Square roots with follow-up application

Questions that ask to find square roots and then use them to solve another equation, find cube roots, or perform additional calculations.

9 questions

OCR FP1 2007 June Q10
10
  1. Use an algebraic method to find the square roots of the complex number \(16 + 30 \mathrm { i }\).
  2. Use your answers to part (i) to solve the equation \(z ^ { 2 } - 2 z - ( 15 + 30 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\).
OCR FP1 2008 June Q9
9
  1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\).
  2. Find \(( 3 - 2 \mathrm { i } ) ^ { 2 }\).
  3. Hence solve the quartic equation \(x ^ { 4 } - 10 x ^ { 2 } + 169 = 0\).
OCR FP1 2010 January Q8
8 The complex number \(a\) is such that \(a ^ { 2 } = 5 - 12 \mathrm { i }\).
  1. Use an algebraic method to find the two possible values of \(a\).
  2. Sketch on a single Argand diagram the two possible loci given by \(| z - a | = | a |\).
OCR FP1 2010 June Q10
10 The complex number \(z\), where \(0 < \arg z < \frac { 1 } { 2 } \pi\), is such that \(z ^ { 2 } = 3 + 4 \mathrm { i }\).
  1. Use an algebraic method to find \(z\).
  2. Show that \(z ^ { 3 } = 2 + 11 \mathrm { i }\). The complex number \(w\) is the root of the equation $$w ^ { 6 } - 4 w ^ { 3 } + 125 = 0$$ for which \(- \frac { 1 } { 2 } \pi < \arg w < 0\).
  3. Find \(w\).
OCR FP1 2015 June Q7
7
  1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\). You must show sufficient working to justify your answers.
  2. Hence solve the quadratic equation \(x ^ { 2 } - 4 x - 1 - 12 \mathrm { i } = 0\).
OCR Further Pure Core AS 2022 June Q5
5 In this question you must show detailed reasoning.
  1. Use an algebraic method to find the square roots of \(- 16 + 30 \mathrm { i }\).
  2. By finding the cube of one of your answers to part (a) determine a cube root of \(\frac { - 99 + 5 i } { 4 }\). Give your answer in the form \(a + b \mathrm { i }\).
OCR MEI Further Pure Core 2019 June Q10
10 In this question you must show detailed reasoning.
  1. You are given that \(- 1 + \mathrm { i }\) is a root of the equation \(z ^ { 3 } = a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Find \(a\) and \(b\).
  2. Find all the roots of the equation in part (a), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
  3. Chris says "the complex roots of a polynomial equation come in complex conjugate pairs". Explain why this does not apply to the polynomial equation in part (a).
SPS SPS FM 2022 January Q7
7.
  1. Use an algebraic method to find the square roots of the complex number \(16 + 30 \mathrm { i }\).
  2. Use your answers to part (i) to solve the equation \(z ^ { 2 } - 2 z - ( 15 + 30 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\).
    [0pt] [BLANK PAGE]
OCR FP1 AS 2018 March Q7
7 In this question you must show detailed reasoning.
  1. Find the square roots of the number \(528 + 46 \mathrm { i }\) giving your answers in the form \(a + b \mathrm { i }\).
  2. \(\quad 3 + 2 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - a x + 78 = 0\), where \(a\) is a real number. Find the value of \(a\).