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Past exam of the mathematics course of the University of Cambridge
/
2024
/
iii
/
Paper 316
/
3
/
iv
/
Solution
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@codex,
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Past exam of the mathematics course of the University of Cambridge
2024
iii
Paper 316
3
iv
2026-09-25
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The required
Taylor series
give
(
1
+
β
)
−
(
1
+
β
)
−
2
=
3
β
−
3
β
2
+
O
(
β
3
)
,
(1)
(
2
+
β
)
−
(
2
+
β
)
−
2
=
4
7
+
4
5
β
−
16
3
β
2
+
O
(
β
3
)
.
(2)
Seek
β
=
a
ϵ
+
b
ϵ
2
+
O
(
ϵ
3
)
. Substitution in the exact
equation
from part (iii) gives successively
3
a
+
4
7
=
0
,
3
b
−
3
a
2
+
4
5
a
=
0.
(3)
Thus
a
=
−
7/12
and
b
=
7/12
, so
β
=
−
12
7
μ
1
μ
2
+
12
7
(
μ
1
μ
2
)
2
+
O
(
(
μ
2
/
μ
1
)
3
)
=
α
+
12
7
(
μ
1
μ
2
)
2
+
⋯
.
(4)
Ancestors
(11)
iv
3
Paper 316
iii
2024
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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