Linear equations in z and z*

8

1. Solve the equation \(( 1 + 2 \mathrm { i } ) w + \mathrm { i } w ^ { * } = 3 + 5 \mathrm { i }\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 - 2 \mathrm { i } | \leqslant 1\) and \(\arg ( z - 4 \mathrm { i } ) \geqslant - \frac { 1 } { 4 } \pi\).
   2. Find the least value of \(\operatorname { Im } z\) for points in this region, giving your answer in an exact form.

5. $$2 z + z ^ { * } = \frac { 3 + 4 i } { 7 + i }$$ Find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show all your working.

5. $$2 z + z ^ { * } = \frac { 3 + 4 \mathrm { i } } { 7 + \mathrm { i } }$$ Find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show all your working.

9. Given that \(z = x + \mathrm { i } y\), where \(x \in \mathbb { R } , y \in \mathbb { R }\), find the value of \(x\) and the value of \(y\) such that $$( 3 - i ) z ^ { * } + 2 i z = 9 - i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

6. Given that \(z = x + \mathrm { i } y\), find the value of \(x\) and the value of \(y\) such that $$z + 3 \mathrm { i } z ^ { * } = - 1 + 13 \mathrm { i }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

3 The complex number \(z\) satisfies the equation \(z + 2 \mathrm { i } z ^ { * } = 12 + 9 \mathrm { i }\). Find \(z\), giving your answer in the form \(x + \mathrm { i } y\).

3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$z - 3 \mathbf { i } z ^ { * }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
2. Find the complex number \(z\) such that $$z - 3 \mathrm { i } z ^ { * } = 16$$

2 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$3 \mathrm { i } z + 2 z ^ { * }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
2. Find the complex number \(z\) such that $$3 \mathrm { i } z + 2 z ^ { * } = 7 + 8 \mathrm { i }$$

2 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$( 1 - 2 i ) z - z ^ { * }$$
2. Hence find the complex number \(z\) such that $$( 1 - 2 \mathrm { i } ) z - z ^ { * } = 10 ( 2 + \mathrm { i } )$$

   PART	REFERENCE
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3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right)$$
2. Hence find the complex number \(z\) such that $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right) = 0$$

4 Find the complex number \(z\) such that $$5 \mathrm { i } z + 3 z ^ { * } + 16 = 8 \mathrm { i }$$ Give your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real.
[0pt] [6 marks] \(5 \quad\) A curve \(C\) has equation \(y = x ( x + 3 )\).

1. Find the gradient of the line passing through the point ( \(- 5,10\) ) and the point on \(C\) with \(x\)-coordinate \(- 5 + h\). Give your answer in its simplest form.
2. Show how the answer to part (a) can be used to find the gradient of the curve \(C\) at the point \(( - 5,10 )\). State the value of this gradient.
   [0pt] [2 marks] \(6 \quad\) A curve \(C\) has equation \(y = \frac { 1 } { x ( x + 2 ) }\).
3. Write down the equations of all the asymptotes of \(C\).
4. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is - 1 .
   1. Find the \(y\)-coordinate of the stationary point.
   2. Sketch the curve \(C\).
5. Solve the inequality $$\frac { 1 } { x ( x + 2 ) } \leqslant \frac { 1 } { 8 }$$

3 The complex number \(z\) satisfies the equation \(5 ( z - \mathrm { i } ) = ( - 1 + 2 \mathrm { i } ) z ^ { * }\).
Determine \(z\), giving your answer in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.

1. Solve the equation \(2 z - 5 \mathrm { i } z ^ { * } = 12\)
[0pt] [4 marks]

8

1. The complex number \(z\) is given by \(z = x + i y\) where \(x , y \in \mathbb { R }\) 8
2. 1. Write down the complex conjugate \(z ^ { * }\) in terms of \(x\) and \(y\) 8
3. (ii) Hence prove that \(z z ^ { * }\) is real for all \(z \in \mathbb { C }\)
   8
4. The complex number \(w\) satisfies the equation $$3 w + 10 \mathrm { i } = 2 w ^ { \star } + 5$$ 8
5. 1. Find \(w\)
      8
6. (ii) Calculate the value of \(w ^ { 2 } \left( w ^ { * } \right) ^ { 2 }\)

8

1. Solve the equation \(\omega + 2 + 7 \mathrm { i } = 3 \omega ^ { * } - \mathrm { i }\).
2. Prove algebraically that, for non-zero \(z , z = - z ^ { * }\) if and only if \(z\) is purely imaginary.
3. The complex numbers \(z\) and \(z ^ { * }\) are represented on an Argand diagram by the points \(A\) and \(B\) respectively.
   1. State, for any \(z\), the single transformation which transforms \(A\) to \(B\).
   2. Use a geometric argument to prove that \(z = z ^ { * }\) if and only if \(z\) is purely real.

1. The complex number \(z\) satisfies the equation \(z + 2 \mathrm { i } z ^ { * } + 1 - 4 \mathrm { i } = 0\).

You are given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
Determine the values of \(x\) and \(y\).