4.02k Argand diagrams: geometric interpretation

446 questions

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OCR MEI FP2 2013 January Q2
18 marks Challenging +1.3
    1. Show that $$1 + \mathrm { e } ^ { \mathrm { j } 2 \theta } = 2 \cos \theta ( \cos \theta + \mathrm { j } \sin \theta )$$
    2. The series \(C\) and \(S\) are defined as follows. $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos 2 \theta + \binom { n } { 2 } \cos 4 \theta + \ldots + \cos 2 n \theta \\ & S = \binom { n } { 1 } \sin 2 \theta + \binom { n } { 2 } \sin 4 \theta + \ldots + \sin 2 n \theta \end{aligned}$$ By considering \(C + \mathrm { j } S\), show that $$C = 2 ^ { n } \cos ^ { n } \theta \cos n \theta$$ and find a corresponding expression for \(S\).
    1. Express \(\mathrm { e } ^ { \mathrm { j } 2 \pi / 3 }\) in the form \(x + \mathrm { j } y\), where the real numbers \(x\) and \(y\) should be given exactly.
    2. An equilateral triangle in the Argand diagram has its centre at the origin. One vertex of the triangle is at the point representing \(2 + 4 \mathrm { j }\). Obtain the complex numbers representing the other two vertices, giving your answers in the form \(x + \mathrm { j } y\), where the real numbers \(x\) and \(y\) should be given exactly.
    3. Show that the length of a side of the triangle is \(2 \sqrt { 15 }\).
CAIE P3 2012 June Q7
8 marks Standard +0.3
  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 the complex numbers \(u , 1 + 2 \mathrm { i }\) and \(1 - 3 \mathrm { i }\) respectively.
  3. By considering the arguments of \(1 + 2 \mathrm { i }\) and \(1 - 3 \mathrm { i }\), show that $$\tan ^ { - 1 } 2 + \tan ^ { - 1 } 3 = \frac { 3 } { 4 } \pi$$
CAIE P3 2016 June Q9
9 marks Challenging +1.2
  1. Sketch this diagram and state fully the geometrical relationship between \(O B\) and \(A C\).
  2. Find, in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex number \(\frac { u } { v }\).
  3. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
CAIE P3 2013 November Q7
10 marks Standard +0.3
  1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(| z + \mathrm { i } | = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg ( w - 2 ) = \frac { 3 } { 4 } \pi\). Find the least value of \(| z - w |\) for points on these loci.
CAIE P3 2016 November Q9
10 marks Standard +0.3
  1. Solve the equation \(( 1 + 2 \mathrm { i } ) w ^ { 2 } + 4 w - ( 1 - 2 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 1 - \mathrm { i } | \leqslant 2\) and \(- \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).
CAIE P3 2016 November Q7
9 marks Standard +0.3
  1. Find the modulus and argument of \(z\).
  2. Express each of the following in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact:
    1. \(z + 2 z ^ { * }\);
    2. \(\frac { z ^ { * } } { \mathrm { i } z }\).
    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 \(A O B\) is equal to \(\frac { 1 } { 6 } \pi\).
CAIE P3 2019 November Q6
7 marks Standard +0.3
  1. Express \(w\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The complex number \(v\) is such that \(| v | = 2 | u |\) and \(\arg v = \arg u + \frac { 1 } { 3 } \pi\).
  2. Sketch an Argand diagram showing the points representing \(u\) and \(v\).
  3. Explain why \(v\) can be expressed as \(2 u w\). Hence find \(v\), giving your answer in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.
OCR FP3 2007 June Q7
10 marks Standard +0.3
  1. Show that \(\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1\).
  2. Write down the seven roots of the equation \(z ^ { 7 } = 1\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) and show their positions in an Argand diagram.
  3. Hence express \(z ^ { 7 } - 1\) as the product of one real linear factor and three real quadratic factors.
OCR MEI Further Pure Core 2020 November Q10
7 marks Standard +0.3
  1. Write down, in exponential ( \(r \mathrm { e } ^ { \mathrm { i } \theta }\) ) form, the complex numbers represented by the points \(\mathrm { A } , \mathrm { B }\), \(\mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
  2. When these complex numbers are multiplied by the complex number \(w\), the resulting complex numbers are represented by the points G, H, I, J, K and L. Find \(w\) in exponential form.
  3. You are given that \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L represent roots of the equation \(z ^ { 6 } = p\). Find \(p\).
