Mathematics · May / June 2022 · Paper 11 · Question 7
YMA01
As printed in the paper
The question as text
7. The curve C has equation y = f(x), x > 0 Given that f′(x) = 2 x + A • x2 + 3 , where A is a constant • f′′(x) = 0 when x = 4
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
7(a)[4]
−
−−
f '( )
x
=
2
x
+
A
x
−
2
+
3
f ''(
x
)
=
...
x
+ −
.
.
.
x
−−
2 1
2
2
3
3
1
−
−
f '
'(
x
)
=
2
×−
x
+ −
2
Ax
−
3
= −
x
−
2
Ax
−
3
2
2
2
3
−
f ''(4)
=
0
−
4
2
−
2
A
×
4
−
3
=
0
A
=
...
1
2
A
−
−
=
0
A
= −
4
8
64
1 1
7(b)[5]
−+
1
2
x
Ax
−+
2 1
2
−
f
( )
x
=
2
x
+
Ax
−
2
+
3
d
x
=
+
+
3 (
x
+
c
)
2
1 1
−+
2
1
−+
2
1
A
=
4
x −
+
3 (
x
+
c
)
2
x
A
f(12)
=
8 3
4
12
−
+
36
+
c
=
8 3
c
=
...
1
2
4
109
A
8
3
4 12
36
36
c
=
−
−
−
= −
or follow through
c =
−
12
3
12
1
4
109
So (
)
f( )
x
=
4
x
+
+
3
x
−
2
oe
3
x
Notes: Mark parts (a) and (b) together
(a)
M1: Correct method of differentiation, at least one correct power reduced by 1.
The indices or coefficients do not have to be processed/simplified.
A1: Correct differentiation, need not be simplified, but the indices and coefficients must be processed
correctly.
dM1: Sets f ''(4)
=
0
and proceeds to find a value for A. It is dependent upon the previous M1
Don't be overly concerned with the mechanics of the solution here
A = −
4
A1:
(b)
M1: Attempts to integrate f’(x). Look for a correct power increased by 1 on at least one term. The indices and
coefficients do not have to be processed/simplified. Constant of integration is not needed for this mark.
A1ft: Correct integration, either with A or follow through their value of A. The indices and coefficients must
be processed correctly. Constant of integration is not needed for this mark.
dM1: Sets f(12) = 8 3 and proceeds to find a value for a constant of integration '' c'' using their value
of A. It is dependent upon the previous M1
A
36
A1ft: Correct value for c found. Follow through their value of A , so
c =
−
.
12
Accept awrt 1 dp
1
4
109
4
x
+
+
3x
−
A1: Correct answer,
2
, not follow through.
x
3
If they go on to multiply by 3 etc it is A0
Withθ being the angle subtended by arc AB
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