Mathematics · May / June 2024 · Paper 31 · Question 2
YMA01
As printed in the paper
The question as text
2 5 8 2. x g( ) x 2
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
2(a)[3]
Attempts all 3 constants. E.g. Sets
x
=
2
C
=
..., 2
−
B
+
C
=
8
B
=
...
dM1
and compares
2x terms gives
...
A =
6
2
1
x
−+
−
2
x
6
(
)
2
2
−+
1
d
=
−
+
6ln
−
2
2(b)[4]
x
x
x
x
x
x
−
2
8
8
6
(
)
2
2
x
−+
1
d
x
=
x
−
x
+
6ln
x
−
2
=
56
+
6ln6 12
−
−
6ln 2
−
2
x
4
4
=
44
+
6ln3
=
2,
= −
5,
=
2
, and/or 28
2ln3
ANSWERS IN (a) of
A
B
C
+
in (b) please send to review
(a)
Note that correct values for A, B, and C imply the first two marks
M1:
Attempts to find one of the constants via
(
)
2
an identity e.g.2
x
−
5
x
+
8
=
Ax
+
B
(
x
−
2)
+
C
Note that if an identity is used it must be correct. Condone arithmetical slips when
evaluating.
The identity may be implied by correct equations
2 :
e.g.
=
2,
:
−
2
+
= −
5, constant :
−
2
+
=
8
x
A
x
A
B
B
C
division is used look for a quotient as far as 2x2
1
x
−
2
x
−
2 2
x
−
5
x
+
8
....................
6
Note that the division may appear in many different formats or may be by inspection so 2x is
sufficient to score this mark.
2
2
−
5
+
8
2 (
−
2)
+
4
−
5
+
8
−
(
−
2)
−
2
+
8
6
x
x
x x
x
x
x
e.g.
=
=
2
x
+
=
2
x
−+
1
x
−
2
x
−
2
x
−
2
x
−
2
e.g.
2x
−
1
2x
2
−
6
x
x
−
2
−
4x
2
Via an identity: If three simultaneous equations are formed allow this mark to be scored for
proceeding to values for A, B and C. You do not need to check this. Allow sign slips only in
forming the equations. Via division: Look for a quotient of 2x
± and a constant remainder
c
6
A1:
2
−+
1
The expression must be written; it cannot be awarded for just stating the correct
x
x
−
2
values of A, B and C, unless it is later written correctly as an expression in (b) (may be seen within
an integral)
(b)
C
M1:
Attempts to integrate the term
proceeding to
ln
x −
2
C does not have to be equal D (do
D
x −
2
not be concerned with brackets or modulus sign)
A1ft: Correct integration of g( )x for their A, B and C in part (a).
2
C
Ax
They must be integrating an expression of the form
Ax
+
B
+
+
Bx
+
C
ln
−
2
2
x
(where
A B C ≠
, ,
0
) which may be unsimplified with or without a constant of integration.
Do not be concerned by the absence of the bracket or modulus on the
2
(as both values
x −
substituted in will give positive lns)
dM1: Attempts to proceed to the form
+
ln
by
p
q
r
2
using the limits 8 and 4 in an expression of the formEx
+
Fx
+
D
ln
x
−
2
o.e. (where
, ,
0
) and subtracting either way round
D E F ≠
combining their ln terms correctly. e.g. this mark cannot be scored forln 6
ln 6
−
ln 2
=
=
ln3
ln 2
but treat as a sign slip in front of the log rather than incorrect log work e.g.
6ln6
6ln 2
6ln3
−
= −
Condone slips when evaluating or multiplying out brackets only (condone a spurious + c present
for this mark). The substitution and combining of their ln terms can be implied by their
provided no incorrect log work is seen. Note that ln r does not have to be ln 3 e.g. it may be ln 729.
A1:
Withhold the final mark if there is an integral and dx around the answer or + c
44
+
6ln3
Alt (b) Attempts via substitution
Note there may be attempts using substitution e.g.
=
−
2
on some or all of their expression.
u
x
8
6
6
6
e.g.
[
]
6
d
x
→
d
u
=
6ln
u
x
−
2
u
2
4
2
Correct attempt to integrate C
M1:
u to
D
ln
u C does not have to be equal D
2
2
C
Au
Ax
A1ft:
(
2)
(2
)
ln
or condone
ln
A u
+
+
B
+
+
A
+
B x
+
C
u
+
Bx
+
C
u
2
2
u
dM1: Attempts to either substitute in the limits 6 and 2, subtracts either way round and combines their ln
terms correctly, or substitutes back to an expression in terms of x and proceeds as in the main
2
scheme and notes. If they are working in a mixture of variables,
−
+
6ln
then the correct
x
x
u
limits must be substituted into the appropriate terms.
A1:
44
+
6ln3
Withhold the final mark if there is an integral and dx around the answer or + c
6
10
6
=
log
=
log 10
−
log
3
log
=
6
−
3log
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