4.02k Argand diagrams: geometric interpretation

CAIE P3 2002 November Q8

9 marks Moderate -0.3

8

1. Find the two square roots of the complex number \(- 3 + 4 \mathrm { i }\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. The complex number \(z\) is given by $$z = \frac { - 1 + 3 \mathrm { i } } { 2 + \mathrm { i } } .$$
   1. Express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
   2. Show on a sketch of an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(- 1 + 3 \mathrm { i } , 2 + \mathrm { i }\) and \(z\) respectively.
   3. State an equation relating the lengths \(O A , O B\) and \(O C\).

CAIE P3 2003 November Q7

9 marks Standard +0.8

7 The complex number \(u\) is given by \(u = \frac { 7 + 4 \mathrm { i } } { 3 - 2 \mathrm { i } }\).

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the complex number \(z\) such that \(| z - u | = 2\).
3. Find the greatest value of \(\arg z\) for points on this locus.

CAIE P3 2005 November Q7

8 marks Moderate -0.3

7 The equation \(2 x ^ { 3 } + x ^ { 2 } + 25 = 0\) has one real root and two complex roots.

1. Verify that \(1 + 2 \mathrm { i }\) is one of the complex roots.
2. Write down the other complex root of the equation.
3. Sketch an Argand diagram showing the point representing the complex number \(1 + 2 \mathrm { i }\). Show on the same diagram the set of points representing the complex numbers \(z\) which satisfy $$| z | = | z - 1 - 2 \mathrm { i } |$$

CAIE P3 2006 November Q8

10 marks Standard +0.3

8 Let \(\mathrm { f } ( x ) = \frac { 7 x + 4 } { ( 2 x + 1 ) ( x + 1 ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence show that \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 2 + \ln \frac { 5 } { 3 }\).

CAIE P3 2008 November Q10

12 marks Standard +0.8

10 The complex number \(w\) is given by \(w = - \frac { 1 } { 2 } + \mathrm { i } \frac { \sqrt { } 3 } { 2 }\).

1. Find the modulus and argument of \(w\).
2. The complex number \(z\) has modulus \(R\) and argument \(\theta\), where \(- \frac { 1 } { 3 } \pi < \theta < \frac { 1 } { 3 } \pi\). State the modulus and argument of \(w z\) and the modulus and argument of \(\frac { z } { w }\).
3. Hence explain why, in an Argand diagram, the points representing \(z , w z\) and \(\frac { z } { w }\) are the vertices of an equilateral triangle.
4. In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the origin. One of the vertices represents the complex number \(4 + 2 \mathrm { i }\). Find the complex numbers represented by the other two vertices. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.

CAIE P3 2009 November Q7

10 marks Standard +0.3

7 The complex number \(- 2 + \mathrm { i }\) is denoted by \(u\).

1. Given that \(u\) is a root of the equation \(x ^ { 3 } - 11 x - k = 0\), where \(k\) is real, find the value of \(k\).
2. Write down the other complex root of this equation.
3. Find the modulus and argument of \(u\).
4. Sketch an Argand diagram showing the point representing \(u\). Shade the region whose points represent the complex numbers \(z\) satisfying both the inequalities $$| z | < | z - 2 | \quad \text { and } \quad 0 < \arg ( z - u ) < \frac { 1 } { 4 } \pi$$

CAIE P3 2009 November Q7

9 marks Moderate -0.3

7 The complex numbers \(- 2 + \mathrm { i }\) and \(3 + \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.

1. Find, in the form \(x + \mathrm { i } y\), the complex numbers
   1. \(u + v\),
   2. \(\frac { u } { v }\), showing all your working.
   3. State the argument of \(\frac { u } { v }\). In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u + v\) respectively.
   4. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
   5. State fully the geometrical relationship between the line segments \(O A\) and \(B C\).

CAIE P3 2010 November Q6

9 marks Moderate -0.8

6 The complex number \(z\) is given by $$z = ( \sqrt { } 3 ) + \mathrm { i } .$$

1. Find the modulus and argument of \(z\).
2. The complex conjugate of \(z\) is denoted by \(z ^ { * }\). Showing your working, express in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real,
   1. \(2 z + z ^ { * }\),
   2. \(\frac { \mathrm { i } z ^ { * } } { z }\).
   3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z\) and \(\mathrm { i } z ^ { * }\) respectively. Prove that angle \(A O B = \frac { 1 } { 6 } \pi\).

CAIE P3 2010 November Q3

6 marks Moderate -0.3

3 The complex number \(w\) is defined by \(w = 2 + \mathrm { i }\).

1. Showing your working, express \(w ^ { 2 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real. Find the modulus of \(w ^ { 2 }\).
2. Shade on an Argand diagram the region whose points represent the complex numbers \(z\) which satisfy $$\left| z - w ^ { 2 } \right| \leqslant \left| w ^ { 2 } \right|$$

CAIE P3 2011 November Q10

10 marks Standard +0.3

10

1. Showing your working, find the two square roots of the complex number \(1 - ( 2 \sqrt { } 6 ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact.
2. On a sketch of an Argand diagram, shade the region whose points represent the complex numbers \(z\) which satisfy the inequality \(| z - 3 i | \leqslant 2\). Find the greatest value of \(\arg z\) for points in this region.

CAIE P3 2011 November Q6

8 marks Standard +0.3

6 The complex number \(w\) is defined by \(w = - 1 + \mathrm { i }\).

1. Find the modulus and argument of \(w ^ { 2 }\) and \(w ^ { 3 }\), showing your working.
2. The points in an Argand diagram representing \(w\) and \(w ^ { 2 }\) are the ends of a diameter of a circle. Find the equation of the circle, giving your answer in the form \(| z - ( a + b \mathrm { i } ) | = k\).

