Mathematics · May / June 2022 · Paper 41 · Question 7
YMA01
As printed in the paper
The question as text
7. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The curve C has parametric equations x = sin t – 3 cos 2 t y = 3 sin t + 2 cos t 0 t 5
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
7(a)[4]
d
x
d
y
t =
cos
t
+
6cos sin
t
t
t =
3cos
t
−
2sin
t
d
d
d
y
d
y
d
t
3cos
t
−
2sin
t
3cos
π
−
2sin
π
=
×
=
=
=
3
*
d
x
d
t
d
x
cos
t
+
6cos sin
t
t
cos
π
+
6cos
π
sin
π
7(b)[3]
When
t
=
π
,
x
= −
3,
y
= −
2
y
" 2"
3(
x
" 3")
−−
=
−−
y
=
3
x
+
7
(
)
7(c)[2]
y
=
3
x
+
7
3sin
t
+
2cos
t
=
3 sin
t
−
3
cos
2
t
+
7 or
(
)
y
=
3
x
+
3cos
2
t
+
2
cos
t
3
x
+
7
=
3
x
+
9cos
2
t
+
2 o
c s
t
9cos
2
t
+
2cos
t
−
7
=
0
*
7(d)[3]
7
cos
t =
9
32
7
4 2
14
y = ×
3
+ ×
2
=
+
9
9
3
9
Notes
(a)
d
x
or cos
t
+
3sin 2
t
B1:
cos
t
+
6cos sin
t
t
=
d
t
d
y
B1:
3cos
t
−
2sin
t
=
d
t
using their d
x
t and their d
y
M1: Attempts d
y
d
y
d
t
=
×
t and substitutes t
=
π
. (May substitute π before
d
d
d
x
d
t
d
x
dividing.)
A1*: Achieves d
y
x =
3
with full working shown and no errors.
d
(b)
B1:
x
= −
3,
y
= −which may be seen within their working
2
M1: Attempts to find the equation of the tangent with gradient 3. If they use y
=
mx
+ they must proceed as
c
far as
...
c =
A1:
y
=
3
x
+
7
(c)
M1: A full attempt to solve simultaneously the given parametric equations with their equation of the tangent
A1*: Achieves
9cos
2
t
+
2cos
t
−
7
=
0
with no errors
(d)
7
B1:
seen or implied.
Allow if seen in (c).
cos
t =
9
M1: Attempts to find the y coordinate Must attempt to evaluate trig terms. If no substitution/working shown,
then score for awrt 3.44 following a correct value for cos t
A1: 4 2
14
or exact equivalent. Withhold if additional answers are given.
+
3
9
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