Mathematics · May / June 2025 · Paper 31 · Question 3

YMA01

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The question as text

3. log10 y log10 x O (–3.5, 0) (0, –2) Figure 1 Figure 1 shows a linear relationship between log10 y and log10 x The line passes through the points (–3.5, 0) and (0, –2) as shown.

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Show the answer and mark scheme

3(a)[2]
2
log
y
= ±
log
x
−
2
10
10
3.5
4
log
y
= −
log
x
−
2
10
10
7
Notes:
2
4
M1: Obtains an equation of the form
log
y
= ±
log
x
−
2
oe e.g.
log
y
= −
log
x
+
log 0.01
10
3.5
10
10
7
10
10
4
2 log
4
Note
y
= −
x
−
2
or
y
= ±
x
−
2
or
log
y
= −
x
−
2
score M0
10
10
7
3.5
7
If this is attempted via simultaneous equations e.g. 0
= −
3.5
m
+
c
,
−
2
=
c
it is scored for reaching a
2
2
gradient of
which may be implied by awrt 0.57 and then
log
y
= ±
log
x
−
2
as above.
±
10
3.5
10
3.5
Condone log or lg for
log
but not ln or any other incorrect base.
10
4
A1:
Correct equation
log
y
= −
log
x
−
2
or equivalent.
10
10
7
4
e.g.
7log
y
+
4log
x
+
14
=
0,
log
y
= −
log
x
+
log 0.01
10
10
10
7
10
10
2
Condone
log
y
= −
log
x
−
2
and isw once a correct equation is seen.
10
3.5
10
Condone log or lg for
log
but not ln
10
log
y
= −
log
x
−
2
10
7
10
4
−
4
log
x
7
−
2
4
−
log
x
−
2
10
−
log
x
10
y
=
10
7
or
y
=
10
or
y
=
10
7
10
×
10
−
2
or e.g.
4
y
−
log
log
2
log
2
y
=
x
7
−
= −
10
10
10
4
−
x
7
or e.g.
4
4
log
y
= −
log
x
−
2
log
yx
= −
2
7
7
10
10
10
4
−
log
2
4
4
x
7
−
10
−
log
x
−
y
10
or
y
10
10
10
−
2
y
x
10
−
2
=
=
7
×
=
7
×
or e.g.
y
log
10
y
4
4
4
10
−
10
−
2
10
−
2
10
log
yx
7
10
−
2
10
−
2
x
7
=
=
or
=
yx
7
=
10
4
−
x
7
1
4
−
y
x
oe
=
7
100
Do not allow marks for part (a) to be scored in part (b)
Main method: Starting from an equation of the form
log
y
=
m
log
x
+
c
where m and c are nonzero.
10
10
M1: Correctly applies
10
lhs
=
10
rhs
to their
log
y
=
m
log
x
+ including the use of
c
10
log
y
=
y
10
10
10
e.g.
log
y
=
m
log
x
+
c
y
=
10m
log
x c
+
or
log
y
=
m
log
x
+
c
y
=
10
m
log
x
×
10
c
10
10
10
10
10
10
or combines their
log
y and
m
log
x correctly to obtain
log
yx
m
= ± or equivalent e.g.
c
10
10
10
y
log
y
=
m
log
x
+
c
log
y
=
log
x
m
+
c
log
y
−
log
x
m
=
c
log
= ±
c
10
10
10
10
10
10
10
x
m
May be implied by their work. Note that e.g.
4log
4
−
x
log
y
= −
log
x
−
2
y
=
10
10
−
10
2
scores M0
7
10
7
10
dM1: Depends on the previous method mark.
Complete and correct work to obtain an equation in x and y only or for obtaining values for p and q
which must not involve logs.
You can condone sign slips with their constant term but the log work must be correct.
1
4
4
4
−
−
−
A1:
Correct equation in the form required
y
=
x
7
or e.g.
y
=
0.01
x
7
or e.g.
y
=
10
−
2
x
7
100
4
−
x
7
8
y
=
from correct work. Allow equivalent exact values for q e.g.
but not e.g.
q = −
q = −
100
14
Note that the correct answer does not imply the method marks and you need to check carefully.
Alternatives for part (b):
Way 2:
M1: Starts with
y
=
px
q
and obtains
log
y
=
log
p
+
log
x
q
oe e.g.
log
y
=
log
p
+
q
log
10
10
10
10
10
dM1: Depends on the previous method mark.
Correct attempt at p and q by comparing their equation of the form
