Mathematics · Oct / Nov 2025 · Paper 41 · Question 7
YMA01
As printed in the paper
The question as text
7. C B θ D A Figure 2 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 2 shows a sketch of a parallelogram ABCD. Given that → = 6i – 2j + 3k • AB → = –4i + 3j – 5k • BC • angle ABC = θ, 0 < θ < 90°
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
7(a)[3]
Attempts BA BC
2
3
=
24
+ +
6 15
=
45
−
3
−
5
Attempts to apply
a b
=
a b
cos
θ
45 = 7
×
50 cos
θ
9 2
cos
θ
= 14
oe
2
9 2
7(b)[3]
Attempts
θ
−
sin
=
1
14
34
Attempts to apply
a b
sin
θ
=
7
×
50
×
14
Area ABCD = 5 17
(a)
6
4
M1:
Attempts BA
BC
. Look for
±−
2
±−3
= ±
24
±±
6 15
The expression is
3
5
sufficient but may be implied by 45
. Condone slips but at least two
±
products should be correct.
dM1: Uses
a b
=
a b
cos
θ
with either of a or b found using a correct method.
Either length may be implied by " 49" oe or " 50" oe. It is dependent on
the previous method mark.
9 2
9
A1:
θ
o.e such as
Must be exact
cos
= 14
cos
θ
=
7 2
9 2
9 2
−
Note that
cos
θ
=
−
or
cos
θ
= 14
from
a b =
45
is M1dM1A0
14
Alt (a) using the cosine rule
2
2
2
M1:
Attempts to find AC
, AB
and BC
(or
,
and
) using a
AC
AB
BC
correct method with at least two lengths correct. May be seen within the
cosine rule formula.
dM1: Uses the cosine rule to form a correct equation using their lengths in the
correct positions. It is dependent on the previous method mark.
7
2
+
(5 2)
2
−
3
2
e.g.
θ=
cos
2(7)(5 2)
9 2
9
θ
A1:
cos
= 14
o.e such as
cos
θ
=
7 2
4
(b)
Note that finding e.g. cos BAC
=
in (a) can still lead to full marks in
21
(b)
Attempts to find the exact value of sinθ examples include:
M1:
2
9 2
34
usingθ
−
oe May be implied by their
14 or exact
sin
=
1
"
"
14
equivalent or score for a correct expression for sinθ with their
cos
substituted. Condone invisible brackets.
drawing a rightangled triangle to find the opposite side and then usingopp
θ=
sin
hyp
If an exact expression is not seen somewhere in their solution then
M0M0A0 but allow a candidate to write exact values but then use decimals
in intermediate working provide exact values are recovered before the final
answer.
9 2
Do not accept
θ=
−
1
for this mark.
sin
sin cos
"
"
14
M1:
Attempts to apply
. The expression is sufficient. Look for e.g.
a b
sinθ
34
×
×
"7" " 50" "
"
14
9 2
−
1
Condone
×
×
for this mark or
"7" " 50" sin cos
"
"
14
(
)
"7" " 50" sin awrt 24.6
×
×
Note there are other methods to find the area. Send to review if you see
something that you think is credit worthy.
A1:
5 17 following an exact value for sinθ seen, both previous method marks
9 2
scored and
cos
θ
=
±
o.e. (units not required)
14
Note: There are other valid methods which if seen then send to review:
Heron’s formula
+
+
7
5 2
3
5 2
s
=
=
5
+
A
=
2
s s
(
−
7)(
s
−
5 2)(
s
−
3)
=
5 17
2
2
Cross product e.g.
i
j
k
−
−
= −−
−
=
−
2
+ −
2
+ −
2
=
6
2
3
i
18
j
10
k
( 1)
( 18)
( 10)
5 17
−
−
4
3
5
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