4.02k Argand diagrams: geometric interpretation

Edexcel F1 2016 January Q7

9 marks Standard +0.3

7. $$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 15 x ^ { 2 } + 99 x - 130$$

1. Given that \(x = 3 + 2 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of the equation \(\mathrm { f } ( x ) = 0\)
2. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.

Edexcel F1 2017 January Q5

8 marks Moderate -0.3

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.

Edexcel F1 2018 January Q2

9 marks Standard +0.3

2. $$f ( z ) = z ^ { 4 } - 6 z ^ { 3 } + 38 z ^ { 2 } - 94 z + 221$$

1. Given that \(z = 2 + 3 i\) is a root of the equation \(f ( z ) = 0\), use algebra to find the three other roots of \(f ( z ) = 0\)
2. Show the four roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

Edexcel F1 2021 January Q6

11 marks Moderate -0.3

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 $$z , z ^ { * } , \frac { z + 3 i } { 2 - 4 i } \text { and } z ^ { 2 }$$

Edexcel F1 2023 January Q3

10 marks Standard +0.3

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$\mathrm { f } ( z ) = 4 z ^ { 3 } + p z ^ { 2 } - 24 z + 108$$ where \(p\) is a constant.
Given that - 3 is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the value of \(p\)
2. using algebra, solve \(\mathrm { f } ( \mathrm { z } ) = 0\) completely, giving the roots in simplest form,
3. determine the modulus of the complex roots of \(\mathrm { f } ( \mathrm { z } ) = 0\)
4. show the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

Edexcel F1 2024 January Q2

9 marks Standard +0.8

2. $$f ( z ) = 2 z ^ { 3 } + p z ^ { 2 } + q z - 41$$ where \(p\) and \(q\) are integers.
The complex number \(5 - 4 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down another complex root of this equation.
2. Solve the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) completely.
3. Determine the value of \(p\) and the value of \(q\). When plotted on an Argand diagram, the points representing the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) form the vertices of a triangle.
4. Determine the area of this triangle.

Edexcel F1 2014 June Q5

7 marks Standard +0.8

1. Given that \(z _ { 1 } = - 3 - 4 \mathrm { i }\) and \(z _ { 2 } = 4 - 3 \mathrm { i }\)
   1. show, on an Argand diagram, the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\)
   2. Given that \(O\) is the origin, show that \(O P\) is perpendicular to \(O Q\).
   3. Show the point \(R\) on your diagram, where \(R\) represents \(z _ { 1 } + z _ { 2 }\)
   4. Prove that \(O P R Q\) is a square.
      

Edexcel F1 2015 June Q7

11 marks Moderate -0.3

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.

Edexcel F1 2017 June Q9

10 marks Moderate -0.3

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.

Edexcel F1 2018 June Q5

8 marks Moderate -0.3

5. Given that $$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 \equiv \left( z ^ { 2 } + 9 \right) \left( z ^ { 2 } + a z + b \right)$$ where \(a\) and \(b\) are real numbers,

1. find the value of \(a\) and the value of \(b\).
2. Hence find the exact roots of the equation $$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 = 0$$
3. Show your roots on a single Argand diagram.

Edexcel F1 2020 June Q3

9 marks Standard +0.3

3. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a , b , c\) and \(d\) are integers.
The complex numbers \(3 + \mathrm { i }\) and \(- 1 - 2 \mathrm { i }\) are roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down the other roots of this equation.
2. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
3. Determine the values of \(a , b , c\) and \(d\).

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   VIAV SIHI NI JIIIM I ION OC

Edexcel F1 2022 June Q1

6 marks Moderate -0.8

1. $$z _ { 1 } = 3 + 3 i \quad z _ { 2 } = p + q i \quad p , q \in \mathbb { R }$$ Given that \(\left| z _ { 1 } z _ { 2 } \right| = 15 \sqrt { 2 }\)

1. determine \(\left| z _ { 2 } \right|\) Given also that \(p = - 4\)
2. determine the possible values of \(q\)
3. Show \(z _ { 1 }\) and the possible positions for \(z _ { 2 }\) on the same Argand diagram.

Edexcel F1 2023 June Q2

7 marks Standard +0.3

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Given that \(x = 2 + 3 \mathrm { i }\) is a root of the equation $$2 x ^ { 4 } - 8 x ^ { 3 } + 29 x ^ { 2 } - 12 x + 39 = 0$$

1. write down another complex root of this equation.
2. Use algebra to determine the other 2 roots of the equation.
3. Show all 4 roots on a single Argand diagram.

Edexcel F1 2024 June Q2

9 marks Standard +0.3

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. $$\mathrm { f } ( z ) = z ^ { 3 } - 13 z ^ { 2 } + 59 z + p \quad p \in \mathbb { Z }$$ Given that \(z = 3\) is a root of the equation \(f ( z ) = 0\)

1. show that \(p = - 87\)
2. Use algebra to determine the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\), giving your answers in simplest form. On an Argand diagram
   • the root \(z = 3\) is represented by the point \(P\)
   • the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are represented by the points \(Q\) and \(R\)
   • the number \(z = - 9\) is represented by the point \(S\)
   • Show on a single Argand diagram the positions of \(P , Q , R\) and \(S\)
   • Determine the perimeter of the quadrilateral \(P Q S R\), giving your answer as a simplified surd.

