Mathematics · May / June 2023 · Paper 31 · Question 6
YMA01
As printed in the paper
The question as text
6. y O x P Figure 2 Figure 2 shows a sketch of the graph y = f(x), where f(x) = 3½x – 2½ – 10 The vertex of the graph is at point P, shown in Figure 2.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
6(a)[2]
ff(0)
=
f( 4)
−
=
...
6(b)[2]
= 8
(
)
Attempts to solve
−
3
x
−
2
−
10
=
5
x
+
10
x
=
...
6(c)[2]
7
x >−
only
4
16
(
)
x
or
x
=
6(d)[3]
3
Attempts (
)
3 |
x
|
−
2
−
10
=
0
|
x
|
=
k
,
k
>
0
or (
)
3
−
x
−
2
−
10
=
0
x
= −
k
or (
)
3
x
−
2
−
10
=
0
x
=
k
x
= −
k
16
16
x
=
3 a
nd
−
with no other values
3
Notes
(a)
B1
For one correct coordinate
For (
)
B1
2,
−
10
. Allow
x
=
...,
y
=
...
Do not accept e.g. 6/3 unless 2 has been seen/identified with
this.
(b)
M1
For a full attempt at f f (0) . Can be scored for f (−4). Allow for use of their f(0) even if incorrect
as long as the process is clear, e.g. f(0) = .. stated or calculated first then used. May be scored by
first attempting ff(x) before substituting. This mark is for showing the correct process of
composites, so may be scored if there are slips or errors with modulus if the intent is clear.
A1cso ff(0)
8
=
only. A0 if other values given.
(c)
(
)
M1
Attempts to solve
3
x
2
10
5
x
10
x
...
Allow with equality or any inequality for the
−
−
−
=
+
=
M mark.
Alternatively, rearranges to |
x
−
2|
=
ax
+
b
,
squares both sides and solves the quadratic.
7
A1
x >−
(oe) only. If another inequality or value is given and not rejected withhold this mark.
4
(d)
Work for (d) must be seen or referred to in (d). Do not accept for work attempted in earlier parts but not
used in (d).
16
B1 For
x =
. Allow when seen even from incorrect working as it could be verified. May be seen on
3
16
sketch as long as referred to in (d). Allow also for
|
x =
|
3
M1
3 |
x −
achieve a value for |x|, or (
)
3
−−
x
2
−
10
=
0
to achieve a value for x, or for reflecting in the yaxis
(making negative) their 16
from an attempt at (
)
3
x −
2
−
10
=
0
. May be part of longer winded
3
attempts. Allow missing brackets for the M.
16
Note it is possible to arrive at an equation leading to
x = ±
from incorrect starting points, and such
3
methods will score M0.
16
16
A1
For
x = −
with no other values (aside their
x =
) . Must give the negative value, not just
3
3
1
6
|
x =
|
. May be stated on a sketch as long as work seen in (d). Do not isw if they clearly reject
3
this value later or if they try to form an inequality from the values, which is A0 as other values are
included.
A =
2 500
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