Mathematics · May / June 2025 · Paper 21 · Question 6

YMA01

As printed in the paper

The question as text

6. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

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

Show the answer and mark scheme

6(i)[5]
x
2
2
2
One correct law applied – examples (not exhaustive):
x
+
11
=
(
)
(
)
2log
4
−
=
log
4
−
2
,
log
log
(
11
)
log 2
x
x
x
+
−
2
2
2
2
2
2
3
=
log 8
2
Correctly combines two original terms e.g.
(
)
(
)
M1
2log
4
−
x
=
log
4
x
+
44
2
2
Correct equation not involving logs (
)
2
4
−
x
=
4
x
+
44
2
(
)(
)
x
−
12
x
−
28
=
0
x
−
14
x
+
2
=
0
x
=
...
dM1
x
= − only
2
x
2
x
+
1
2
x
+
1
x
(ii) Combines
y = ×
6 3
=
3
3
= ×
6 3
and
y
x
3
×=
3
6
x
3
=
2
=
log 2
x
3
log ''2''
y = ×
6 3
3
= ×
6 ''2''
coordinates of P are: (
)
log 2, 12
3
(i)
B1: Sight of one correct log law applied. See scheme. There will be other equivalent examples
+
11
x
=
(
)
e.g.
log
log
x
+
11
−
1
. This mark can still be awarded if they have made errors before
2
2
2
applying the law and do not withhold this mark for any subsequent errors.
(
2 4
−
x
M1: Correctly combines two of the three original terms of the equation. e.g.
log
2
+
11
x
May be implied by a correct equation in any form not involving logs provided it has not come from
(
)
2
log (4
−
)
2
4
−
x
x
incorrect log work. e.g.
2
=
3
=
8
is M0
+
11
x
+
11
x
log
2
2
2
A1: Correct equation in any form not involving logs which has not come from incorrect log work.
dM1: Solves a quadratic equation via factorisation, the formula or completing the square (usual rules apply).
They are not allowed to just state the roots. Dependent on the previous method mark.
A1:
x = − only (the 14 if found must be rejected or not included in their final answer)
2
(ii) Note that if logarithms are written as decimals instead of keeping their values exact then
if they find x first the maximum score is M1dM0A0ddM0A0
if they find y first the maximum score is M1dM1A1ddM0A0
M1: Sets up a correct equation in x or y. Most likely in x but it is possible to form an equation in y
( )
2
2
y
2
x
+
1
x
e.g.
=
3
= ×
3
3
= ×
3
y
6
dM1: A correct method of solving the equation in x (or y) to find either
an unsimplified expression or value for x in terms of logs, or
an expression or value for y not involving logs
It is dependent on the previous method mark. Condone arithmetical slips but any log/index work must
be correct.
There may be some alternative approaches here to solving:
e.g. taking logarithms of base 3:
log (3
2
x
+
1
)
=
log (6 3 )
×
x
2
x
+ =
1
log 6
+
log 3
x
2
x
+ = +
1
1 log 2
+
x
x
=
...
3
3
3
3
3
e.g. forming a quadratic in 3
x (or another variable)
3 3
2
6 3
0
3 (3
2)
0
3
2
...
×
x
−×
x
=
x
x
−
=
x
=
x
=
2
2
y
y
If solving an equation in y then e.g.
y
= ×
3
y
= ×36
3
y
=
6
− or log2
A1: Correct x coordinate (
log 2 or equivalent expressions such as
log 6 1
log3 o.e.) or y coordinate
3
3
which must be 12
ddM1: Uses a correct method to find values for both x and y. Condone arithmetical slips but any log/index
work must be correct.
It is dependent on the previous two method marks.
May be implied by a correct coordinate for their x or y. Do not allow logarithms to be written as
decimals.
A1: (
)
log 2,12 Accept if written separately e.g.
x
=
log 2,
y
=
12
. isw if they incorrectly state the values
3
3
of a and b after correct coordinates are seen.
