Mathematics · May / June 2025 · Paper 31 · Question 1
YMA01
As printed in the paper
The question as text
1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The functions f and g are defined by x 2 x ∈ x 0 x f ( ) x 3 1 x x x ∈ x 0 2 g( ) 4
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
1(a)[2]
( )
( )
2
2 1
2 1
( )
gf 1
=
g
=
4
−
3 1 1
3 1 1
×+
×+
15
oe
=
4
1(b)[2]
2
( )
or ( )
f
x …
0
f
x <
3
2
0
„
f
( )
x <
3
Notes:
(a)
M1: Correct order of operations so requires an attempt to find f (1) and then apply g to their f (1)
2
2
x
Allow for an attempt to substitute x = 1 into
4
−
3
x
+
1
3
A1:
Correct value. Allow equivalents e.g. 3.75,
3 4
Correct answer only scores both marks.
(b)
Allow the use of y or f but not x in place of f (x) for both marks.
2
One correct bound e.g. either ( )
or ( )
B1:
f
x …
0
f
x <
which may be seen embedded in another
3
inequality.
Allow exact equivalents for 2
including 0.6
3
2
2
2
2
B1:
Correct range. Allow equivalent notation e.g.
0
„
f
,
0,
,
0,
,
>
…
0
<
y
∈
y
3
3
3
3
2
Allow written as separate inequalities e.g. ( )
f
x
…
0, f
( )
x
<
,
f
( )
x
…
0 and f
( )
x
<
3
2
but not e.g. ( )
( )
f
x
…
0 or f
x <
3
Allow y or f for f (x).
Allow exact equivalents for 2
including 0.6
3
1(c)[2]
2
x
(
)
y
=
3
xy
+
y
=
2
x
3
xy
−
2
x
= −
y
x
3
y
−
2
= −
y
3
1
x
+
or
2
y
(
)
x
=
3
xy
+
x
=
2
y
3
xy
−
2
y
= −
x
y
3
x
−
2
= −
x
3
y
+
1
x
( )
f
−
1
x
=
2
3
−
x
( )
( )
( )
or ( )
1(d)[2]
f
−
1
x
=
f
x
or
f
−
1
x
=
x
f x
=
x
x
2
x
x
2
x
"
"
or "
x =
"
x
or
=
x
=
2
3
x
3
x
1
2
−
3
3
x
+
1
−
+
(
)
(
)
(
)
(
)
x
3
x
+
1
=
2
x
2 3
−
x
or
x
=
x
2 3
−
x
or
2
x
=
x
3
x
+
1
x
=
...
1
x =
0, 3
(c)
M1: Attempts to change the subject.
If making x the subject, they must proceed as far as getting the two x terms on one side of the equation
and the term not in x on the other and factorise out the x.
If making y the subject, they must swap x and y first and then proceed as far as getting the two y terms
on one side of the equation and the term not in y on the other and factorise out the y.
(
)
x
(
)
−
x
A1:
Correct expression
( )
oe e.g.
( )
f
−
1
x
=
f
−
1
x
=
2
−
3
x
3
x
−
2
Accept equivalent expressions and isw after a correct answer is seen.
Do not be concerned with the lhs and just look for the correct expression in terms of x.
You can also ignore any reference to a domain as this is not required.
(d)
M1: Sets up one of the required equations e.g.
( )
( )
( )
or ( )
theirf
−
1
x
=
f
x
or
their f
−
1
x
=
x
f x
=
x
and proceeds to obtain at least one value for x.
( )
If using their
f
−
1
x
it must be a function of x.
Condone copying slips as long as the intention is clear.
1
. Allow exact equivalents for 1
A1:
Correct values
x =
0, 3
including 0.3
3
Do not isw e.g. if they clearly reject x = 0 then score A0
Ignore any attempts to find the y values.
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