Mathematics · Oct / Nov 2025 · Paper 41 · Question 1
YMA01
As printed in the paper
Show the answer and mark scheme
1(a)[5]
+
2
=
+
2
5
x
1
x
4
2
(
) (
)
(
) (
) (
)
−
2
−
×−
2
−
×−
×−
3
5
5
2
3
5
2
3
4
5
(
)
+
= + −
×
+
×
+
×
1
x
1
2
x
x
x
2
2
2
2
3!
2
1
5
75
125
(
)
−
2
2
3
2
+
5x
=
−
x
+
x
−
x
4
4
16
8
2
1(b)[1]
x <
5
2
4
≈
2
4
−
20
x
+
75
x
or
a
=
4,
b
= −
20
and
c =
75
1(c)[1]
2
+
5
x
(a) Note that B1M1A0A1A0 is a possible mark trait
B1: For taking out a common factor of 1
4 (or 0.25 or accept
2−) so score for
2
1 1
. It cannot be for just 1
(
)
−
2
(
)
−
+
kx
or
0.25 1
+
kx
2
4 on its own
4
M1: Attempts the binomial expansion with index 2
−
Score for the correct binomial coefficient for the 3rd or 4th term with the correct
power of x
Look for (
) (
) (
oe or (
) (
) (
) (
−
×−
−
×−
×−
2
3
2
3
4
)
2
)
3
×
...
x
×
...
x
oe If no working is shown
2
3!
then the coefficient must be consistent with their k. Condone invisible or poor
bracketing for this mark.
−
−
2
2
Do not allow notation such as
,
unless interpreted correctly in further
1
2
work.
−
2
5
A1: Correct binomial expansion for
up to the 4th term. This may be left
1
+
2 x
unsimplified. May be implied by four correct terms. Isw if they subsequently
simplify incorrectly. Invisible
brackets may be implied by further work.
1
5
75
125
2
3
A1: Three correct and simplified terms from
−
x
+
x
−
x
May be written as
4
4
16
8
2
3
decimal equivalents e.g.
0.25 1.25
−
x
+
4.6875
x
−
15.625
x
. May be seen as a list or
5
1
written on different lines. Condone
−
for this mark.
4 x
1
5
75
125
2
3
A1:
−
+
−
oe See notes above for decimal equivalent forms. May be
x
x
x
4
4
16
8
seen as a list or written on different lines. Isw once the four correct terms are
5
1
seen unless they attempt to multiply by e.g. 16 Do not accept
−
4 x
Alt(a) Direct expansion
−
−
−
−
−
(
)(
( 2)( 3)
(
)(
( 2)( 3)( 4)
(
)(
)
)
2
)
3
(2
±
5 )
x
−
2
=
2
−
2
+ −
( 2) 2
−
3
±
5
x
+
2
−
4
±
5
x
+
2
−
5
±
5
x
2
6
1
B1: For
2− oe seen as the constant term e.g.
2
=
+
...
x
+
...
4
M1: For the correct 3rd or 4th term of the expansion. Condone missing brackets on
or
( 5 )
±
x
2
( 5 )x
±
3
A1: Correct expansion up to the 4th term which may be left unsimplified. May be
implied by four
correct terms. Isw if they subsequently simplify incorrectly.
Invisible brackets may be
implied by further work.
A1: As above in main scheme notes
A1: As above in main scheme notes
(b)
2
2
2
2
2
x <
B1:
oe e.g.
−
<
x
<
or e.g.
−
,
isw once a correct inequality is seen.
5
5
5
5
5
2
x <−
Condone
5
(c)
2
1
5
75
4
≈
2
2
B1ft:
4
−
20
x
+
75
x
but follow through on
16 ''
×
−
x
+
x
''
. Must be a
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