4.02e Arithmetic of complex numbers: add, subtract, multiply, divide

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.

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.

9

1. The complex numbers \(u\) and \(w\) are such that $$u - w = 2 \mathrm { i } \quad \text { and } \quad u w = 6$$ Find \(u\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities $$| z - 2 - 2 \mathbf { i } | \leqslant 2 , \quad 0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi \quad \text { and } \quad \operatorname { Re } z \leqslant 3$$
   \includegraphics[max width=\textwidth, alt={}]{c1bd46f8-a33a-4927-af59-718b1c9dd4e1-16_462_709_260_719}
   A tank containing water is in the form of a hemisphere. The axis is vertical, the lowest point is \(A\) and the radius is \(r\), as shown in the diagram. The depth of water at time \(t\) is \(h\). At time \(t = 0\) the tank is full and the depth of the water is \(r\). At this instant a tap at \(A\) is opened and water begins to flow out at a rate proportional to \(\sqrt { h }\). The tank becomes empty at time \(t = 14\). The volume of water in the tank is \(V\) when the depth is \(h\). It is given that \(V = \frac { 1 } { 3 } \pi \left( 3 r h ^ { 2 } - h ^ { 3 } \right)\).
   1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { B } { 2 r h ^ { \frac { 1 } { 2 } } - h ^ { \frac { 3 } { 2 } } } ,$$ where \(B\) is a positive constant.
   2. Solve the differential equation and obtain an expression for \(t\) in terms of \(h\) and \(r\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The complex number \(u\) is defined by \(u = \frac { \sqrt { 2 } - a \sqrt { 2 } \mathrm { i } } { 1 + 2 \mathrm { i } }\), where \(a\) is a positive integer.

1. Express \(u\) in terms of \(a\), in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
   It is now given that \(a = 3\).
2. Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
3. Using your answer to part (b), find the two square roots of \(u\). Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).

5 The complex number \(3 - \mathrm { i }\) is denoted by \(u\).

1. Show, on an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. State the type of quadrilateral formed by the points \(O , A , B\) and \(C\).
2. Express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u ^ { * } } { u }\), or otherwise, prove that \(\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).

5 The complex number \(2 + y \mathrm { i }\) is denoted by \(a\), where \(y\) is a real number and \(y < 0\). It is given that \(\mathrm { f } ( a ) = a ^ { 3 } - a ^ { 2 } - 2 a\).

1. Find a simplified expression for \(\mathrm { f } ( a )\) in terms of \(y\).
2. Given that \(\operatorname { Re } ( \mathrm { f } ( a ) ) = - 20\), find \(\arg a\).

9 The complex numbers \(z\) and \(\omega\) are defined by \(z = 1 - i\) and \(\omega = - 3 + 3 \sqrt { 3 } i\).

1. Express \(z \omega\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real and in exact surd form.
2. Express \(z\) and \(\omega\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\) in each case.
3. On an Argand diagram, the points representing \(\omega\) and \(z \omega\) are \(A\) and \(B\) respectively. Prove that \(O A B\) is an isosceles right-angled triangle, where \(O\) is the origin.
4. Using your answers to part (b), prove that \(\tan \frac { 5 } { 12 } \pi = \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).

4 Solve the equation $$\frac { 5 z } { 1 + 2 \mathrm { i } } - z z ^ { * } + 30 + 10 \mathrm { i } = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

6 The complex number \(u\) is defined by $$u = \frac { 7 + \mathrm { i } } { 1 - \mathrm { i } }$$

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Show on a sketch of an Argand diagram the points \(A , B\) and \(C\) representing \(u , 7 + \mathrm { i }\) and \(1 - \mathrm { i }\) respectively.
3. By considering the arguments of \(7 + \mathrm { i }\) and \(1 - \mathrm { i }\), show that $$\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) = \tan ^ { - 1 } \left( \frac { 1 } { 7 } \right) + \frac { 1 } { 4 } \pi$$

3

1. Given the complex numbers \(u = a + \mathrm { i } b\) and \(w = c + \mathrm { i } d\), where \(a , b , c\) and \(d\) are real, prove that \(( u + w ) ^ { * } = u ^ { * } + w ^ { * }\).
2. Solve the equation \(( z + 2 + \mathrm { i } ) ^ { * } + ( 2 + \mathrm { i } ) z = 0\), giving your answer in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.

5 The complex numbers \(u\) and \(w\) are defined by \(u = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\) and \(w = 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\).

1. Find \(\frac { u ^ { 2 } } { w }\), giving your answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\).
2. State the least positive integer \(n\) such that both \(\operatorname { Im } w ^ { n } = 0\) and \(\operatorname { Re } w ^ { n } > 0\).