CAIE P3 2012 November Q10

11 marks Standard +0.3

10

1. Without using a calculator, solve the equation \(\mathrm { i } w ^ { 2 } = ( 2 - 2 \mathrm { i } ) ^ { 2 }\).
2. 1. Sketch an Argand diagram showing the region \(R\) consisting of points representing the complex numbers \(z\) where $$| z - 4 - 4 i | \leqslant 2$$
   2. For the complex numbers represented by points in the region \(R\), it is given that $$p \leqslant | z | \leqslant q \quad \text { and } \quad \alpha \leqslant \arg z \leqslant \beta$$ Find the values of \(p , q , \alpha\) and \(\beta\), giving your answers correct to 3 significant figures.

CAIE P3 2013 November Q9

10 marks Standard +0.3

9

1. Without using a calculator, use the formula for the solution of a quadratic equation to solve $$( 2 - \mathrm { i } ) z ^ { 2 } + 2 z + 2 + \mathrm { i } = 0$$ Give your answers in the form \(a + b \mathrm { i }\).
2. The complex number \(w\) is defined by \(w = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\). In an Argand diagram, the points \(A , B\) and \(C\) represent the complex numbers \(w , w ^ { 3 }\) and \(w ^ { * }\) respectively (where \(w ^ { * }\) denotes the complex conjugate of \(w\) ). Draw the Argand diagram showing the points \(A , B\) and \(C\), and calculate the area of triangle \(A B C\).

CAIE P3 2014 November Q5

7 marks Standard +0.3

5 The complex numbers \(w\) and \(z\) are defined by \(w = 5 + 3 \mathrm { i }\) and \(z = 4 + \mathrm { i }\).

1. Express \(\frac { \mathrm { i } w } { z }\) in the form \(x + \mathrm { i } y\), showing all your working and giving the exact values of \(x\) and \(y\).
2. Find \(w z\) and hence, by considering arguments, show that $$\tan ^ { - 1 } \left( \frac { 3 } { 5 } \right) + \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right) = \frac { 1 } { 4 } \pi$$

CAIE P3 2015 November Q9

10 marks Standard +0.8

9 The complex number 3 - i is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).

1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)$$

CAIE P3 2015 November Q9

10 marks Standard +0.3

9

1. It is given that \(( 1 + 3 \mathrm { i } ) w = 2 + 4 \mathrm { i }\). Showing all necessary working, prove that the exact value of \(\left| w ^ { 2 } \right|\) is 2 and find \(\arg \left( w ^ { 2 } \right)\) correct to 3 significant figures.
2. On a single Argand diagram sketch the loci \(| z | = 5\) and \(| z - 5 | = | z |\). Hence determine the complex numbers represented by points common to both loci, giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).

CAIE P3 2017 November Q7

8 marks Standard +0.3

7 Throughout this question the use of a calculator is not permitted.
The complex number \(1 - ( \sqrt { } 3 ) \mathrm { i }\) is denoted by \(u\).

1. Find the modulus and argument of \(u\).
2. Show that \(u ^ { 3 } + 8 = 0\).
3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - u | \leqslant 2\) and \(\operatorname { Re } z \geqslant 2\), where \(\operatorname { Re } z\) denotes the real part of \(z\).
   [0pt] [4] \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 8 x ^ { 2 } + 9 x + 8 } { ( 1 - x ) ( 2 x + 3 ) ^ { 2 } }\).

CAIE P3 2018 November Q8

9 marks Standard +0.3

8

1. Showing all necessary working, express the complex number \(\frac { 2 + 3 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the values of \(r\) and \(\theta\) correct to 3 significant figures.
2. On an Argand diagram sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - 3 + 2 i | = 1\). Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.

CAIE P3 2019 November Q10

10 marks Standard +0.3

10

1. The complex number \(u\) is given by \(u = - 3 - ( 2 \sqrt { } 10 )\) i. Showing all necessary working and without using a calculator, find the square roots of \(u\). Give your answers in the form \(a + \mathrm { i } b\), where the numbers \(a\) and \(b\) are real and exact.
2. On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - \mathrm { i } | \leqslant 3 , \arg z \geqslant \frac { 1 } { 4 } \pi\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of the complex number \(z\).
   [0pt] [5] If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE P3 2019 November Q7

9 marks Standard +0.3

7

1. Find the complex number \(z\) satisfying the equation $$z + \frac { \mathrm { i } z } { z ^ { * } } - 2 = 0$$ where \(z ^ { * }\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. 1. On a single Argand diagram sketch the loci given by the equations \(| z - 2 \mathrm { i } | = 2\) and \(\operatorname { Im } z = 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
   2. In the first quadrant the two loci intersect at the point \(P\). Find the exact argument of the complex number represented by \(P\).

CAIE P3 Specimen Q9

10 marks Standard +0.3

9 The complex number \(3 - \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).

1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{d4a7604c-9e2c-47ef-a496-8697bc88fdd4-18_360_758_260_689} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).
4. Find the exact value of the \(x\)-coordinate of \(M\).
5. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.

CAIE P3 2020 June Q10

12 marks Standard +0.3

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.

CAIE P3 2020 June Q8

10 marks Standard +0.3

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.

CAIE P3 2020 June Q9

10 marks Standard +0.3

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.

CAIE P3 2021 June Q5

7 marks Standard +0.8

5

1. Solve the equation \(z ^ { 2 } - 2 p \mathrm { i } z - q = 0\), where \(p\) and \(q\) are real constants.
   In an Argand diagram with origin \(O\), the roots of this equation are represented by the distinct points \(A\) and \(B\).
2. Given that \(A\) and \(B\) lie on the imaginary axis, find a relation between \(p\) and \(q\).
3. Given instead that triangle \(O A B\) is equilateral, express \(q\) in terms of \(p\).