Mathematics · May / June 2023 · Paper 21 · 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]
3cos (tan sin
θ
θ
θ
+
3)
=
11 5cos
−
θ
sin
θ
2
or
3cos
sin
3
11 5cos
3sin
θ
+
9cos
θ
=
11 5cos
−
θ
θ
θ
+
=
−
θ
cos
θ
3(1
−
cos
2
θ
)
+
9cos
θ
=
11
−
5cos
θ
*
3cos
2
θ
−
14cos
θ
+
8
=
0
2
9(b)[4]
cos2
x =
3
−
1 2
cos
3
...
x
=
=
2
x =
awrt 24.1, awrt 155.9, awrt 204.1, awrt 335.9
(a)
sin
θ
M1
Substitutes or uses
tan
θ
=
to achieve an equation in sine and cosine only.
cos
θ
May be in another variable and condone if there is a mix of variables in the same equation
for this mark.
dM1
Attempts to use
±
sin
2
θ
±
cos
2
θ
= ± to achieve an equation in cosine only. It is dependent
1
on the previous method mark. Condone slips in their manipulation of the equation.
May be in another variable and condone if there is a mix of variables in the same equation
for this mark.
A1*
Achieves
3cos
2
θ
−
14cos
θ
+
8
=
0
with no errors seen including brackets. Withhold this mark for
poor notation used such as
3cos
θ
2
−
14cos
θ
+
8
=
0
Allow a candidate to work consistently in one variable and then change to θ but withhold
this final mark if there is a mixture of variables within the same equation/line of working.
(b)
Note that answers only scores 0 marks. May work in any variable and do not be
concerned as to whether they are writing e.g. x or 2x in their working
) 2
(
B1
cos...
=
(ignore any reference to 4)
3
M1
Solves using the correct order of operations to find at least one value for x. You may need to
2
check this on your calculator. May be implied by a correct value for
cos2
" "
x =
3
A1
Two of awrt 24, awrt 156, awrt 204, awrt 336 . (Condone in radians two of awrt 0.42, awrt
2.7, awrt 3.6, awrt 5.9)
A1
awrt 24.1, awrt 155.9, awrt 204.1, awrt 335.9 and no others in the range
Minimum acceptable to be able to score full marks:
2
−
1 2
or (
)
e.g.
cos...
=
...
=
cos
3
3
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