Mathematics · May / June 2025 · Paper 21 · Question 8
YMA01
As printed in the paper
The question as text
8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
8(i)[3]
x
+
=
x
+
=
5
5
3
7
13
π
π
π
π
2
x
+
=
,
,
x
=
...
5
6
6
6
29
59
π
π
x
=
,
60
60
8(ii)[5]
5sin tan
θ
θ
=
cos
θ
+
4
sin
θ
sin
θ
tan
5sin
cos
4
θ
=
→
θ
×
=
θ
+
cos
θ
cos
θ
(
)
5 1 cos
−
2
θ
=
cos
2
θ
+
4cos
θ
2
6cos
θ
+
4cos
θ
−=
5
0
−±
4
16 120
+
cos
θ
=
(
=
0 63849
.
...)
θ
=
12
θ
=
50 3
.
°
,
309 7
.
°
In both (i) and (ii) condone poor notation, omission of θ or x in their working provided the intention
is clear or recovered/implied by further work.
(i)
π
3
π
π
M1: Correct order of operations
2
x
+
=
arctan
implied by 2
or any correct angle.
x
+
=
5
3
5
6
Condone decimal values here e.g. 2
x+
0.628...
=
0.52....
Allow to work consistently in degrees. e.g. 2
x+
36
=
30
sin
θ
Alternatively, attempts to use
tan
θ
=
, squares both sides , attempts to use
cos
θ
±
sin
2
θ
±
cos
2
θ
= ± and proceeds to a quadratic equation in sine or cosine
1
e.g.
π
3sin 2
+
x
5
π
π
π
=
2
3
3sin 2
x
+
=
3cos 2
x
+
9sin
2
x
+
=
3 1 sin
−
π
5
5
5
cos 2
+
x
5
1
π
π
π
π
12sin
2
2
+
=
3
sin 2
+
= ±
( )
2
+
=
,...
x
x
x
5
5
2
5
6
π
π
7
π
13
π
dM1: Correct order of operations
2
+
=
,
,
to find a value for x which does not need to be in
x
5
6
6
6
π
the given range (e.g. condone
−
)
60
Condone decimal answers here (awrt 0.05
−
, awrt 1.5, awrt 3.1). It is dependent on the previous
30
36
±
method mark. Allow to work consistently in degrees. e.g.
2
x
+
36
=
30
x
=
2
) 29
π
59
π
A1: (
and no other angles in the range. Must be exact and in radians.
x
=
,
60
60
Alternative: Using compound angle formula
3
π
−
tan
3
5
M1:
Forms the equation
tan 2
=
Allow to work consistently in degrees
x
3
π
1
+
tan
3
5
dM1: Correct order of operations (arctan followed by dividing by 2)
Allow to work consistently in degrees
) 29
59
π
π
A1: (
x
=
,
and no other angles in the range. Must be exact and in radians.
60
60
(ii)
Note that there may be other credit worthy methods relying on identities covered in later
units. If seen then send to review.
sin
θ
M1: Attempts to use
tan
θ
=
leading to an equation in sinθ and cosθ
cos
θ
sin
θ
e.g.
5sin
θ
×
=
cos
θ
+
4
Condone slips.
cos
θ
dM1: Multiplies through by cosθ (on at least 2 of the 3 terms in the equation) and attempts to use
±
sin
2
θ
±
cos
2
θ
= ± to set up a quadratic equation in cosθ (may be in sinθ if they make an earlier
1
slip so allow this to score as well)
Condone slips e.g. not multiplying both terms by 5 if
5(1 cos
2
)
is not seen on the left hand side
−
θ
of the equation.
Terms do not need to be collected for this mark.
Condone arithmetical slips in their rearrangement.
6cos
2
θ
+
4cos
θ
−= o.e The = 0 may be implied by further work.
5
0
A1:
Requires 3 terms but allow the constant term to be on the other side of the equation.
ddM1: Solves their threeterm quadratic equation in cosθ (or condone sinθ) and proceeds to a value
for θ
Allow use of a calculator to solve the quadratic otherwise usual rules apply for factorising, using
the formula or completing the square.
We must see
a correct method used to solve the quadratic, or can be implied by a correct root to 1dp for their
quadratic equation
a correct angle for their root by taking arccos of the root (allow to the nearest integer in
degrees) You may need to check this.
A correct angle on its own does not imply this mark.
A1: (
) awrt 50 3
.
,
awrt 309 7
.
and no others in the range. Must be in degrees.
θ=
°
°
4
4
2
Attempts
12
=
3x
×
L
L
=
or
=
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