log
y
=
m
log
x
+
c
from part (a)
10
10
or e.g. identifies the gradient and yintercept from the sketch to obtain values for p and q not involving
logs e.g.
log
p
=
c
p
=
10c
and
log
qx
=
q
log
x
q
=
m
10
10
10
4
1
4
−
−
−
A1:
Correct equation in the form required
y
=
x
or e.g.
y
=
0.01
x
or e.g.
y
=
10
−
2
x
7
7
100
8
2
correct work. Allow equivalent exact values for q e.g.
q = −
but not e.g.
q = −
14
3.5
Note that correct values for p and q with no incorrect work seen scores M1dM1
Note that a correct equation with no incorrect work seen scores M1dM1A1
Way 3:
M1:
7log
y
+
4log
x
= −
14
log
y x
7
4
= −
14
y x
7
4
=
10
−
14
10
10
10
Multiplies up to obtain integer coefficients, combines the log terms correctly then applies
10
10
−
14
dM1:
y x
7
4
=
10
−
14
y
=
7
x
4
Complete and correct attempt to obtain y in terms of x.
1
4
4
4
−
−
−
A1:
Correct equation in the form required
y
=
x
7
or e.g.
y
=
0.01
x
7
or e.g.
y
=
10
−
2
x
7
100
4
−
x
7
8
y
=
from correct work. Allow equivalent exact values for q e.g.
q = −
but not e.g.
100
14
Way 4:
M1: (
) (
)
2
2
log
x
, log
y
=
0,
−
2
x
=
1,
y
=
10
−
p
=
10
−
10
10
Uses (
)
0,
−
2
correctly to establish the value of p.
dM1: (
) (
)
log
x
, log
y
= −
3.5, 0
y
=
1,
x
=
10
−
3.5
=
1 10
−
2
×
10
−
3.5
q
−−
2 3.5
q
=
0
q
=
...
10
10
Uses (
)
−
3.5, 0
correctly with their value of p to obtain a value for q.
1
4
4
−
−
−
A1: Correct equation in the form required
y
=
x
7
or e.g.
y
=
0.01
x
7
or e.g.
y
=
10
−
2
x
100
4
−
x
7
8
y
=
from correct work. Allow equivalent exact values for q e.g.
q = −
but not e.g.
q = −
100
14
Fully correct answer in (b) with no working scores full marks in (b).
Partially correct answer in (b) with no working scores no marks.
(
)
3(b)[3]
2
log
y
= ±
log
x
−
2
10
10
3.5
4
log
y
= −
log
x
−
2
10
10
7
Notes:
2
4
M1: Obtains an equation of the form
log
y
= ±
log
x
−
2
oe e.g.
log
y
= −
log
x
+
log 0.01
10
3.5
10
10
7
10
10
4
2 log
4
Note
y
= −
x
−
2
or
y
= ±
x
−
2
or
log
y
= −
x
−
2
score M0
10
10
7
3.5
7
If this is attempted via simultaneous equations e.g. 0
= −
3.5
m
+
c
,
−
2
=
c
it is scored for reaching a
2
2
gradient of
which may be implied by awrt 0.57 and then
log
y
= ±
log
x
−
2
as above.
±
10
3.5
10
3.5
Condone log or lg for
log
but not ln or any other incorrect base.
10
4
A1:
Correct equation
log
y
= −
log
x
−
2
or equivalent.
10
10
7
4
e.g.
7log
y
+
4log
x
+
14
=
0,
log
y
= −
log
x
+
log 0.01
10
10
10
7
10
10
2
Condone
log
y
= −
log
x
−
2
and isw once a correct equation is seen.
10
3.5
10
Condone log or lg for
log
but not ln
10
log
y
= −
log
x
−
2
10
7
10
4
−
4
log
x
7
−
2
4
−
log
x
−
2
10
−
log
x
10
y
=
10
7
or
y
=
10
or
y
=
10
7
10
×
10
−
2
or e.g.
4
y
−
log
log
2
log
2
y
=
x
7
−
= −
10
10
10
4
−
x
7
or e.g.
4
4
log
y
= −
log
x
−
2
log
yx
= −
2
7
7
10
10
10
4
−
log
2
4
4
x
7
−
10
−
log
x
−
y
10
or
y
10
10
10
−
2
y
x
10
−
2
=
=
7
×
=
7
×
or e.g.
y
log
10
y
4
4
4
10
−
10
−
2
10
−
2
10
log
yx
7
10
−
2
10
−
2
x
7
=
=
or
=
yx
7
=
10
4
−
x
7
1
4
−
y
x
oe
=
7
100
Do not allow marks for part (a) to be scored in part (b)
Main method: Starting from an equation of the form
log
y
=
m
log
x
+
c
