Mathematics · May / June 2021 · Paper 31 · Question 7
YMA01
As printed in the paper
The question as text
7. Given that π x = 6 sin2 2 y 0 < y < 4 show that d y 1 x = d A ( Bx − x ) 2 where A and B are integers to be found. (5)
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
d
x
x
=
6sin 2
2
y
⇒
=
24sin 2 cos2
y
y
d
y
Attempts to use d
y
d
x
= ÷
1
d
d
x
y
x
x
Attempts to use both sin 2
y =
and
cos2
y
=
1 sin 2
−
2
y
=
1
−
6
6
d
y
1
1
1
=
=
=
d
x
24sin 2 cos2
y
y
x
x
4 6
x
−
x
2
24
×
×
1
−
6
6
Way 2
d
x
x
=
6sin 2
2
y
= −
3 3cos4
y
⇒
=
12sin 4
y
d
y
Attempts to use d
y
d
x
= ÷
1
d
x
d
y
6
x
−
x
2
sin 4
y
=
Attempts to use
3
d
y
1
1
1
=
=
=
d
x
12sin 4
y
6
x
−
x
2
4 6
x
−
x
2
12
×
3
Way 3
d
x
x
=
6sin 2
2
y
=
24sin
2
y
cos
2
y
⇒
=
48sin
y
cos
3
y
−
48sin
3
y
cos
y
d
y
Attempts to use d
y
d
x
= ÷
1
d
x
d
y
d
y
1
1
=
=
(
)
d
x
48sin
y
cos
3
y
−
48sin
3
y
cos
y
48sin
cos
cos
2
sin
2
y
y
y
−
y
d
y
1
1
=
=
d
x
24sin 2 cos2
y
y
x
x
24
×
×
1
−
6
6
1
=
4 6x
−
x
2
In general, apply the following marking guidance for this question:
M1: Attempts to differentiate to obtain d
x
in a correct form
d
y
A1: Correct derivative
M1: Attempts to use d
y
d
x
= ÷
1
.
d
x
d
y
M1: Change fully to a function of x.
A1: All correct
E.g.
M1: Attempts to use the chain rule on the rhs to achieve ksin 2y cos 2y
A1: Fully correct derivative d
x
=
24sin 2 cos2
y
y
d
y
M1: Attempts to use d
y
d
x
= ÷
1
.
d
x
d
y
Condone a slip on the coefficient. E.g. d
x
d
y
24
=
24sin 2 cos2
y
y
⇒
=
d
y
d
x
sin 2 cos2
y
y
Do not condone slips/errors on the variable. E.g. d
x
d
y
24sin 2 cos2
y
y
=
⇒
=
d
y
d
x
24sin 2 cos2
M1: Attempts to change both sin 2y and cos 2y to functions in x.
Expect to see sin 2y
=
p x
and cos2
y
=
1
−
qx
d
y
1
A1: CSO
=
d
x
4 6
x
−
x
2
Way 2
M1: Differentiates to obtain ksin4y
A1: Fully correct derivative d
x
=
12sin 4
y
d
y
M1: Attempts to use d
y
d
x
= ÷
1
.
d
x
d
y
Condone a slip on the coefficient. E.g. d
x
d
y
12
=
12sin 4
y
⇒
=
d
y
d
x
sin 4
y
M1: Attempts to change both sin 4y to a function in x.
(
)
Expect to see
sin 4
y
=
p
6
x
−
x
2
or equivalent e.g.
sin 4
y
=
p
9
−
9 6
−
x
+
x
2
d
y
1
=
A1: CSO
d
x
4 6
x
−
x
2
Way 3
M1: Differentiates to obtain
A
sin
y
cos
3
y
+
B
sin
3
y
cos
y
d
x
A1: Fully correct derivative
y =
48sin
y
cos
3
y
−
48sin
3
y
cos
y
d
M1: Attempts to use d
y
d
x
= ÷
1
.
d
x
d
y
M1: Attempts to change both sin 2y and cos 2y to functions in x.
Expect to see sin 2y
=
p x
and cos2
y
=
1
−
qx
d
y
1
A1: CSO
=
d
x
4 6
x
−
x
2
A possible alternative:
x
x
1
x
x
=
6sin 2
2
y
⇒
sin 2
y
=
⇒
2
y
=
sin
−
1
⇒
y
=
sin
−
1
6
6
2
6
M1: For a correct attempt to make y or 2y the subject
A1: Correct rearrangement
1
−
d
y
1
1
1
x
2
=
×
×
d
x
2
1
12 6
−
x
6
1
A
x
−
2
M1: For
×
...
M1: For
...
×
B
1
−
6
x
6
d
y
1
A1: CSO
=
d
x
4 6
x
−
x
2
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