Mathematics · Oct / Nov 2021 · Paper 41 · Question 4
YMA01
As printed in the paper
The question as text
4. f (x) = 1 − 4 2 2
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
4(a)[4]
1
−
4
x
2
= −
1
×
4
x
2
2
(
)
(
)
1
1
2
1
1
3
3
×−
× −
4
x
2
×−
×−
× −
4
x
2
2
2
2
2
2
or
2
3!
= −
1 2
x
2
−
2
x
4
−
4
x
6
+
...
4(b)[2]
1
2
4
6
3
1
1
1
x =
Substitutes
into both sides of (a) e.g.
≈−×
1 2
−
2
−
4
4
4
4
4
4
3
≈
1.7324
cao
(a)
1
B1: Correct first two terms which does not need to be simplified, so
1
4
x
2
is fine.
+
×−
2
M1: Correct attempt at the binomial expansion for term 3 or term 4.
(
)
(
)
1
1
1
1
3
2
3
2
2
×−
× ±
4
x
×−
×−
× ±
4
x
2
2
2
2
2
Look for
or
but condone a failure to square or cube the 4
2
3!
(i.e. missing brackets)
Also award this mark for candidates who mistakenly attempt the binomial expansion of
1
1
1
1
3
×−
×−
×−
(
)
2
(
)
2
2
2
2
2
3
1 4x
−
±
4
x
4
x
to produce either
or
±
2
3!
1
1
1
1
3
×−
×−
×−
2
3
2
2
2
2
2
Or
1
−
x
2
to produce either
±
x
2
or
±
x
2
2
3!
A1: Two correct and simplified terms of
−
2
x
2
,
−
2
x
4
,
−
4
x
6
(
)
(
)
A1:
1 2
−
x
2
−
2
x
4
−
4
x
6
or exact simplified equivalent such as
1
+ −
2
x
2
+ −
2
x
4
+ −
Ignore any additional terms. This may be given separately as a list.
1,
−
2
x
2
,
−
2
x
4
,
(b)
1
3
3
M1:
Substitutes
x =
into both sides of (a) and achieves LHS of
or
4
4
2
1
2
1
4
1
6
It would be implied by (
)
3 =
2 their'' 1 2
×
−×
−
2
−
4
''
4
4
4
A1:
1.7324 Correct answer only here. This is not awrt
Note that the calculator answer is 1.7321
Alt (a) It is possible to attempt part (a) as follows but it would be very unusual to get the terms up to
1
1
1
1
1
1
(
)
(
)
1
−
4
x
2
=
1
−
2
x
×
1
+
2
x
=
1
−
x
−
x
2
−
x
3
...
×
1
+
x
−
x
2
+
x
3
2
2
2
2
2
2
Score M1 for such an attempt and award the other marks as in the main scheme. As is the main scheme look for
a correct attempt at a term 3 or 4 in either bracket followed by an attempt to multiply out
d
y
d
y
16sec
2
t
tan
t
d
t
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