Mathematics · Oct / Nov 2024 · Paper 21 · Question 4

YMA01

As printed in the paper

The question as text

4. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The curve C has equation 1 1 y 4 x 9 x 3 x 0 2 2

Read automatically from the paper; the printed page above is exact.

Show the answer and mark scheme

4(a)[2]
1
−
1
y
=
4
x
+
9
x
+
3
2
2
d
9
y
1
3
=
2
x
−
−
x
−
2
2
d
2
x
4(b)[2]
d
9
y
−
1
−
3
=
0
2
x
−
x
=
0
4
x
−
9
=
0
x
=
...
2
2
d
x
2
9
x =
oe e.g. 2.25
4
(a)
M1:
1
−
1
−
1
−
3
1
1
2 1
−
−
1
For
x
→
x
or
x
→
x
2.
Allow unprocessed e.g.
x
→
x
or
x
→
2
2
2
2
2
Correct simplified derivative. d
y
A1:
x = is not required. Isw once a correct answer is seen.
d
1
3
Allow equivalent simplified expressions e.g.
2
−
−
4.5
−
x
2
x
2
9
9
2
−
1
−
−
3
+ scores A0 as does
2
−
1
+ −
−
3
x
2
x
2
c
x
2
x
2
2
2
(b)
−
1
−
3
M1:
Starts from a derivative of the form
Ax
−
Bx
where
A
×
B
>
0,
sets = 0 (seen or
2
2
implied) and solves via a correct method to obtain an expression or value of
9
A1:
x =
oe with no incorrect working seen in their method.
4
Condone if e.g. d
y
x >
0
is seen as long as they reach the correct value for x.
d
There must be no other values. Must come from a correct derivative but see * note
below.
9
9
The minimum acceptable for both marks is
2
−
1
−
−
3
=
0
=
x
x
x
2
2
2
4
The following is condoned:
9
2
9
4
81
9
2
−
1
−
−
3
=
0
−
=
0
−
=
0
=
x
x
x
2
2
2
4
3
4
1
2
3
x
x
x
2
x
2
Ignore any attempts to find the y coordinate.
*Allow full recovery in (b) if the working is correct in (a) but isw has been applied e.g.
d
9
y
−
1
−
3
−
1
−
3
(a)
=
2
x
−
x
=
4
x
−
9
x
M1A1 (isw)
2
2
2
2
d
x
2
9
(b)
4
−
1
−
9
−
3
=
0
=
M1A1
x
2
x
2
x
4
y
−
3
−
5
−
3
−
5
= −
x
+
x
oe e.g.
−
x
+
6.75
x
2
2
2
2
d
x
2
4
(ii)
d
2
9
−
3
27 9
−
5
16 0.5925...
y
2
2
(
)
= −
+
=
d
2
4
4
4
27
x
9
=
x
4
d
2
>
y
0
so (local) minimum
d
2
x
4(c)[2]
1
−
1
y
=
4
x
+
9
x
+
3
2
2
d
9
y
1
3
=
2
x
−
−
x
−
2
2
d
2
x
d
9
y
−
1
−
3
=
0
2
x
−
x
=
0
4
x
−
9
=
0
x
=
...
2
2
d
x
2
9
x =
oe e.g. 2.25
4
(a)
M1:
1
−
1
−
1
−
3
1
1
2 1
−
−
1
For
x
→
x
or
x
→
x
2.
Allow unprocessed e.g.
x
→
x
or
x
→
2
2
2
2
2
Correct simplified derivative. d
y
A1:
x = is not required. Isw once a correct answer is seen.
d
1
3
Allow equivalent simplified expressions e.g.
2
−
−
4.5
−
x
2
x
2
9
9
2
−
1
−
−
3
+ scores A0 as does
2
−
1
+ −
−
3
x
2
x
2
c
x
2
x
2
2
2
(b)
−
1
−
3
M1:
Starts from a derivative of the form
Ax
−
Bx
where
A
×
B
>
0,
sets = 0 (seen or
2
2
implied) and solves via a correct method to obtain an expression or value of
9
A1:
x =
oe with no incorrect working seen in their method.
4
Condone if e.g. d
y
x >
0
is seen as long as they reach the correct value for x.
d
There must be no other values. Must come from a correct derivative but see * note
below.
9
9
The minimum acceptable for both marks is
2
−
1
−
−
3
=
0
=
x
x
x
2
2
2
4
The following is condoned:
