Mathematics · May / June 2024 · Paper 11 · Question 5
YMA01
As printed in the paper
The question as text
5. E Diagram NOT accurately drawn 12.2 m 6 m D 6 m 1.3 rad 10.8 m A C B Figure 2 Figure 2 shows the plan view of a garden. The shape of the garden ABCDEA consists of a triangle ABE and a right‑angled triangle BCD joined to a sector BDE of a circle with radius 6 m and centre B. The points A, B and C lie on a straight line with AB = 10.8 m π radians, angle EBD = 1.3 radians and AE = 12.2 m Angle BCD = 2
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Show the answer and mark scheme
5(a)[2]
1
2 ×
6
2
×
1.3
=
...
=
23.4
(m2)
5(b)[2]
(
)
12.2
2
=
6
2
+
10.8
2
−××
2 6 10.8cos ABE
6
2
+
10.8
2
−
12.2
2
19
(
)
cos
ABE
=
=
2 6 10.8
××
648
ABE =
1.54
5(c)[3]
1
(
)
Area
ABE
=
×
10.8 6sin
×
ABE
2
1
(
)
(
)
Area
BCD
=
×
6cos
π
−
1.3
−
"1.54"
×
6sin
π
−
1.3
−
"1.54"
2
or e.g.
1
(
)
2
Area
BCD
=
×
6sin
(
π
−
1.3
−
"1.54"
)
×
6
2
−
6sin
(
π
−
1.3
−
"1.54"
)
2
Total area
60.9m
2
=
Notes:
(a) Marks for part (a) must be seen in part (a)
1
θ
M1: Attempts to evaluate
2 r θ with r = 6 and θ = 1.3 or equivalent e.g.
2
×
π
×
6
2
with
θ
=
1.3
360
may be implied by use of e.g. 74° or 75°
A1: Correct value. Allow equivalent values e.g. 117
2
, 23
5 .
5
Condone lack of units but if any are given they should be correct. Correct answer only scores both marks.
(b)
M1: Correct numerical application of the cosine rule for the required angle.
A1: For awrt 1.54
(c) Allow equivalent work in degrees.
1 10.8 6sin their
(
)
M1: Correct method for area ABE e.g.
×
×
ABE
2
M1: Correct method for area BCD
Must be a correct method including the ½
6
CD
Note that CD may be found using the sine rule e.g.
CD
...
=
=
(
)
sin
π
sin
π
−
1.3 "1.54"
−
2
6
BC
Note that BC may be found using the sine rule e.g.
=
BC
=
...
sin
π
π
sin 1.3 "1.54"
2
+
−
2
Having found CD or BC as above, BC or CD may be found using Pythagoras.
In such cases the trigonometry must be correct for their values.
Do not allow mixing of degrees and radians e.g. 180 1.3 "1.54"
−
−
for
π−
1.3 "1.54"
−
A1: Correct total area. Allow awrt 60.9 (Condone lack of units)
Some values for reference:
Angle DBC = 0.300…(17.1…°), Angle BDC = 1.27…(72.7…°),
Area ABE = 32.38…, Area BCD = 5.083…, CD = 1.77…, BC = 5.73…
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