Edexcel F1 2021 October Q4

7 marks Moderate -0.3

4. $$f ( z ) = 2 z ^ { 3 } - z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are integers. The complex number \(- 1 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down another complex root of this equation.
2. Determine the value of \(a\) and the value of \(b\).
3. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

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   VIAV SIHI NI III IM I ON OC

Edexcel F1 2018 Specimen Q8

9 marks Moderate -0.3

8. $$\mathrm { f } ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are real constants.
Given that \(- 3 + 8 \mathrm { i }\) is a complex root of the equation \(\mathrm { f } ( z ) = 0\)

1. write down another complex root of this equation.
2. Hence, or otherwise, find the other roots of the equation \(\mathrm { f } ( z ) = 0\)
3. Show on a single Argand diagram all four roots of the equation \(\mathrm { f } ( z ) = 0\)

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   VJ4V SIHI NI JIIYM ION OC

Edexcel FP1 Q9

Challenging +1.2

9. Given that \(z _ { 1 } = 3 + 2 i\) and \(z _ { 2 } = \frac { 12 - 5 i } { z _ { 1 } }\),

1. find \(z _ { 2 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.
2. Show on an Argand diagram the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\).
3. Given that \(O\) is the origin, show that \(\angle P O Q = \frac { \pi } { 2 }\). The circle passing through the points \(O , P\) and \(Q\) has centre \(C\). Find
4. the complex number represented by C,
5. the exact value of the radius of the circle.

Edexcel FP1 2009 January Q9

10 marks Standard +0.8

9. Given that \(z _ { 1 } = 3 + 2 i\) and \(z _ { 2 } = \frac { 12 - 5 i } { z _ { 1 } }\),

1. find \(z _ { 2 }\) in the form \(a + i b\), where \(a\) and \(b\) are real.
2. Show on an Argand diagram the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\).
3. Given that \(O\) is the origin, show that \(\angle P O Q = \frac { \pi } { 2 }\). The circle passing through the points \(O , P\) and \(Q\) has centre \(C\). Find
4. the complex number represented by C,
5. the exact value of the radius of the circle.

Edexcel FP1 2012 January Q5

6 marks Moderate -0.3

5. The roots of the equation $$z ^ { 3 } - 8 z ^ { 2 } + 22 z - 20 = 0$$ are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

1. Given that \(z _ { 1 } = 3 + \mathrm { i }\), find \(z _ { 2 }\) and \(z _ { 3 }\).
2. Show, on a single Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

Edexcel FP1 2013 January Q5

7 marks Moderate -0.8

5. $$f ( x ) = \left( 4 x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 6 x + 34 \right)$$

1. Find the four roots of \(\mathrm { f } ( x ) = 0\) Give your answers in the form \(x = p + \mathrm { i } q\), where \(p\) and \(q\) are real.
2. Show these four roots on a single Argand diagram.

Edexcel FP1 2009 June Q1

8 marks Moderate -0.8

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

$$z _ { 1 } = 2 - i \quad \text { and } \quad z _ { 2 } = - 8 + 9 i$$

1. Show \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram. Find, showing your working,
2. the value of \(\left| z _ { 1 } \right|\),
3. the value of \(\arg z _ { 1 }\), giving your answer in radians to 2 decimal places,
4. \(\frac { Z _ { 2 } } { Z _ { 1 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real.

Edexcel FP1 2010 June Q1

7 marks Moderate -0.5

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

1. Show that \(z ^ { 2 } = - 5 - 12 \mathrm { i }\). Find, showing your working,
2. the value of \(\left| z ^ { 2 } \right|\),
3. the value of \(\arg \left( z ^ { 2 } \right)\), giving your answer in radians to 2 decimal places.
4. Show \(z\) and \(z ^ { 2 }\) on a single Argand diagram.

Edexcel FP1 2011 June Q2

8 marks Moderate -0.8

2. $$z _ { 1 } = - 2 + \mathrm { i }$$

1. Find the modulus of \(z _ { 1 }\).
2. Find, in radians, the argument of \(z _ { 1 }\), giving your answer to 2 decimal places. The solutions to the quadratic equation $$z ^ { 2 } - 10 z + 28 = 0$$ are \(z _ { 2 }\) and \(z _ { 3 }\).
3. Find \(z _ { 2 }\) and \(z _ { 3 }\), giving your answers in the form \(p \pm i \sqrt { } q\), where \(p\) and \(q\) are integers.
4. Show, on an Argand diagram, the points representing your complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

Edexcel FP1 2013 June Q3

8 marks Moderate -0.5

3. $$z _ { 1 } = \frac { 1 } { 2 } ( 1 + \mathrm { i } \sqrt { } 3 ) , z _ { 2 } = - \sqrt { } 3 + \mathrm { i }$$

1. Express \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) giving exact values of \(r\) and \(\theta\).
   (4)
2. Find \(\left| z _ { 1 } z _ { 2 } \right|\).
3. Show and label \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram.
   (2)

Edexcel FP1 2014 June Q3

8 marks Moderate -0.3

3. Given that 2 and \(1 - 5 \mathrm { i }\) are roots of the equation $$x ^ { 3 } + p x ^ { 2 } + 30 x + q = 0 , \quad p , q \in \mathbb { R }$$

1. write down the third root of the equation.
2. Find the value of \(p\) and the value of \(q\).
3. Show the three roots of this equation on a single Argand diagram.