Condone missing brackets but do not allow the coordinates to be the wrong way round.
6(ii)[5]
x
2
2
2
One correct law applied – examples (not exhaustive):
x
+
11
=
(
)
(
)
2log
4
−
=
log
4
−
2
,
log
log
(
11
)
log 2
x
x
x
+
−
2
2
2
2
2
2
3
=
log 8
2
Correctly combines two original terms e.g.
(
)
(
)
M1
2log
4
−
x
=
log
4
x
+
44
2
2
Correct equation not involving logs (
)
2
4
−
x
=
4
x
+
44
2
(
)(
)
x
−
12
x
−
28
=
0
x
−
14
x
+
2
=
0
x
=
...
dM1
x
= − only
2
x
2
x
+
1
2
x
+
1
x
(ii) Combines
y = ×
6 3
=
3
3
= ×
6 3
and
y
x
3
×=
3
6
x
3
=
2
=
log 2
x
3
log ''2''
y = ×
6 3
3
= ×
6 ''2''
coordinates of P are: (
)
log 2, 12
3
(i)
B1: Sight of one correct log law applied. See scheme. There will be other equivalent examples
+
11
x
=
(
)
e.g.
log
log
x
+
11
−
1
. This mark can still be awarded if they have made errors before
2
2
2
applying the law and do not withhold this mark for any subsequent errors.
(
2 4
−
x
M1: Correctly combines two of the three original terms of the equation. e.g.
log
2
+
11
x
May be implied by a correct equation in any form not involving logs provided it has not come from
(
)
2
log (4
−
)
2
4
−
x
x
incorrect log work. e.g.
2
=
3
=
8
is M0
+
11
x
+
11
x
log
2
2
2
A1: Correct equation in any form not involving logs which has not come from incorrect log work.
dM1: Solves a quadratic equation via factorisation, the formula or completing the square (usual rules apply).
They are not allowed to just state the roots. Dependent on the previous method mark.
A1:
x = − only (the 14 if found must be rejected or not included in their final answer)
2
(ii) Note that if logarithms are written as decimals instead of keeping their values exact then
if they find x first the maximum score is M1dM0A0ddM0A0
if they find y first the maximum score is M1dM1A1ddM0A0
M1: Sets up a correct equation in x or y. Most likely in x but it is possible to form an equation in y
( )
2
2
y
2
x
+
1
x
e.g.
=
3
= ×
3
3
= ×
3
y
6
dM1: A correct method of solving the equation in x (or y) to find either
an unsimplified expression or value for x in terms of logs, or
an expression or value for y not involving logs
It is dependent on the previous method mark. Condone arithmetical slips but any log/index work must
be correct.
There may be some alternative approaches here to solving:
e.g. taking logarithms of base 3:
log (3
2
x
+
1
)
=
log (6 3 )
×
x
2
x
+ =
1
log 6
+
log 3
x
2
x
+ = +
1
1 log 2
+
x
x
=
...
3
3
3
3
3
e.g. forming a quadratic in 3
x (or another variable)
3 3
2
6 3
0
3 (3
2)
0
3
2
...
×
x
−×
x
=
x
x
−
=
x
=
x
=
2
2
y
y
If solving an equation in y then e.g.
y
= ×
3
y
= ×36
3
y
=
6
− or log2
A1: Correct x coordinate (
log 2 or equivalent expressions such as
log 6 1
log3 o.e.) or y coordinate
3
3
which must be 12
ddM1: Uses a correct method to find values for both x and y. Condone arithmetical slips but any log/index
work must be correct.
It is dependent on the previous two method marks.
May be implied by a correct coordinate for their x or y. Do not allow logarithms to be written as
decimals.
A1: (
)
log 2,12 Accept if written separately e.g.
x
=
log 2,
y
=
12
. isw if they incorrectly state the values
3
3
of a and b after correct coordinates are seen.
Condone missing brackets but do not allow the coordinates to be the wrong way round.

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