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2007
P1 Q3
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Complex Numbers
07 P1 Q3
2007 H2 Mathematics Paper 1 Question 3
Complex Numbers
Answers
(a)
Out of syllabus
(b)
w
=
−
1
+
2
i
{w=- 1 + 2 \mathrm{i}}
w
=
−
1
+
2
i
Full solutions
(a)
Out of syllabus
(b)
(
a
+
b
i
)
(
a
−
b
i
)
+
2
(
a
+
b
i
)
=
3
+
4
i
a
2
+
b
2
+
2
a
+
2
b
i
=
3
+
4
i
\begin{gather*} (a + b\mathrm{i})(a - b\mathrm{i}) + 2(a + b\mathrm{i}) = 3 + 4 \mathrm{i} \\ a^2 + b^2 + 2a + 2b\mathrm{i} = 3 + 4 \mathrm{i} \end{gather*}
(
a
+
b
i
)
(
a
−
b
i
)
+
2
(
a
+
b
i
)
=
3
+
4
i
a
2
+
b
2
+
2
a
+
2
b
i
=
3
+
4
i
Comparing real and imaginary parts,
a
2
+
2
a
+
b
2
=
3
2
b
=
4
\begin{align} && \quad a^2 + 2a + b^2 &= 3 \\ && \quad 2b &= 4 \\ \end{align}
a
2
+
2
a
+
b
2
2
b
=
3
=
4
b
=
2
b = 2
b
=
2
a
2
+
2
a
+
2
2
=
3
a
2
+
2
a
+
1
=
0
(
a
+
1
)
2
=
0
a
=
−
1
\begin{align*} a^2 + 2a + 2^2 &= 3 \\ a^2 + 2a + 1 &= 0 \\ (a + 1)^2 &= 0 \\ a &= - 1 \end{align*}
a
2
+
2
a
+
2
2
a
2
+
2
a
+
1
(
a
+
1
)
2
a
=
3
=
0
=
0
=
−
1
w
=
−
1
+
2
i
■
w=- 1 + 2 \mathrm{i} \; \blacksquare
w
=
−
1
+
2
i
■
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