CAIE P3 2013 November Q8
10 marks Standard +0.3
8 Throughout this question the use of a calculator is not permitted.
  1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(| z + \mathrm { i } | = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg ( w - 2 ) = \frac { 3 } { 4 } \pi\). Find the least value of \(| z - w |\) for points on these loci.
OCR FP1 AS 2018 March Q4
6 marks Standard +0.8
4 Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geqslant 9\).
OCR Further Pure Core 1 2018 September Q2
6 marks Standard +0.3
2 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 1 | = 5\) and \(\arg ( z + 4 + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\) respectively.
  1. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Indicate by shading on your Argand diagram the following set of points. $$\{ z : | z - 1 | \leqslant 5 \} \cap \left\{ z : 0 \leqslant \arg ( z + 4 + 4 i ) \leqslant \frac { 1 } { 4 } \pi \right\}$$
OCR Further Pure Core 1 2017 Specimen Q10
10 marks Standard +0.3
10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\). \includegraphics[max width=\textwidth, alt={}, center]{bc258133-b0d6-49bb-96a7-a5ef7f9c31fc-04_885_851_482_516}
  1. Justify each of the following statements.
    1. \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
    2. \(A B C D E\) is a regular pentagon.
    3. \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
    4. \(b ^ { * } = e\)
    5. \(a + b + c + d + e = 0\)
    6. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).
AQA FP2 2006 January Q3
12 marks Moderate -0.3
3 The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by $$z _ { 1 } = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } } \quad \text { and } \quad z _ { 2 } = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$$
  1. Show that \(z _ { 1 } = \mathrm { i }\).
  2. Show that \(\left| z _ { 1 } \right| = \left| z _ { 2 } \right|\).
  3. Express both \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  4. Draw an Argand diagram to show the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 1 } + z _ { 2 }\).
  5. Use your Argand diagram to show that $$\tan \frac { 5 } { 12 } \pi = 2 + \sqrt { 3 }$$
AQA FP2 2006 January Q5
9 marks Standard +0.8
5 The complex number \(z\) satisfies the relation $$| z + 4 - 4 i | = 4$$
  1. Sketch, on an Argand diagram, the locus of \(z\).
  2. Show that the greatest value of \(| z |\) is \(4 ( \sqrt { 2 } + 1 )\).
  3. Find the value of \(z\) for which $$\arg ( z + 4 - 4 \mathrm { i } ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).
AQA FP2 2007 January Q2
8 marks Standard +0.3
2
  1. Sketch on one diagram:
    1. the locus of points satisfying \(| z - 4 + 2 \mathrm { i } | = 2\);
    2. the locus of points satisfying \(| z | = | z - 3 - 2 \mathrm { i } |\).
  2. Shade on your sketch the region in which
    both $$| z - 4 + 2 i | \leqslant 2$$ and $$| z | \leqslant | z - 3 - 2 \mathrm { i } |$$
AQA FP2 2008 January Q3
11 marks Standard +0.3
3 A circle \(C\) and a half-line \(L\) have equations $$| z - 2 \sqrt { 3 } - \mathrm { i } | = 4$$ and $$\arg ( z + i ) = \frac { \pi } { 6 }$$ respectively.
  1. Show that:
    1. the circle \(C\) passes through the point where \(z = - \mathrm { i }\);
    2. the half-line \(L\) passes through the centre of \(C\).
  2. On one Argand diagram, sketch \(C\) and \(L\).
  3. Shade on your sketch the set of points satisfying both $$| z - 2 \sqrt { 3 } - \mathrm { i } | \leqslant 4$$ and $$0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 6 }$$
AQA FP2 2009 January Q2
8 marks Standard +0.8
2
  1. Indicate on an Argand diagram the region for which \(| z - 4 \mathrm { i } | \leqslant 2\).
  2. The complex number \(z\) satisfies \(| z - 4 \mathrm { i } | \leqslant 2\). Find the range of possible values of \(\arg z\).
AQA FP2 2009 January Q8
11 marks Standard +0.8
8
  1. Show that $$\left( z ^ { 4 } - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z ^ { 4 } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 8 } - 2 z ^ { 4 } \cos \theta + 1$$ (2 marks)
  2. Hence solve the equation $$z ^ { 8 } - z ^ { 4 } + 1 = 0$$ giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \phi }\), where \(- \pi < \phi \leqslant \pi\).
