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 $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right) = 0$$

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.

8

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

Solve the equation \(2z - 5iz^* = 12\) [4 marks]

Given that \(z\) is the complex number \(x + iy\) and satisfies $$|z| + z = 6 - 2i$$ find the value of \(x\) and the value of \(y\). [4]

Solve the equation \(2z - 5iz^* = 12\). [4]

The complex number \(z\) satisfies the equation \(z + 2iz^* = 12 + 9i\). Find \(z\), giving your answer in the form \(x + iy\). [5]

The complex number \(z\) satisfies the equation \(z + 2iz^* + 1 - 4i = 0\). You are given that \(z = x + iy\), where \(x\) and \(y\) are real numbers. Determine the values of \(x\) and \(y\). [4]