Square roots of complex numbers

A question is this type if and only if it asks to find the two square roots of a given complex number in Cartesian form, showing all working without a calculator.

7 questions · Standard +0.1

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CAIE P3 2015 June Q7
9 marks
7 The complex number \(u\) is given by \(u = - 1 + ( 4 \sqrt { } 3 ) \mathrm { i }\).
  1. Without using a calculator and showing all your working, find the two square roots of \(u\). Give your answers in the form \(a + \mathrm { i } b\), where the real numbers \(a\) and \(b\) are exact.
  2. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the relation \(| z - u | = 1\). Determine the greatest value of \(\arg z\) for points on this locus. \(8 \quad\) Let \(f ( x ) = \frac { 5 x ^ { 2 } + x + 6 } { ( 3 - 2 x ) \left( x ^ { 2 } + 4 \right) }\).
CAIE P3 2016 June Q10
11 marks Standard +0.3
10
  1. Showing all your working and without the use of a calculator, find the square roots of the complex number \(7 - ( 6 \sqrt { } 2 ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    1. On an Argand diagram, sketch the loci of points representing complex numbers \(w\) and \(z\) such that \(| w - 1 - 2 \mathrm { i } | = 1\) and \(\arg ( z - 1 ) = \frac { 3 } { 4 } \pi\).
    2. Calculate the least value of \(| w - z |\) for points on these loci.
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 2007 November Q8
10 marks Standard +0.3
8
  1. The complex number \(z\) is given by \(z = \frac { 4 - 3 \mathrm { i } } { 1 - 2 \mathrm { i } }\).
    1. Express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    2. Find the modulus and argument of \(z\).
  2. Find the two square roots of the complex number 5-12i, giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
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.
OCR Further Pure Core 1 2021 June Q2
4 marks Moderate -0.8
2 In this question you must show detailed reasoning.
  1. Determine the square roots of 25 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
  2. Illustrate the number 25 i and its square roots on an Argand diagram.
OCR Further Pure Core 2 2021 June Q4
9 marks Standard +0.8
4 In this question you must show detailed reasoning.
The complex number \(- 4 + i \sqrt { 48 }\) is denoted by \(z\).
  1. Determine the cube roots of \(z\), giving the roots in exponential form. The points which represent the cube roots of \(z\) are denoted by \(A , B\) and \(C\) and these form a triangle in an Argand diagram.
  2. Write down the angles that any lines of symmetry of triangle \(A B C\) make with the positive real axis, justifying your answer.