Mathematics · Oct / Nov 2023 · Paper 41 · Question 1
YMA01
As printed in the paper
Show the answer and mark scheme
1(a)[4]
2
−
1
5
x
8
−
2
(2
−
5 )
x
=
1
−
or e.g.
4
2
2
5
x
4 1
−
2
2
3
(
2
) (
3
)
(
2
) (
3
) (
4
)
1
5
x
−
×−
5
x
−
×−
×−
5
x
(
)
= ×
8
1
+ −
2
×−
+
×−
+
×−
...
4
2
2!
2
3!
2
8
75
2
3
=
2 10
+
x
+
x
+
125
x
...
2
(
2
5x
)
2
−
Alt (a) by
−
2
8 (2
×
−
5 )
x
direct
−×−
2
3
2
−×−×−
2
3
4
3
B1, M1A1
expansion
2
3
1
4
5
−
(
)
−
(
)
−
(
)
−
(
)
= ×
8
2
,
+ −
2 2
−
5
x
+
2
−
5
x
+
2
−
5
x
2!
3!
8
75
2
3
=
2 10
+
x
+
x
+
125
x
...
(
)
2
2
2
−
5x
1(b)[1]
2
x <
o.e.
5
(a)
1 1 ...
1
2
−
2
to obtain e.g. (
)
−
−
2
(
)
−
B1: For taking out a factor of
2 or 4
−
2
from
(2
−
5 )
x
±
, 2
1 ...
±
4
May be implied by a constant term of 2 or by e.g. (
2 1 ... −
) 2
±
The “8” is likely to be present but it is not required for this mark.
M1: For the form of the binomial expansion (
)
1 ax
+
−
2
Requires the correct structure for either term three or term four. Allow a slip on the sign.
So allow for either (
) (
) (
or (
) (
) (
) (
−
2
×−
3
−
2
×−
3
×−
4
ax
)
2
ax
)
3
where a ≠ 1, could be −5 but must
±
±
2
3!
ax −
) 2
be the “a” in their
"2" 1
(
.
±
Condone missing brackets around the “ax” for this mark.
x −
5
A1: Any unsimplified or simplified but correct form of the binomial expansion for
1
−
2
Ignore the factor preceding the bracket for this mark and ignore any extra terms if found.
2
3
(
2
) (
3
)
(
2
) (
3
) (
4
)
5
x
−
×−
5
x
−
×−
×−
5
x
(
)
Score for
1
+ −
2
×−
+
×−
+
×−
o.e.
2
2!
2
3!
2
2
5
x
5
x
Brackets must be present unless they are implied by later work. Allow
for
−
2
2
75
125
2
3
The simplified form is
1 5
+
x
+
x
+
x
+
...
Allow as a list of terms.
4
2
75
2
3
A1: cao
2 10
+
x
+
x
+
125
x
...
This must be simplified and allow as a list of terms.
2
2 e.g. 37.5 and ignore any extra terms.
75
2
Do not isw and mark the final answer. E.g. Fully correct work leading to
2 10
+
x
+
x
2
2
3
followed by
loses the final mark.
=
4
+
20
x
+
75
x
+
250
x
...
Alternative to (a) by direct expansion
(2 5 )
−
x
−
2
=
2
−
2
+
....
B1: This is awarded for
which may be implied by a final answer of 2 + …
(2
−
5 )x
−
2
M1: For the form of the expansion of
Requires the correct structure for either term three or term four. Allow a slip on the sign.
So allow for either (
) (
)( ) (
−
2
×−
3
or (
) (
) (
) ( ) (
−
2
×−
3
×−
4
)
2
−
4
±
5
x
2
−
5
)
3
2
±
5
x
2
3!
Condone missing brackets around the 5x.
−
A1: Any unsimplified or simplified but correct form of the binomial expansion for
(2
−
5 )x
Ignore the factor preceding the bracket for this mark and ignore any extra terms if found.
−×−
2
3
2
−×−×−
2
3
4
3
−
2
(
)
−
3
(
)
1
−
4
(
)
−
5
(
)
Score for
2
+ −
2 2
−
5
x
+
2
−
5
x
+
2
−
5
x
o.e.
2!
3!
Brackets must be present unless they are implied by later work.
1
5
75
125
The simplified form is
2
3
Allow as a list of terms.
+
x
+
x
+
x
+
...
4
4
16
8
75
2
3
A1: cao
2 10
+
x
+
x
+
125
x
...
This must be simplified and allow as a list of terms.
2
Allow equivalents for 75
2 e.g. 37.5 and ignore any extra terms.
75
2
Do not isw and mark the final answer. E.g. Fully correct work leading to
2 10
+
x
+
x
2
2
3
followed by
=
4
+
20
x
+
75
x
+
250
x
...
loses the final mark.
Note: Correct or partially correct answers with no working in (a) should be sent to review.
(b)
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