Mathematics · Oct / Nov 2023 · Paper 11 · Question 4
YMA01
As printed in the paper
The question as text
4. y C O x Figure 1 1 Figure 1 shows a sketch of part of the curve C with equation y = x + 2
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
4(a)[1]
x = −
2
4(b)[2]
3
4
2
4
(
2)
2
x
+
x
+
x
=
x x
+
4(c)[3]
( 2, 0)
−
O
4(d)[1]
2 as the graphs intersect (each other) twice
1
(since
(
x
+
2)(
x
3
+
4
x
2
+
4 )
x
= is the same as
1
3
4
2
4
)
x
+
x
+
x
=
x
+
2
Note: Check for answers by the questions or on the Figure (rather than the diagram). If there is a
contradiction between the answers by the question and the main body of the solution then the main
body of the solution takes precedence.
(a)
x = − only. Accept labelled on the diagram but do not accept just 2
−. If other equations are given
B1:
2
(other than
y = ) then B0.
0
x = −is one of the solutions to the cubic so the equation of the asymptote must be seen
2
Beware that
in (a), on the diagram or by the question.
(b)
(
2
±
4
±
4)
(
+
2)(
2
±
2 )
M1:
Attempts to take out a linear factor eg
x x
x
or
x
x
x
. May be implied by the
correct answer.
x x+
(
2)
2
A1:
or (
+
2)(
+
2)
. Accept (
+
2)(
+
2)
and condone a missing trailing bracket on the
x x
x
x
x
x
x x +
(
2
2
final linear factor e.g. (
+
2)(
+ but
2
is A0.
x x
x
(
x
+
0)(
x
+
2)
2
Condone
. Isw once a correct expression is seen and ignore a spurious = 0
(c)
Note: If there are multiple attempts then score the highest one
B1:
Sketches a positive cubic anywhere on a set of axes. Do not penalise poor curvature provided the
intention is clear and ignore if the curvature looks asymptotic. The cubic does not need to have two turning
points for this mark.
Examples of acceptable cubic shapes (including where poor curvature would be condoned):
(If you are unsure then send to review)
B1:
A cubic which
has a turning point where the xaxis and vertical asymptote intersectpasses through the origin (from quadrants 3 to 1 or from quadrants 2 to 4 – it cannot start or stopat the origin) It also cannot be a turning point.
Do not penalise poor curvature provided the intention is clear and ignore if the curvature looks asymptotic.
B1:
( 2, 0)
−
indicated on the graph where their graph crosses or turns on the negative xaxis Do not be
concerned regarding any other points where the graph crosses or turns on the xaxis including the
origin.
Condone 2 labelled on the xaxis or the coordinates the wrong way round as (0,
−
2)
missing brackets.
2
The asymptote labelled
x = − does not score this mark and do not be concerned with the point of
intersection relative to the asymptote.
(d)
This mark can only be scored provided B1B1B0 or B1B1B1 is scored in (c)
B1:
2 (real) roots and a valid reason e.g. (graphs/curves/functions/equations/they)
intersect/meet/cross/touch (each other) twice. e.g. “intersect twice” or “2 intersections” can score
this mark.
May also refer to an intersection in the first and third quadrants.
Only withhold the mark if it is clear that they are referring to intersections between the cubic graph
and the xaxis.
Do not accept responses relating to the roots of the equation, use of the discriminant or other
algebraic methods which do not use the graphs.
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