Mathematics · Oct / Nov 2025 · Paper 11 · Question 2
YMA01
As printed in the paper
The question as text
2. Find 2 ( x 3 ) ( 2 x 5 ) d x writing the answer in simplest form. (5)
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
2
−
+
(
x
3) (2
x
5) d
x
2
(
x
−
3) (2
x
+
5)
=
x
2
−
6
x
+
9 (2
x
+
5)
=
...
or e.g.
2
(
x
−
3) (2
x
+
5)
=
(
x
−
3) 2
x
2
−
x
−
15
=
...
=
3
−
2
−
+
2
x
7
x
12
x
45
1
7
−
2
+
=
4
−
3
−
2
+
+
(
x
3) (2
x
5) d
x
x
x
6
x
45
x
c
2
3
Notes
M1: Attempts to multiply out. Look for an attempt to multiply two brackets together to form a
quadratic before the result is combined with the third bracket.
Condone slips e.g. (
)
x
−
3
2
=
x
2
±
9
and it may be left unsimplified but it must lead to an
expression that could be simplified to the form
ax
3
+
bx
2
+
cx
+
d
where , , ,
a b c d ≠
Condone an answer appearing without an intermediate step only if it is of the form (or can be
expressed in the form)
2
x
3
+
px
2
+
qx
±
45
where
p q ≠
,
0
3
A1:
For
2
x
−
7
x
2
−
12
x
+
45
but allow this unsimplified with the terms uncollected
e.g.
2
x
3
+
5
x
2
−
12
x
2
−
30
x
+
18
x
+
45
and award this mark as soon as a correct expression is
seen. Inclusion of “+ c” here scores A0 but the final mark is still available.
−
2
+
=
3
−
2
−
+
+ is A0 but
So
(
x
3) (2
x
5)
2
x
7
x
12
x
45
c
2
(
x
−
3) (2
x
+
5)d
x
=
2
x
3
−
7
x
2
−
12
x
+
45 d
x
+
c
is ok for A1
M1:
For an attempt to integrate any 2 terms with different powers of their simplified or unsimplified
expression by increasing the power of x by 1 following an attempt to expand all 3 brackets.
It is for any 2 of,
...
x
3
→
...
x
4
, ...
x
2
→
...
x
3
, ...
x
→
...
x
2
,
k
→
kx
3
→
3 1
+
Allow indices to be unprocessed e.g.
...
x
...
x
There must have been an attempt to expand all 3 brackets so do not award for e.g.
x
3
(
)
2
x
−
3
=
x
2
−
6
x
+
9
→
−
3
x
2
+
9
x
3
) (
)
1
or e.g. (
) (
2
)
(
2
or (
) (
)
(
) (
x
−
3
2
x
+
5
→
x
−
3
x
2
+
5
x
x
−
3
2
2
x
+
5
→
x
−
3
3
2
x
+
3
A1ft: For 2 different correctly integrated and simplified terms with different powers with indices
processed.
Follow through on any two different terms with different powers of their simplified or
unsimplified expansion.
1
7
4
2
A1:
x
−
x
3
−
6
x
+
45
x
+ or exact simplified equivalent including the + c all on one line.
c
2
3
Apply isw once a correct expression, including “+ c” is seen.
Condone the presence of spurious integral signs and/or dx’s.
Alternative attempts to integrate such as integration by parts or by
substitution – send to review.
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