Mathematics · Oct / Nov 2024 · Paper 31 · Question 6

YMA01

As printed in the paper

The question as text

6. The functions f and g are defined by 21 f (x) = 6 – x  0 x + 2 3 g (x) = x 2 + 5 x ∈ 

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

Show the answer and mark scheme

6(a)[2]
2
gf 2
( )
g 3
( )
3
5
=
=
+
= 14
6(b)[3]
Correct attempt at inverse
21
21
21
=
6
−
2
+
3
=
=
−
3
÷
2
y
x
x
2
+
3
6
−
6
−
x
y
y
3
+
3
21
3
x
( )
or
( )
f
−
1
x
=
f
−
1
x
=
−
2 6
(
)
2 6
(
)
2
−
x
−
x
1
−
„
x
<
6
6(c)[3]
(
)
2
gg( )
=
126
2
+
5
+
5
=
126
2
+
5
=
121
x
x
x
( )
2
=
''6''
= ±
6
x
x
= ±
6
x
(a)
( )
M1: Full attempt at
gf 2 applying the rules in the correct order
Look for one of
21
An attempt at
followed by an attempt at
k +
2
5
6
−
→
k
2
×
2
+
3
21
2
An attempt at substituting x = 2 into
6
−
+
5
2
+
3
x
A1: 14
Allow just 14 with no incorrect working for both marks
(b)
M1: For an attempt at the method of finding the inverse. It is the process you are scoring here.
Look for a process that starts with y as a function of x and proceeds to x as a function of just y.
21
ax
+
b
Starting with
=
6
−
, or possibly a changed function of the form
=
, which proceeds to a
y
y
2
+
3
2
+
3
x
x
21
form x = function of just y. Some candidates may swap x and y first. E.g
=
6
−
x
2
y
function y = function of just x which is fine.
3
+
3
21
3
A1: Correct inverse
( )
x
or
( )
o.e. but condone y or
f − for
f
−
1
=
f
−
1
=
−
x
x
(
)
(
)
2 6
−
2 6
−
2
x
x
−
3
x
−
3
10.5
21
Allow other equivalents such as
,
and
=
=
−
1.5
=
−
y
y
y
(
)
(
)
2
x
−
6
6
−
x
6
−
x
ISW following a correct answer
, [
B1: Correct domain written in correct form. E.g. 1
1, 6
)
−
„
<
6
−
x
(c)
M1: For an attempt to ''undo'' g once in an attempt to find the value of g( ),
2x or
x +
2
5
.
x
This may be scored for any of the following methods (condoning slips)
(
)
2
x
2
+
5
+
5
=
126
x
2
+
5
=
121
(
)
(
)(
)
2
2
4
2
2
2
2
x
+
5
+
5
=
126
x
+
10
x
−
96
=
0
x
−
6
x
+
16
=
0
x
=
...
Condone (
)
2
2
2
2
5
5
126
4
10
96
0
awrt 2.45
OR
x
+
+
=
x
+
x
−
=
x
=
x =
g( )
x =
126
−
5
Do not award this mark if the candidate believes that to find x you only need to undo g once.
So, answers like
x =
11
or
x =
6
will score 0,0,0 unless further work is seen
dM1: For a complete attempt to find the exact solution to the equation. Examples of this would include
(
)
2
2
+
5
+
5
=
126
2
+
5
=
121
=
121
±
5
x
x
x
(
)
(
)(
)
2
2
2
2
2
2
+
5
+
5
=
126
4
+
10
−
96
=
0
−
6
+
16
=
0
=
6
x
x
x
x
x
x
)
)
(
or (
2
2
x
2
+
5
+
5
=
126
x
=
6
x
2
+
5
+
5
=
126
x
=
−
6
(
)
2
2
4
2
x
+
5
+
5
=
126
x
+
10
x
−
96
=
0
x
=
6
x =
'11' 5
−
6
A1:
x = ±
only. Ignore any reference to 4i
±
if included.
Sight of
±
6
without incorrect working scores all 3 marks

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