degrees, of angle BÂC.
18M.2.SL.TZ1.T_1e:
Show that Farmer Brown is incorrect.
17M.2.SL.TZ1.T_2d:
Find the length of the perimeter of ABCDE.
18N.2.SL.TZ0.T_5a:
Show that the size of angle BÂC is 20.2°, correct to 3 significant figures.
18M.2.SL.TZ2.T_5a.i:
Use Giovanni’s diagram to show that angle ABC, the angle at which the Tower is leaning relative to the
horizontal, is 85° to the nearest degree.
18M.2.SL.TZ2.T_5a.ii:
Use Giovanni's diagram to calculate the length of AX.
18M.2.SL.TZ2.T_5a.iii:
Use Giovanni's diagram to find the length of BX, the horizontal displacement of the Tower.
18M.2.SL.TZ2.T_5b:
Find the percentage error on Giovanni’s diagram.
17N.2.SL.TZ0.T_3a:
Show that ${\text{BD}} = 93{\text{ m}}$ correct to the nearest metre.
17N.2.SL.TZ0.T_3b:
Calculate angle ${\rm{B\hat CD}}$ .
17N.2.SL.TZ0.T_3d.i:
Calculate Abdallah’s estimate for the area.
17N.2.SL.TZ0.T_3d.ii:
Find the percentage error in Abdallah’s estimate.
19M.2.SL.TZ1.T_2a:
Find the length ${\text{AB}}$ , giving your answer to four significant figures.
19M.2.SL.TZ1.T_2b:
Find the total area of the two rectangles that make up the roof.
19M.2.SL.TZ1.T_2d:
Find the percentage error in her estimate.
19M.2.SL.TZ1.T_2e:
Olivia investigates arranging the panels, such that the longer edge of each panel is parallel to ${\text{AB}}$ or ${\text{AC}}$ .

State whether this new arrangement will allow Olivia to fit more solar panels to the roof. Justify your answer.
19M.2.SL.TZ2.T_2a:
Find the distance from A to C.
19M.2.SL.TZ2.T_2b.i:
Find the size of angle ${\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{C}}$ .
19M.2.SL.TZ2.T_2b.ii:
Find the size of angle ${\text{C}}\mathop {\text{A}}\limits^ \wedge {\text{D}}$ .
19M.2.SL.TZ2.T_2c:
Find the size of angle ${\text{A}}\mathop {\text{C}}\limits^ \wedge {\text{D}}$ .
18M.2.SL.TZ1.T_1a:
Calculate the area of triangle EAD.
18M.2.SL.TZ1.T_1b:
Calculate the total volume of the barn.
18M.2.SL.TZ1.T_1f:
Calculate the total length of metal required for one support.
17M.2.SL.TZ1.T_2a.ii:
Show that the equation in part (a)(i) simplifies to $3{x^2} + 19x - 414 = 0$ .
17M.2.SL.TZ1.T_2b:
Find the length of CD.
17M.2.SL.TZ1.T_2e:
Calculate the length of CF.
18N.2.SL.TZ0.T_5d:
A vertical pole, TB, is constructed at point B and has height 25 m.

Calculate the angle of elevation of T from, M, the midpoint of the side AC.
17M.2.SL.TZ2.T_4a:
Find the length of the path BD.
17M.2.SL.TZ2.T_4c:
Find the area of the park.
16N.2.SL.TZ0.T_5c:
the area of triangle ABD;
16N.2.SL.TZ0.T_5e:
the length of AP;
20N.2.SL.TZ0.T_3a:
Calculate the length of $BD$ .
20N.2.SL.TZ0.T_3c:
Calculate the area of triangle $BDC$ .

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