Mathematics · Oct / Nov 2024 · Paper 11 · Question 9
YMA01
As printed in the paper
The question as text
9. y C O x Figure 4 Figure 4 shows a sketch of the curve C with equation y = f(x), where f(x) = ( x x x )( 2 ) 5 3 4 20
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
9(a)[1]
)(
)
f( )
x
=
(
x
+
5
3
x
2
−
4
x
+
20
=
3
x
3
+
11
x
2
+
100
9(b)[3]
f ( )
′
x
=
9
x
2
+
22
x
(
)
( )
f ( 4)
′−
9
4
2
22
4
56
Finds
= ×−
+
×−=
9(c)[4]
Sets
2
f ( )
′
x
=
"9
x
+
22 "
x
=
''56''
14
(
)
9
2
22
56
0
,
4
x
+
x
−
=
x
=
−
9
(
)
−
1, 84
(d)(i)
(
)
−
4, 336
(ii)
Note: mark the question as a whole – do not be concerned if there is incorrect part labelling.
(a)
B1: Correct range of values given,
x
…
−
5
or exact equivalent in set notation. Must be
clearly identified as the answer, do not isw if a different range is given later but allow if
x = − is stated as well as the range.
5
(b)
M1: Attempts to multiply out to form a cubic (which may be unsimplified), achieving at
least
3
x +
3
.... 100
±
with at least one x or x2 term between, and allow if there are extra
constant terms between too.
M1: Differentiates to form a quadratic. Power decreased by 1 at least twice.
2
A1cso:
f ( )
′
x
=
9
x
+
22
x
Must have come from a correct expansion.
The product rule may be attempted:
M1: For
f ( )
′
x
=
(1)(3
x
2
−
4
x
+
20)
+
(
x
+
5)(
Ax
+
B
)
M1: Expands and simplifies to a quadratic as long as a sum of terms was attempted.
A1:
f ( )
′
x
=
9
x
2
+
22
x
(c)
M1: Finds f ( 4)
′−
for their f ( )x
′
dM1: Sets
f ( )
′
x
"9
x
2
22 "
x
''56''
with their derivative and value.
=
+
=
ddM1: Solves their
2
which must be a 3 term quadratic. May be implied by
"9
x
+
22
x
=
56''
the correct answer for their equation.
14
A1cso: For
x =
from fully correct work. Ignore references to x = 4
− if left in but A0 if
9
any erroneous extra solutions are given. The derivative must have been correct. Note
the same answer will follow if they have a constant term in their (otherwise correct)
derivative.
9(d)[2]
(
)
(
)
(i) B1:
−
1, 84
(ii)
B1:
−
4, 336
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