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Past exam of the mathematics course of the University of Cambridge
/
2019
/
iii
/
Paper 125
/
3
/
a
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Mathematics course of the University of Cambridge
Past exam of the mathematics course of the University of Cambridge
2019
iii
Paper 125
3
2026-10-03
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Solution
a
Solution
0
0
0
a
On
y
2
=
x
3
+
5
x
2
+
4
x
, the
slope
through
P
1
=
(
−
1
,
0
)
and
P
2
=
(
−
2
,
2
)
is
−
2
. The
elliptic-curve addition formula
gives
x
(
P
1
+
P
2
)
=
(
−
2
)
2
−
5
−
(
−
1
)
−
(
−
2
)
=
2
,
y
(
P
1
+
P
2
)
=
(
−
2
)
(
−
1
−
2
)
=
6.
(1)
For doubling
P
2
, the
tangent
slope
is
λ
=
2
y
3
x
2
+
10
x
+
4
(
−
2
,
2
)
=
−
1
,
(2)
and hence
P
1
+
P
2
=
(
2
,
6
)
,
2
P
2
=
(
0
,
0
)
.
(3)
Ancestors
(10)
3
Paper 125
iii
2019
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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