Complex number parameter problems

10

1. The complex number \(u\) is defined by \(u = \frac { 3 \mathrm { i } } { a + 2 \mathrm { i } }\), where \(a\) is real.
   1. Express \(u\) in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are in terms of \(a\).
   2. Find the exact value of \(a\) for which \(\arg u ^ { * } = \frac { 1 } { 3 } \pi\).
2. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 \mathbf { i } | \leqslant | z - 1 - \mathbf { i } |\) and \(| z - 2 - \mathbf { i } | \leqslant 2\).
   2. Calculate the least value of \(\arg z\) for points in this region.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The complex number \(u\) is defined by \(u = \frac { 3 + 2 \mathrm { i } } { a - 5 \mathrm { i } }\), where \(a\) is real.

1. Express \(u\) in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are in terms of \(a\).
2. Given that \(\arg u = \frac { 1 } { 4 } \pi\), find the value of \(a\).

8 It is given that \(\frac { 2 + 3 a \mathrm { i } } { a + 2 \mathrm { i } } = \lambda ( 2 - \mathrm { i } )\), where \(a\) and \(\lambda\) are real constants.

1. Show that \(3 a ^ { 2 } + 4 a - 4 = 0\).
2. Hence find the possible values of \(a\) and the corresponding values of \(\lambda\).

7. (i) Given that $$\frac { 2 w - 3 } { 10 } = \frac { 4 + 7 i } { 4 - 3 i }$$ find \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show your working.
(ii) Given that $$z = ( 2 + \lambda i ) ( 5 + i )$$ where \(\lambda\) is a real constant, and that $$\arg z = \frac { \pi } { 4 }$$ find the value of \(\lambda\).
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5. (i) Given that $$\frac { 2 z + 3 } { z + 5 - 2 i } = 1 + i$$ find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants.
(ii) Given that $$w = ( 3 + \lambda \mathrm { i } ) ( 2 + \mathrm { i } )$$ where \(\lambda\) is a real constant, and that $$| w | = 15$$ find the possible values of \(\lambda\).

1. Given that

$$\frac { 3 z - 1 } { 2 } = \frac { \lambda + 5 i } { \lambda - 4 i }$$ where \(\lambda\) is a real constant,

1. determine \(z\), giving your answer in the form \(x + y i\), where \(x\) and \(y\) are real and in terms of \(\lambda\). Given also that \(\arg z = \frac { \pi } { 4 }\)
2. find the possible values of \(\lambda\).

5. $$z = 5 + \mathrm { i } \sqrt { 3 } , \quad w = \sqrt { 3 } - \mathrm { i }$$

1. Find the value of \(| w |\). Find in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real constants,
2. \(z w\), showing clearly how you obtained your answer,
3. \(\frac { z } { w }\), showing clearly how you obtained your answer. Given that $$\arg ( z + \lambda ) = \frac { \pi } { 3 } , \quad \text { where } \lambda \text { is a real constant, }$$
4. find the value of \(\lambda\).

7. $$z = 2 - \mathrm { i } \sqrt { } 3$$

1. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. Use algebra to express
2. \(z + z ^ { 2 }\) in the form \(a + b \mathrm { i } \sqrt { } 3\), where \(a\) and \(b\) are integers,
3. \(\frac { z + 7 } { z - 1 }\) in the form \(c + d \mathrm { i } \sqrt { } 3\), where \(c\) and \(d\) are integers. Given that $$w = \lambda - 3 \mathrm { i }$$ where \(\lambda\) is a real constant, and \(\arg ( 4 - 5 \mathrm { i } + 3 w ) = - \frac { \pi } { 2 }\),
4. find the value of \(\lambda\).

9. The complex number \(w\) is given by $$w = 10 - 5 \mathrm { i }$$

1. Find \(| w |\).
2. Find arg \(w\), giving your answer in radians to 2 decimal places. The complex numbers \(z\) and \(w\) satisfy the equation $$( 2 + i ) ( z + 3 i ) = w$$
3. Use algebra to find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Given that $$\arg ( \lambda + 9 i + w ) = \frac { \pi } { 4 }$$ where \(\lambda\) is a real constant,
4. find the value of \(\lambda\).

7. $$z _ { 1 } = 2 + 3 \mathrm { i } , \quad z _ { 2 } = 3 + 2 \mathrm { i } , \quad z _ { 3 } = a + b \mathrm { i } , \quad a , b \in \mathbb { R }$$

1. Find the exact value of \(\left| z _ { 1 } + z _ { 2 } \right|\). Given that \(w = \frac { z _ { 1 } z _ { 3 } } { z _ { 2 } }\),
2. find \(w\) in terms of \(a\) and \(b\), giving your answer in the form \(x + \mathrm { i } y , \quad x , y \in \mathbb { R }\) Given also that \(w = \frac { 17 } { 13 } - \frac { 7 } { 13 } \mathrm { i }\),
3. find the value of \(a\) and the value of \(b\),
4. find \(\arg w\), giving your answer in radians to 3 decimal places.

4. The complex number \(z\) is given by $$z = \frac { p + 2 \mathrm { i } } { 3 + p \mathrm { i } }$$ where \(p\) is an integer.