where m and c are nonzero.
10
10
M1: Correctly applies
10
lhs
=
10
rhs
to their
log
y
=
m
log
x
+ including the use of
c
10
log
y
=
y
10
10
10
e.g.
log
y
=
m
log
x
+
c
y
=
10m
log
x c
+
or
log
y
=
m
log
x
+
c
y
=
10
m
log
x
×
10
c
10
10
10
10
10
10
or combines their
log
y and
m
log
x correctly to obtain
log
yx
m
= ± or equivalent e.g.
c
10
10
10
y
log
y
=
m
log
x
+
c
log
y
=
log
x
m
+
c
log
y
−
log
x
m
=
c
log
= ±
c
10
10
10
10
10
10
10
x
m
May be implied by their work. Note that e.g.
4log
4
−
x
log
y
= −
log
x
−
2
y
=
10
10
−
10
2
scores M0
7
10
7
10
dM1: Depends on the previous method mark.
Complete and correct work to obtain an equation in x and y only or for obtaining values for p and q
which must not involve logs.
You can condone sign slips with their constant term but the log work must be correct.
1
4
4
4
−
−
−
A1:
Correct equation in the form required
y
=
x
7
or e.g.
y
=
0.01
x
7
or e.g.
y
=
10
−
2
x
7
100
4
−
x
7
8
y
=
from correct work. Allow equivalent exact values for q e.g.
but not e.g.
q = −
q = −
100
14
Note that the correct answer does not imply the method marks and you need to check carefully.
Alternatives for part (b):
Way 2:
M1: Starts with
y
=
px
q
and obtains
log
y
=
log
p
+
log
x
q
oe e.g.
log
y
=
log
p
+
q
log
10
10
10
10
10
dM1: Depends on the previous method mark.
Correct attempt at p and q by comparing their equation of the form
log
y
=
m
log
x
+
c
from part (a)
10
10
or e.g. identifies the gradient and yintercept from the sketch to obtain values for p and q not involving
logs e.g.
log
p
=
c
p
=
10c
and
log
qx
=
q
log
x
q
=
m
10
10
10
4
1
4
−
−
−
A1:
Correct equation in the form required
y
=
x
or e.g.
y
=
0.01
x
or e.g.
y
=
10
−
2
x
7
7
100
8
2
correct work. Allow equivalent exact values for q e.g.
q = −
but not e.g.
q = −
14
3.5
Note that correct values for p and q with no incorrect work seen scores M1dM1
Note that a correct equation with no incorrect work seen scores M1dM1A1
Way 3:
M1:
7log
y
+
4log
x
= −
14
log
y x
7
4
= −
14
y x
7
4
=
10
−
14
10
10
10
Multiplies up to obtain integer coefficients, combines the log terms correctly then applies
10
10
−
14
dM1:
y x
7
4
=
10
−
14
y
=
7
x
4
Complete and correct attempt to obtain y in terms of x.
1
4
4
4
−
−
−
A1:
Correct equation in the form required
y
=
x
7
or e.g.
y
=
0.01
x
7
or e.g.
y
=
10
−
2
x
7
100
4
−
x
7
8
y
=
from correct work. Allow equivalent exact values for q e.g.
q = −
but not e.g.
100
14
Way 4:
M1: (
) (
)
2
2
log
x
, log
y
=
0,
−
2
x
=
1,
y
=
10
−
p
=
10
−
10
10
Uses (
)
0,
−
2
correctly to establish the value of p.
dM1: (
) (
)
log
x
, log
y
= −
3.5, 0
y
=
1,
x
=
10
−
3.5
=
1 10
−
2
×
10
−
3.5
q
−−
2 3.5
q
=
0
q
=
...
10
10
Uses (
)
−
3.5, 0
correctly with their value of p to obtain a value for q.
1
4
4
−
−
−
A1: Correct equation in the form required
y
=
x
7
or e.g.
y
=
0.01
x
7
or e.g.
y
=
10
−
2
x
100
4
−
x
7
8
y
=
from correct work. Allow equivalent exact values for q e.g.
q = −
but not e.g.
q = −
100
14
Fully correct answer in (b) with no working scores full marks in (b).
Partially correct answer in (b) with no working scores no marks.
(
)

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