Mathematics · May / June 2021 · Paper 11 · Question 3
YMA01
As printed in the paper
The question as text
3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. B Not drawn to scale 2 2 m p m 60° C q m A Figure 1 Figure 1 shows the plan view of a flower bed. The flowerbed is in the shape of a triangle ABC with ● AB = p metres ● AC = q metres ● BC = 2 2 metres ● angle BAC = 60°
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
3(a)[2]
(
)
2
2 2
=
p
2
+
q
2
−
2
pq
cos60
° oe
p
2
+
q
2
−
pq
= *
8
3(b)[5]
q
=
p
+
2
⇒
8
=
p
2
+
("
p
+
2")
2
−
p
("
p
+
2")
p
2
+
2
p
−
4
=
0
or
q
2
−
2
q
−
4
=
0
2
(
2
)
2
4 1 ( 4)
−±
2
2
2
−× × −
4 1 ( 4)
±
−
−× × −
p
=
or
q
=
2
2
p = −+
1
5
or
q = +
1
5
p = −+
1
5
and
q = +
1
5
only
3(c)[2]
1
Area
=
×
(" 1
−+
5")("1
+
5") sin 60
×
°
2
Area =
3 (m2)
Alt(a)
Forming a line BX which is perpendicular to AC where X is on the line AC.
p
AX
=
p
cos60
=
2
p
2
3
or
BX
=
p
2
−
=
p
BX
=
p
sin60
2
2
2
2
2
3
p
(
)
p
(
)
2
or (
)
2
2
p
+
q
−
=
2 2
p
sin60
+
q
−
=
2 2
2
2
2
3
p
2
p
2
+
q
2
−
pq
+
=
8
4
4
= *
p
2
+
q
2
−
pq
8
(a)
2
They may carry this out in two stages by forming two right angled triangles with BX being
perpendicular to AC (see Alt(a)). To score this mark they must proceed as far as
2
2
2
3
p
(
)
p
(
)
2
or (
)
2
2
p
+
q
−
=
2 2
p
sin60
+
q
−
=
2 2
2
2
2
Condone missing brackets for M1.
A1*
Achieves
2
2
= with no errors including omission of brackets. One of the lines
p
+
q
−
pq
8
above must have been seen for M1A1. If they state = 8 without showing any working then
A1 cannot be scored.
(b)
M1
Substitutes
q
=
p
+
2
(oe) into the given equation
2
2
4
0
A1
p
+
p
−
=
or the equivalent equation in q (
q
2
−
2
q
−
4
=
0
)
M1
Attempts to solve to find p using the formula or completing the square using their values.
Alternatively, they achieve a quadratic equation in q and attempt to solve to find q using
their values. Usual rules for solving quadratics apply. If they state the roots or factorise then
M0. If they use the quadratic formula then the values must be embedded.
B1
p = −+
1
5
or
q = +
1
5
(ignore any other solutions) Must be exact. This is independent
of the previous method mark so if the roots are just stated this mark can be scored.
(Note this is A1 on EPEN)
A1
p = −+
1
5
and
q = +
1
5
only cso (all other marks must have been scored to award A1)
(c)
Attempts to find the area of the triangle using 1
(" 1
5")("1
5") sin 60
M1
2 ×
−+
+
×
°. Must see
at least one stage of working using their p and their q
A1
3 (m2) condone lack of units. Do not accept rounded answers.
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