9
2
9
4
81
9
2
−
1
−
−
3
=
0
−
=
0
−
=
0
=
x
x
x
2
2
2
4
3
4
1
2
3
x
x
x
2
x
2
Ignore any attempts to find the y coordinate.
*Allow full recovery in (b) if the working is correct in (a) but isw has been applied e.g.
d
9
y
−
1
−
3
−
1
−
3
(a)
=
2
x
−
x
=
4
x
−
9
x
M1A1 (isw)
2
2
2
2
d
x
2
9
(b)
4
−
1
−
9
−
3
=
0
=
M1A1
x
2
x
2
x
4
y
−
3
−
5
−
3
−
5
= −
x
+
x
oe e.g.
−
x
+
6.75
x
2
2
2
2
d
x
2
4
(ii)
d
2
9
−
3
27 9
−
5
16 0.5925...
y
2
2
(
)
= −
+
=
d
2
4
4
4
27
x
9
=
x
4
d
2
>
y
0
so (local) minimum
d
2
x
9
0
<
x
<
4
(c)(i)
B1ft: Correct simplified second derivative.
Follow through their first derivative provided there are 2 terms with different
fractional (noninteger) powers.
d
2
y
x = is not required.
d
2
27
27
Condone
−
1
−
3
+
−
5
for
−
−
3
+
−
5
x
2
x
2
x
2
x
2
4
4
Inclusion of “ + c” scores B0ft as with the first derivative.
(ii)
B1:
Fully correct working, reasoning and conclusion. This requires:
a correct second derivative
9
to obtain 16
substitutes
27 or awrt 0.6 or truncated 0.5
x =
4
reference to the sign (> 0 or positive). Must see the evaluation as it is not
obviously positive but allow if expression is not fully evaluated as long as it is
clear it is positive (e.g. 8
8
−
>
0
).
9
27
reference to minimum
d
2
y
Condone work that doesn’t specifically reference
e.g.
d
2
x
3
5
9
−
27 9
−
16
2
2
−
+
=
>
0
so minimum is fine for this mark.
4
4
4
27
(d)
9
9
B1ft: For
0
<
<
. Allow
0
<
„
and allow equivalent statements e.g.
x
x
4
4
9
9
0,
,
0,
.
4
4
Follow through their positive x value from part (b). Condone an answer of
9
9
x „
Condone
0
„
„
x
4
4
9
Do not allow
−∞<
<
x
4
4(d)[1]
9
0
<
x
<
4
(c)(i)
B1ft: Correct simplified second derivative.
Follow through their first derivative provided there are 2 terms with different
fractional (noninteger) powers.
d
2
y
x = is not required.
d
2
27
27
Condone
−
1
−
3
+
−
5
for
−
−
3
+
−
5
x
2
x
2
x
2
x
2
4
4
Inclusion of “ + c” scores B0ft as with the first derivative.
(ii)
B1:
Fully correct working, reasoning and conclusion. This requires:
a correct second derivative
9
to obtain 16
substitutes
27 or awrt 0.6 or truncated 0.5
x =
4
reference to the sign (> 0 or positive). Must see the evaluation as it is not
obviously positive but allow if expression is not fully evaluated as long as it is
clear it is positive (e.g. 8
8
−
>
0
).
9
27
reference to minimum
d
2
y
Condone work that doesn’t specifically reference
e.g.
d
2
x
3
5
9
−
27 9
−
16
2
2
−
+
=
>
0
so minimum is fine for this mark.
4
4
4
27
(d)
9
9
B1ft: For
0
<
<
. Allow
0
<
„
and allow equivalent statements e.g.
x
x
4
4
9
9
0,
,
0,
.
4
4
Follow through their positive x value from part (b). Condone an answer of
9
9
x „
Condone
0
„
„
x
4
4
9
Do not allow
−∞<
<
x
4

Practise this paper →

Make a free account to save your answers and see your weak topics and ask the AI why a wrong answer is wrong.