1. Express \(z\) in the form \(a + b \mathrm { i }\) where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\).
2. Given that \(\arg ( z ) = \theta\), where \(\tan \theta = 1\) find the possible values of \(p\).

1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by

$$z _ { 1 } = p + 2 i \text { and } z _ { 2 } = 1 - 2 i$$ where \(p\) is an integer.

1. Find \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(a + b\) i where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\). Given that \(\left| \frac { z _ { 1 } } { z _ { 2 } } \right| = 13\),
2. find the possible values of \(p\).

7. A complex number \(z\) is given by $$z = a + 2 i$$ where \(a\) is a non-zero real number.

1. Find \(z ^ { 2 } + 2 z\) in the form \(x +\) iy where \(x\) and \(y\) are real expressions in terms of \(a\). Given that \(z ^ { 2 } + 2 z\) is real,
2. find the value of \(a\). Using this value for \(a\),
3. find the values of the modulus and argument of \(z\), giving the argument in radians, and giving your answers to 3 significant figures.
4. Show the points \(P , Q\) and \(R\), representing the complex numbers \(z , z ^ { 2 }\) and \(z ^ { 2 } + 2 z\) respectively, on a single Argand diagram with origin \(O\).
5. Describe fully the geometrical relationship between the line segments \(O P\) and \(Q R\).
   

4. (i) The complex number \(w\) is given by $$w = \frac { p - 4 \mathrm { i } } { 2 - 3 \mathrm { i } }$$ where \(p\) is a real constant.

1. Express \(w\) in the form \(a + b i\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that \(\arg w = \frac { \pi } { 4 }\)
2. find the value of \(p\).
   (ii) The complex number \(z\) is given by $$z = ( 1 - \lambda i ) ( 4 + 3 i )$$ where \(\lambda\) is a real constant. Given that $$| z | = 45$$ find the possible values of \(\lambda\).
   Give your answers as exact values in their simplest form.
   II

1. 1. Given that

$$\frac { 3 w + 7 } { 5 } = \frac { p - 4 \mathrm { i } } { 3 - \mathrm { i } } \quad \text { where } p \text { is a real constant }$$

1. express \(w\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that arg \(w = - \frac { \pi } { 2 }\)
2. find the value of \(p\).
   (ii) Given that $$( z + 1 - 2 i ) ^ { * } = 4 i z$$ find \(z\), giving your answer in the form \(z = x + i y\), where \(x\) and \(y\) are real constants.
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1 The complex number \(3 + a \mathrm { i }\), where \(a\) is real, is denoted by \(z\). Given that \(\arg z = \frac { 1 } { 6 } \pi\), find the value of \(a\) and hence find \(| z |\) and \(z ^ { * } - 3\).

2. The complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) are given by $$\mathrm { z } _ { 1 } = 2 - \mathrm { i } \quad \mathrm { z } _ { 2 } = p - \mathrm { i } \quad \mathrm { z } _ { 3 } = p + \mathrm { i }$$ where \(p\) is a real number.

1. Find \(\frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\) in the form \(a + b \mathrm { i }\) where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\). Given that \(\left| \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } } \right| = 2 \sqrt { 5 }\)
2. find the possible values of \(p\).

1. The complex number \(z\) is given by \(z = 3 + \lambda \mathrm { i }\), where \(\lambda\) is a positive constant. The complex conjugate of \(z\) is denoted by \(\bar { z }\).

Given that \(z ^ { 2 } + \bar { z } ^ { 2 } = 2\), find the value of \(\lambda\).

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

10. The complex number \(z\) is given by \(z = k + 3 i\), where \(k\) is a negative real number. Given that \(z + \frac { 12 } { z }\) is real, find \(k\) and express \(z\) in exact modulus-argument form.
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13. $$z = \frac { a + 3 \mathrm { i } } { 2 + a \mathrm { i } } , \quad a \in \mathbb { R }$$

1. Given that \(a = 4\), find \(| z |\).
2. Show that there is only one value of \(a\) for which \(\arg z = \frac { \pi } { 4 }\), and find this value.

6.

1. $$z _ { 1 } = a + b \mathrm { i } \text { and } z _ { 2 } = c + d \mathrm { i }$$ where \(a , b , c\) and \(d\) are real constants.
   Given that
   • \(b > d\)
   • \(z _ { 1 } + z _ { 2 }\) is real
   • \(\left| z _ { 1 } \right| = \sqrt { 13 }\)
   • \(\left| z _ { 2 } \right| = 5\)
   • \(\operatorname { Re } \left( z _ { 2 } - z _ { 1 } \right) = 2\)
     show that \(a = 2\) and determine the value of each of \(b , c\) and \(d\)
   • (a) On the same Argand diagram
   • sketch the locus of points \(z\) which satisfy \(| z - 12 | = 7\)
   • sketch the locus of points \(w\) which satisfy \(| w - 5 \mathrm { i } | = 4\) showing the coordinates of any points of intersection with the axes.
     (b) Determine the range of possible values of \(| z - w |\)
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