4.02k Argand diagrams: geometric interpretation

2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 1 - i | \leqslant 1\) and \(\arg ( z - 1 ) \leqslant \frac { 3 } { 4 } \pi\).

10 The complex number \(- 1 + \sqrt { 7 } \mathrm { i }\) is denoted by \(u\). It is given that \(u\) is a root of the equation $$2 x ^ { 3 } + 3 x ^ { 2 } + 14 x + k = 0$$ where \(k\) is a real constant.

1. Find the value of \(k\).
2. Find the other two roots of the equation.
3. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - u | = 2\).
4. Determine the greatest value of \(\arg z\) for points on this locus, giving your answer in radians.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 \(\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).

3

1. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(| z + 3 - 2 \mathrm { i } | = 2\).
2. Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.

3 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - \mathrm { i } | \leqslant 3\) and \(| z | \geqslant | z - 4 \mathrm { i } |\).

7

1. On a single Argand diagram sketch the loci given by the equations \(| z - 3 + 2 i | = 2\) and \(| w - 3 + 2 \mathrm { i } | = | w + 3 - 4 \mathrm { i } |\) where z and \(w\) are complex numbers.
2. Hence find the least value of \(| \mathbf { z } - \mathbf { w } |\) for points on these loci. Give your answer in an exact form.

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 \(\tan \frac { 5 } { 12 } \pi = \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).

2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 2 - 3 \mathrm { i } | \leqslant 2\) and \(\arg z \leqslant \frac { 3 } { 4 } \pi\).

2

1. On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(- \frac { 1 } { 3 } \pi \leqslant \arg ( z - 1 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 3 } \pi\) and \(\operatorname { Re } z \leqslant 3\).
2. Calculate the least value of \(\arg z\) for points in the region from (a). Give your answer in radians correct to 3 decimal places.

5

1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 4 - 2 i | \leqslant 3\) and \(| z | \geqslant | 10 - z |\).
2. Find the greatest value of \(\arg z\) for points in this region.

2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | \geqslant 2\) and \(| z - 1 + \mathrm { i } | \leqslant 1\).

10 The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + 4 x + b\), where \(a\) and \(b\) are real constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\).

1. Find the values of \(a\) and \(b\).
2. State a second complex root of this equation.
3. Find the real factors of \(\mathrm { p } ( x )\).
4. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - u | \leqslant \sqrt { 5 }\) and \(\arg z \leqslant \frac { 1 } { 4 } \pi\).
   2. Find the least value of \(\operatorname { Im } z\) for points in the shaded region. Give your answer in an exact form.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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

1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - 2 \mathbf { i } | \leqslant 1\) and \(\operatorname { Im } z \geqslant 2\).
2. Find the greatest value of \(\arg z\) for points in the shaded region, giving your answer in degrees.

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

1. 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 \(\theta\).
2. Hence show that \(u ^ { 6 }\) is real and state its value.
3. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(0 \leqslant \arg ( z - u ) \leqslant \frac { 1 } { 4 } \pi\) and \(\operatorname { Re } z \leqslant 2\).
   2. Find the greatest value of \(| z |\) for points in the shaded region. Give your answer correct to 3 significant figures.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | \leqslant 3 , \operatorname { Re } z \geqslant - 2\) and \(\frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \pi\).

5

1. Solve the equation \(z ^ { 2 } - 6 \mathrm { i } z - 12 = 0\), giving the answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. On a sketch of an Argand diagram with origin \(O\), show points \(A\) and \(B\) representing the roots of the equation in part (a).
3. Find the exact modulus and argument of each root.
4. Hence show that the triangle \(O A B\) is equilateral.

5

1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 2 | \leqslant 2\) and \(\operatorname { Im } z \geqslant 1\).
2. Find the greatest value of \(\arg z\) for points in the shaded region.

2 On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 \mathrm { i } | \leqslant | z + 2 - \mathrm { i } |\) and \(0 \leqslant \arg ( z + 1 ) \leqslant \frac { 1 } { 4 } \pi\).

4

1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 4 - 3 \mathrm { i } | \leqslant 2\) and \(\operatorname { Re } z \leqslant 3\).
2. Find the greatest value of \(\arg z\) for points in this region.

2 On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 1 + 2 i | \leqslant | z |\) and \(| z - 2 | \leqslant 1\).

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 \(\frac { 1 } { z }\).

1 The complex number \(z\) satisfies \(| z | = 2\) and \(0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).

1. On the Argand diagram below, sketch the locus of the points representing \(z\).
2. On the same diagram, sketch the locus of the points representing \(z ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1074_1363_628_351} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1002_26_1820_2017}

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 \(0 < \theta < 2 \pi\).

4. $$f ( x ) = x ^ { 4 } + 3 x ^ { 3 } - 5 x ^ { 2 } - 19 x - 60$$

1. Given that \(x = - 4\) and \(x = 3\) are roots of the equation \(\mathrm { f } ( x ) = 0\), use algebra to solve \(\mathrm { f } ( x ) = 0\) completely.
2. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.