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 \(\lambda\).

8

1. Given that \(z = 1 + y \mathrm { i }\) and that \(y\) is a real number, express \(\frac { 1 } { z }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are functions of \(y\).
2. Show that \(\left( a - \frac { 1 } { 2 } \right) ^ { 2 } + b ^ { 2 } = \frac { 1 } { 4 }\), where \(a\) and \(b\) are the functions of \(y\) found in part (a). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-14_2716_35_108_2012}
3. On a single Argand diagram, sketch the loci given by the equations \(\operatorname { Re } ( z ) = 1\) and \(\left| z - \frac { 1 } { 2 } \right| = \frac { 1 } { 2 }\), where \(z\) is a complex number.
4. The complex number \(z\) is such that \(\operatorname { Re } ( z ) = 1\). Use your answer to part (b) to give a geometrical description of the locus of \(\frac { 1 } { z }\).

4 Find the complex number \(z\) satisfying the equation $$\frac { z - 3 \mathrm { i } } { z + 3 \mathrm { i } } = \frac { 2 - 9 \mathrm { i } } { 5 }$$ Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

2 Find the roots of the equation \(( z + 5 i ) ^ { 3 } = 4 + 4 \sqrt { 3 } i\), giving your answers in the form \(r \cos \theta + i ( r \sin \theta - 5 )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).

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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3. Given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers, solve the equation $$( z - 2 i ) \left( z ^ { * } - 2 i \right) = 21 - 12 i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

1. $$z = 3 + 2 \mathrm { i } , \quad w = 1 - \mathrm { i }$$ Find in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,

1. \(z w\)
2. \(\frac { z } { w ^ { * } }\), showing clearly how you obtained your answer. Given that $$| z + k | = \sqrt { 53 } \text {, where } k \text { is a real constant }$$
3. find the possible values of \(k\).

1. The complex number \(z\) is given by

$$z = - 7 + 3 i$$ Find

1. \(| z |\)
2. \(\arg z\), giving your answer in radians to 2 decimal places. Given that \(\frac { z } { 1 + \mathrm { i } } + w = 3 - 6 \mathrm { i }\)
3. find the complex number \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. You must show all your working.
4. Show the points representing \(z\) and \(w\) on a single Argand diagram.

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 \(\lambda\).

6. The complex number \(z\) is defined by $$z = - \lambda + 3 i \quad \text { where } \lambda \text { is a positive real constant }$$ Given that the modulus of \(z\) is 5

1. write down the value of \(\lambda\)
2. determine the argument of \(z\), giving your answer in radians to one decimal place. In part (c) you must show detailed reasoning.
   Solutions relying on calculator technology are not acceptable.
3. Express in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are real,
   1. \(\frac { z + 3 i } { 2 - 4 i }\)
   2. \(\mathrm { Z } ^ { 2 }\)
4. Show on a single Argand diagram the points \(A\), \(B\), \(C\) and \(D\) that represent the complex numbers $$z , z ^ { * } , \frac { z + 3 i } { 2 - 4 i } \text { and } z ^ { 2 }$$

7. $$z = - 3 k - 2 k \mathrm { i } , \text { where } k \text { is a real, positive constant. }$$

1. Find the modulus and the argument of \(z\), giving the argument in radians to 2 decimal places and giving the modulus as an exact answer in terms of \(k\).
2. Express in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and are given in terms of \(k\) where necessary,
   1. \(\frac { 4 } { z + 3 k }\)
   2. \(z ^ { 2 }\)
3. Given that \(k = 1\), plot the points \(A , B , C\) and \(D\) representing \(z , z ^ { * } , \frac { 4 } { z + 3 k }\) and \(z ^ { 2 }\) respectively on a single Argand diagram.

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.

9. $$z = \frac { 1 } { 5 } - \frac { 2 } { 5 } \mathrm { i }$$

1. Find the modulus and the argument of \(z\), giving the modulus as an exact answer and giving the argument in radians to 2 decimal places. Given that $$\mathrm { zw } = \lambda \mathrm { i }$$ where \(\lambda\) is a real constant,
2. find \(w\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real. Give your answer in terms of \(\lambda\).
3. Given that \(\lambda = \frac { 1 } { 10 }\)
   1. find \(\frac { 4 } { 3 } ( z + w )\),
   2. plot the points \(A , B , C\) and \(D\), representing \(z , z w , w\) and \(\frac { 4 } { 3 } ( z + w )\) respectively, on a single Argand diagram.