  3. Indicate the roots on an Argand diagram.
AQA FP2 2006 June Q4
7 marks Standard +0.3
4
  1. On one Argand diagram, sketch the locus of points satisfying:
    1. \(| z - 3 + 2 \mathrm { i } | = 4\);
    2. \(\quad \arg ( z - 1 ) = - \frac { 1 } { 4 } \pi\).
  2. Indicate on your sketch the set of points satisfying both $$| z - 3 + 2 i | \leqslant 4$$ and $$\arg ( z - 1 ) = - \frac { 1 } { 4 } \pi$$ (1 mark)
AQA FP2 2007 June Q5
9 marks Standard +0.3
5 The sketch shows an Argand diagram. The points \(A\) and \(B\) represent the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) respectively. The angle \(A O B = 90 ^ { \circ }\) and \(O A = O B\). \includegraphics[max width=\textwidth, alt={}, center]{847295e3-d806-43b1-8d25-688c5558bfe1-3_533_869_852_632}
  1. Explain why \(z _ { 2 } = \mathrm { i } z _ { 1 }\).
  2. On a single copy of the diagram, draw:
    1. the locus \(L _ { 1 }\) of points satisfying \(\left| z - z _ { 2 } \right| = \left| z - z _ { 1 } \right|\);
    2. the locus \(L _ { 2 }\) of points satisfying \(\arg \left( z - z _ { 2 } \right) = \arg z _ { 1 }\).
  3. Find, in terms of \(z _ { 1 }\), the complex number representing the point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
AQA FP2 2009 June Q6
12 marks Standard +0.8
6
  1. Two points, \(A\) and \(B\), on an Argand diagram are represented by the complex numbers \(2 + 3 \mathrm { i }\) and \(- 4 - 5 \mathrm { i }\) respectively. Given that the points \(A\) and \(B\) are at the ends of a diameter of a circle \(C _ { 1 }\), express the equation of \(C _ { 1 }\) in the form \(\left| z - z _ { 0 } \right| = k\).
  2. A second circle, \(C _ { 2 }\), is represented on the Argand diagram by the equation \(| z - 5 + 4 \mathrm { i } | = 4\). Sketch on one Argand diagram both \(C _ { 1 }\) and \(C _ { 2 }\).
  3. The points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and are such that \(\left| z _ { 1 } - z _ { 2 } \right|\) has its maximum value. Find this maximum value, giving your answer in the form \(a + b \sqrt { 5 }\).
AQA FP2 2015 June Q5
9 marks Standard +0.3
5 The locus of points, \(L\), satisfies the equation $$| z - 2 + 4 \mathrm { i } | = | z |$$
  1. Sketch \(L\) on the Argand diagram below.
  2. The locus \(L\) cuts the real axis at \(A\) and the imaginary axis at \(B\).
    1. Show that the complex number represented by \(C\), the midpoint of \(A B\), is $$\frac { 5 } { 2 } - \frac { 5 } { 4 } \mathrm { i }$$
    2. The point \(O\) is the origin of the Argand diagram. Find the equation of the circle that passes through the points \(O , A\) and \(B\), giving your answer in the form \(| z - \alpha | = k\).
      [0pt] [2 marks] \section*{(a)}
      \includegraphics[max width=\textwidth, alt={}]{bc3aaed2-4aef-4aec-b657-098b1e581e55-10_1173_1242_1217_463}
AQA FP3 Q5
18 marks Standard +0.8
5 The complex number \(z\) satisfies the relation $$| z + 4 - 4 i | = 4$$
  1. Sketch, on an Argand diagram, the locus of \(z\).
  2. Show that the greatest value of \(| z |\) is \(4 ( \sqrt { 2 } + 1 )\).
  3. Find the value of \(z\) for which $$\arg ( z + 4 - 4 i ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).
AQA Further AS Paper 1 2022 June Q12
6 marks Standard +0.3
12
  1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$| z - 2 \mathrm { i } | = 2$$
    \includegraphics[max width=\textwidth, alt={}]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-18_1219_1260_477_392}
    12
  2. Sketch, also on the Argand diagram above, the locus of points satisfying the equation $$\arg z = \frac { \pi } { 3 }$$ [1 mark] 12
  3. For the complex number \(w\) find the maximum value of \(| w |\) such that $$| w - 2 \mathrm { i } | \leq 2 \quad \text { and } \quad 0 \leq \arg w \leq \frac { \pi } { 3 }$$ $$y = \frac { 2 x + 7 } { 3 x + 5 }$$