Mathematics · Oct / Nov 2023 · Paper 11 · Question 5
YMA01
As printed in the paper
The question as text
5. B Diagram NOT accurately drawn 5 m x m θ° A C (1 + x) m D Figure 2 Figure 2 shows the plan view of a frame for a flat roof. The shape of the frame consists of triangle ABD joined to triangle BCD. Given that • BD = x m • CD = (1 + x) m • BC = 5 m • angle BCD = θ°
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Show the answer and mark scheme
5(a)[2]
5
2
+
(1
+
x
)
2
−
x
2
25 1
+ +
2
x
+
x
2
−
x
2
e.g. (
)
cos
θ
=
=
2 5(1
×
+
)
2 5(1
×
+
)
x
x
2
=
5
2
+
(1
+
)
2
−×
2 5(1
+
)cos
θ
or e.g.
x
x
x
13
+
x
cos
θ
=
*
5
+
5
x
5(b)[5]
awrt 42
° (42.470747….)
θ=
Attempts to find AB, AD or AC:
5
AB
e.g.
...
=
AB
=
sin"42"
sin30
or
e.g. ∠ABC 180 30 "42.5"
=
−
−
=
"107.6"
sin
DBC
sin"42.5"
=
∠
=
"60.5",
∠
=
"107.6" "60.5"
−
=
"47.1"
DBC
ABD
1
2 3
2 3
+
2 3
AD
=
AD
=
...
sin("47.1")
sin30
or
AC
5
e.g.
=
=
...
AC
sin"108"
sin30
AB =
6.75...
or
AD =
5.07...
or
AC =
9.53...
1
Area
5 "6.75" sin 180
(
30
"42.5"
)
or
=
××
×
−
−
2
1
(
)
=
××
5 "9.54" sin "42.5"
×
2
=
awrt16.1
(m2)
(a)
Uses the cosine rule to form an equation for cosθ. Condone just an expression for cosθ.
M1:
(You may not see cosθ=)
Condone invisible brackets for this mark.
Be aware of longer versions which may involve splitting triangle BCD into two right angled
triangles. They would still need to proceed to an equation involving cosθ to score this
mark.
A1*: Achieves the given answer with no errors seen including invisible brackets. There must be at
least one stage of intermediate working between their starting equation and achieving the
given answer. Condone cosθnot appearing in the answer line, provided it is seen correctly
as the subject on an earlier line.
(b) Note there are a variety of different methods to finding the area of triangle ABC
7.54m
2
47.1°
60.5°
8.56m
2
6.75m
3.46m
3.38m
(
)
2 3m
42.5°
42.5°
102.9°77.1°
30°
5.07m
4.46m
(
)
9.53m
1
+
2 3m
B1:
θ=
awrt 42
° seen or implied. Accept 42.5. May work in radians (awrt 0.73/0.74 radians)
M1:
Attempts to find AB, AD or AC using a correct method. Angles and lengths must be in
the correct positions in the relevant formula or formulae. Condone slips in any
x =
2 3
rearrangement, calculations and substituting in
provided the method is correct.
Condone working in radians provided the angles are consistently in degrees or radians
within the expression or formula.
A1:
awrt 6.75/6.76 or awrt 9.53/9.54 or awrt 5.07/5.08 (may be implied by further work).
dM1: Correct full method to find the total area (the expression is sufficient. It is dependent on the
previous method mark. Condone use of incorrectly rounded angles/slips and may work in
radians.
1
π
e.g. Area
=
××
5 "6.75" sin
×
π
−
−
"0.74"
2
6
May find the areas of the two separate triangles ABD and BDC and add them together.
1
1
e.g. Area
=
×
"5.07" "6.75" sin30
×
×
+
×
"4.46" 5 sin"42.5"
××
2
2
Use the diagrams above to help with the various methods. Invisible brackets may be implied
by further work or their answer.
Note that if, as part of their method to find the total area, they find angle ADB but
incorrectly deduce this as an acute angle then this is dM0
awrt 16.1 m2 (condone lack of units)
A1:
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