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Check In – 4.02 Complex Numbers Questions 1. Write down the imaginary part of the complex number 3 4i 6(2 8i) . 2. Solve the equation x 2 4 x 9 0 simplifying your answers as far as possible. 3. It is given that a 3bi a 2 b3i where a and b are real numbers. Find the possible values of a and b . 4. z 3 bi and arg( z ) . 6 Find the exact value of b . 5. Solve the equation 16 x 4 625 . 6. Show that, for any complex number z a bi , z z* is always real. 7. The complex number z is given by z cos i sin . (i) Show that z 1 for all values of and explain what this means geometrically. (ii) Given that arg( z ) 4 , find z in cartesian form. 8. Show that Im (a bi)3 3a2b b3 . 9. When plotted on an Argand diagram the solutions of the equation z 3 z 2 z 3 form a triangle. Find the perimeter of that triangle in exact form. 10. z is a complex number such that Im( z ) 0 and z 1 . Show that w z 1 z 1 is an imaginary number. Extension ‘Thousand Words’ by nrich.maths.org https://nrich.maths.org/2374 ‘These draft qualifications have not yet been accredited by Ofqual. They are published (along with specimen assessment materials, summary brochures and sample resources) to enable teachers to have early sight of our proposed approach. Further changes may be required and no assurance can be given at this time that the proposed qualifications will be made available in their current form, or that they will be accredited in time for first teaching in 2017 and first award in 2019 (2018 for AS Level qualifications).’ Version 1 1 © OCR 2016 Qualification Awaiting Accreditation Worked solutions 1. Im{3 4i 6(2 8i)} Im{3 12 (4 48)i} 4 48 44 2. ( x 2)2 4 9 0 ( x 2)2 5 x 2 i 5 3. Equating real parts: a a2 a a 1 0 a 0 or a 1 Equating imaginary parts: 3b b 3 b3 3b 0 b(b2 3) 0 b 0 or b2 3 This last equation has no real solutions, so b 0 only. 4. b 3 tan 6 b 3tan 5. 6. 6 3 3 625 16 625 x4 0 16 25 25 x2 x2 0 4 4 5 5 5 5 x x x i x i 0 2 2 2 2 5 5 x or i 2 2 x4 z z* a bi a bi 2a which is real because a is real. Version 1 Qualification Awaiting Accreditation 2 © OCR 2016 7. z cos 2 sin 2 (i) 1 1 The complex number z always lies on the unit circle, centre 0 0i when plotted on an Argand diagram. z cos (ii) 8. 4 isin 4 2 2 i 2 2 (a bi)3 a3 3a 2 (bi) 3a(b i)2 (bi)3 but i3 ii 2 i so Im (a bi)3 3a2b b3 9. Spot that z 1 is a solution. Factorising z3 z 2 z 3 0 ( z 1)( z 2 2 z 3) 0 z 1 or (z 1)2 1 3 0 z 1 or z 1 i 2 By Pythagoras, the distance from 1 to 1 i 2 is So the perimeter 2 6 2 6 2( 2 6) 42 6 Im 2i 1i √2 0 -2 -1 Re 0 1 2 √2 -1 i -2 i Version 1 Qualification Awaiting Accreditation 3 © OCR 2016 10. z 1 a bi 1 z 1 a bi 1 a 1 bi a 1 bi (a 1 bi)(a 1 bi) (a 1 bi)(a 1 bi) a 2 a abi a 1 bi abi bi b 2i 2 (a 1) 2 b 2i 2 a 2 b2 1 2bi a 2 b 2 2a 1 1 1 2bi because z 1 a 2 b 2 1 2 2 a b 2a 1 2bi 2a 2 bi a 1 Given that this last expression is simply (real constant) i then w is indeed imaginary. Alternative solution z 1 z 1 z 1 zz z z 1 z 1 z 1 z 1 zz z z 1 Now, zz z 1 , z z 2Im( z )i and z z 2Re( z ) so, 2 z 1 2Im( z )i Im( z )i bi z 1 2Re( z ) 2 Re( z ) 1 a 1 Extension ‘Thousand Words’ by nrich.maths.org https://nrich.maths.org/2374/solution We’d like to know your view on the resources we produce. By clicking on ‘Like’ or ‘Dislike’ you can help us to ensure that our resources work for you. When the email template pops up please add additional comments if you wish and then just click ‘Send’. Thank you. If you do not currently offer this OCR qualification but would like to do so, please complete the Expression of Interest Form which can be found here: www.ocr.org.uk/expression-of-interest OCR Resources: the small print OCR’s resources are provided to support the teaching of OCR specifications, but in no way constitute an endorsed teaching method that is required by the Board, and the decision to use them lies with the individual teacher. Whilst every effort is made to ensure the accuracy of the content, OCR cannot be held responsible for any errors or omissions within these resources. © OCR 2016 - This resource may be freely copied and distributed, as long as the OCR logo and this message remain intact and OCR is acknowledged as the originator of this work. OCR acknowledges the use of the following content: n/a Please get in touch if you want to discuss the accessibility of resources we offer to support delivery of our qualifications: [email protected] Version 1 Qualification Awaiting Accreditation 4 © OCR 2016