Mathematics · May / June 2025 · Paper 31 · Question 2
YMA01
As printed in the paper
The question as text
2. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. f (x) = 7 cos x – 24 sin x
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
2(a)[3]
R =
25
B1
24
tan
α
=
α
=
...
M1
7
α=
1.287
A1
(b)(i)
5
Min
=
M1
(
)
90
−×
3 "25"
×−
1
1
=
A1
33
(b)(ii)
(
)
2
x
+
"1.287"
=
π
,...
x
=
...
M1
π−
"1.287"
x
0.927
=
=
A1
2
Notes:
(a)
(
( )
)
(
)
B1:
Correct value. May be seen embedded in e.g.
f
x
=
25cos
x
+
... .
7
2
(
24
)
2
25
Condone e.g.
R =
+ −
=
Do not allow 25
±
unless the −25 is rejected.
24
7
7
24
M1: Allow for:
tan
α
= ±
, tan
α
= ±
, cos
α
= ±
, sin
α
= ±
leading to a value for α.
7
24
their
R
their
R
If no method is shown it may be implied by their value for α as awrt 1.3 rads or awrt 74°
A1:
Awrt 1.287 Isw once the mark is awarded. Must be in radians.
(
)
( )
(
)
f
x
=
...cos
x
+
1.287 .
May be seen embedded in e.g.
(
( )
)
(
)
Apply isw if correct R and α are seen followed by e.g.
f
x
=
25cos
x
−
1.287 .
Mark (b)(i) and (ii) together.
(b)(i)
5
.
M1: Award for
90
3 Their R
+ ×
May be implied by their value (which may be a decimal) and may be seen embedded in e.g. coordinates
π−
1.287
1
,
which would score (i) M1 A1 and (ii) M1
2
33
For 1
A1:
only.
33
Note that both marks are available in (b)(i) even if their α is incorrect.
(b)(ii) Note that (b)(ii) can be marked independently.
M1: For an attempt to solve 2
x
±
"1.287"
=
π
to obtain a value for x. May be implied by awrt 0.927
Condone a sign slip but the order of operations must be essentially correct e.g.
π
±
"1.287"
2
x
±
"1.287"
=
π
2
x
=
π
±
"1.287"
x
=
2
Condone an attempt to solve e.g. 2
x±
"74 " 180
°=
° but not mixed units e.g. 2
x±
"1.287" 180
=
A1:
For awrt 0.927
2(b)[4]
R =
25
B1
24
tan
α
=
α
=
...
M1
7
α=
1.287
A1
(b)(i)
5
Min
=
M1
(
)
90
−×
3 "25"
×−
1
1
=
A1
33
(b)(ii)
(
)
2
x
+
"1.287"
=
π
,...
x
=
...
M1
π−
"1.287"
x
0.927
=
=
A1
2
Notes:
(a)
(
( )
)
(
)
B1:
Correct value. May be seen embedded in e.g.
f
x
=
25cos
x
+
... .
7
2
(
24
)
2
25
Condone e.g.
R =
+ −
=
Do not allow 25
±
unless the −25 is rejected.
24
7
7
24
M1: Allow for:
tan
α
= ±
, tan
α
= ±
, cos
α
= ±
, sin
α
= ±
leading to a value for α.
7
24
their
R
their
R
If no method is shown it may be implied by their value for α as awrt 1.3 rads or awrt 74°
A1:
Awrt 1.287 Isw once the mark is awarded. Must be in radians.
(
)
( )
(
)
f
x
=
...cos
x
+
1.287 .
May be seen embedded in e.g.
(
( )
)
(
)
Apply isw if correct R and α are seen followed by e.g.
f
x
=
25cos
x
−
1.287 .
Mark (b)(i) and (ii) together.
(b)(i)
5
.
M1: Award for
90
3 Their R
+ ×
May be implied by their value (which may be a decimal) and may be seen embedded in e.g. coordinates
π−
1.287
1
,
which would score (i) M1 A1 and (ii) M1
2
33
For 1
A1:
only.
33
Note that both marks are available in (b)(i) even if their α is incorrect.
(b)(ii) Note that (b)(ii) can be marked independently.
M1: For an attempt to solve 2
x
±
"1.287"
=
π
to obtain a value for x. May be implied by awrt 0.927
Condone a sign slip but the order of operations must be essentially correct e.g.
π
±
"1.287"
2
x
±
"1.287"
=
π
2
x
=
π
±
"1.287"
x
=
2
Condone an attempt to solve e.g. 2
x±
"74 " 180
°=
° but not mixed units e.g. 2
x±
"1.287" 180
=
A1:
For awrt 0.927
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