Mathematics · Oct / Nov 2025 · Paper 31 · Question 9
YMA01
As printed in the paper
The question as text
9. 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
9(a)[3]
4sin cos
θ
θ
=
2sin2
θ
θ
cos2
(
)
e.g.
2
θ
θ
+
θ
=
−
θ
+
θ
6sin
cot 2
2sin 2
3 3cos2
2sin 2
θ
sin 2
9(b)[3]
θ
−
=
2
θ
θ
+
θ
θ
3cot 2
14
6sin
cot 2
4sin
cos
(
)
e.g.
3cot 2 sin2
θ
θ
−
14sin2
θ
=
3 3cos2
−
θ
cos2
θ
+
2sin 2
2
θ
−
14sin2
θ
= −
3(1 sin 2 )
−
2
θ
+
2sin 2
2
θ
2
θ
+
θ
−=
*
5sin 2
14sin2
3
0
9(c)[3]
) 1
(
sin 2
x
=
x
=
...
5
°
x =
awrt5.8 , awrt84.2
°
(a)
B1:
4sin cos
θ
θ
=
2sin2
θ
(seen or implied)
θ
cos2
M1:
Attempts to use
θ
= ±±
2
θ
and
cot 2
θ
=
or alternative
cos2
1
2sin
sin 2
θ
versions of these formulae to get to an expression in just
and
Other routes are possible, but any trig identities must be correct up to
sign error to reach an expression in just
and
.
) cos2
θ
(
A1:
−
θ
+
θ
or equivalent. isw once a correct expression is
3
3cos2
2sin 2
θ
sin 2
seen. Alternatives are possible here.
(b)
M1:
Uses their expression from part (a) in the given equation and attempts to
multiply both sides by sin2θ. Alternatively they may rework part (a) again in
this part but must reach an expression in terms of
one line.
M1:
Cancels 3cos2θ from both sides, attempts to use
±
2
θ
±
2
sin 2
cos 2
and proceeds to a 3TQ in sin2θ only (terms do not need to be collected
on the same side)
A1*:
Achieves the given answer with no errors seen (including invisible
brackets).
(c)
M1:
Solves the quadratic (usual rules, accept calculator usage) and attempts
to use arcsin find a value for x. Accept if radians used or they neglect to
divide by 2 as long as a value for x is reached. Condone use of
throughout this part.
1
−
1 1
A1:
One of awrt 5.8 or awrt 84.2 (provided
x =
or
2
x
=
sin
is seen)
sin 2
5
5
−
1 1
1
A1:
awrt5.8, awrt84.2 (provided
or
=
is seen) and no others
sin 2
x =
2
x
sin
5
5
in the given range.
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