Mathematics · May / June 2024 · Paper 41 · Question 4
YMA01
As printed in the paper
The question as text
4. R Q 5 cm θ O P Figure 1 Figure 1 shows a sketch of a segment PQRP of a circle with centre O and radius 5 cm. Given that • angle POR is θ radians • θ is increasing, from 0 to π, at a constant rate of 0.1 radians per second • the area of the segment PQRP is A cm 2
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
4(a)[2]
A
=
2
θ
−
sin
θ
=
1 co
±
s
θ
2
d
θ
2
d
A
25
(
)
θ=
1
−
cos
θ
d
2
d
θ=
4(b)[4]
0.1
dt
d
A
d
A
d
θ
=
θ×
(oe)
d
t
d
d
t
d
A
25 1 cos
π
=
−
×
0
.
1
d
t
2
3
5
=
(cm2 per second)
8
(a)
M1: Attempts formula for area of segment with r = 5 and differentiates with respect to θ.
d
A
1
(
)
(
)
Look for
α
25
θ
−
sin
θ
=
β
1 cos
±
θ
Allow with either
α=
or 1
d
θ
2
d
A =
1
α=
You may see
A
=α
5
2
θ
−
α
5 sin
2
θ
β
±
β
c
o
s
θ
again with
or 1
d
θ
2
A1: Reaches correct result with no errors seen. Allow with 12.5.
Must follow M1 (the form is given). It cannot just appear without sight of some correct working
(b)
B1: States or uses d
θ=
0.1
dt
This may be seen anywhere in the question, even in part (a) or next to the Figure
B1: States or uses a correct version of the chain rule for the given problem.
May be seen in (a) or in the question itself.
E.g d
d
d
or d
d
d
A
A
θ
θ
A
θ
=
θ×
×
=
are two correct versions
d
t
d
d
t
d
A
d
t
d
t
This may be implied by an equation involving their d
A
d
A
(
)
and their d
K
1
cos
θor
=
−
θ
d
d
θ
M1: Attempts the chain rule with their (a), a correct d
θ=
π
to find a value for d
0.1
and
θ=
dt
3
If they don’t attempt (a), or use a made up K, condone d
1
1
, but d
A
=
0.1
×
'
K
'
=
'
K
'
d
t
2
20
(
)
the form
K
1 cos
θ via this route.
−
A1: CSO. units not required. Decimal answer is 0.625.
Cannot be scored following an incorrect part (a) but allow if they have
d
A
r
2
d
A
25 1 cos
π
5
(
)
(a) left as
=
1
−
c
os
θ
followed by (b)
=
−
×
0.1
=
which scores 00 1111.
d
θ
2
d
t
2
3
8
2
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