Mathematics · Oct / Nov 2021 · Paper 31 · Question 4
YMA01
As printed in the paper
The question as text
4. In this question you should show detailed reasoning. 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
4(a)[4]
2sin
θ
−
30
° =
5cos
θ
⇒
2sin
θ
cos30
°−
2cos sin30
θ
° =
5cos
θ
÷cosθ
⇒
2tan
θ
cos30
°−
2sin30
° =
5
3
1
2tan
2
5
⇒
θ
×
−×
=
2
2
⇒
3 tan
θ
=
6
⇒
tan
θ
=
2 3
*
4(b)[3]
Attempts arctan 2 3 ..and then subtracts 20°
⇒
x
=
awrt 53.9 , 233.9
°
°
(a)
(
)
(
)
(
)
M1
Attempts to use
sin
θ
−
30
° =
sin
θ
cos
±
30
° ±
cos
θ
sin
±
30
° within the given
equation
Condone the omission of a 2 on the second term and a slip on the 5 of 5cosθ
Divides by cosθ to set up an equation in just tanθ .
dM1
They may collect terms in sinθ and cosθ before dividing by cosθ to set up an
equation in just tanθ
An equation with cos30° and sin30° still not processed is acceptable.
Fully correct equation in tanθ with the cos30°and sin30° processed
A1
(
3sin
θ
=
6cos
θ
⇒
tan
θ
=
2 3
is acceptable for both A marks)
(Note If they proceed directly to the final answer from
5
+
2sin30
°
tan
θ
=
⇒
tan
θ
=
2 3
then maximum M1dM1A0A0 unless
2cos30
°
5 1
+
tan
θ
=
or equivalent is seen before the final given answer.
3
A1*
Correctly proceeds to given answer.
(b) Answers with no working scores 0 marks
M1
Attempts to find a value for x.
Allow arctan 2 3 ...followed by adding or subtracting 20°. Which may be implied
by
tan(
x
+
20)
=
2 3
⇒
x
=
arctan(2 3)
±
20
=
...
(
)
sin
x
−
10
° =
sin cos10
x
°±
cos sin10
x
° within
Alternatively, attempts to use
the given equation, divides by cos x to set up an equation in just tan x and
proceeds to find an angle for x
5cos20
+
2sin10
tan
x
=
⇒
x
=
...
2cos10
+
5sin 20
A1
One value provided M1 has been scored. Allow either awrt 54° or 234° (or in
radians awrt 0.94 or 4.08)
A1
x =
awrt 53.9 , 233.9
°
° and no others inside the range provided M1 has been
scored. Ignore any angles outside the range. Must be in degrees
∫
8
−
2
(
)
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