F08 - Further Complex Numbers - Assignment 1

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Assignment 1) Page 1 of 2 National Junior College 2016 – 2017 H2 Further Mathematics Topic F7: Further Complex Numbers (Assignment 1) Name: ____________________________ Suggested Duration: 65 min 1. Find the fourth roots of  8 8 3 i  exactly in the form i e .r  [3] Hence find the roots of the equation 8 4 16 256 0z z   exactly in the form i e .r  [3] 2. (i) Without using a calculator, show that   12 11 3i 4096    . [2] (ii) Hence solve the equation   126 1 3i 1 0z    , giving the roots in the form ier  , where 0r  and π π   . Show the roots on an Argand diagram. [4] (iii) Show that    i i 2 2e e 2 cosz r z r z rz r       . [2] (iv) Use your answers in parts (ii) and (iii) to express   126 1 3i 1z   as the product of three quadratic factors with real coefficients, giving each factor in non-trigonometrical form. [3] (2014/PJC/P1/Q8) 3. (i) Use the formula for the sum of a geometric series to show that    n k kzzz 1 2 )(  = )1( )1(1 2 2 nz z z z zn    , z  1. [3] (ii) Given that z = cos  + i sin , show that       2sini2cos 2sin2 i 1  z z . [2] Deduce that    n k k 1 )sin2sin(sin   = 2sin4 )1(sinsin)1( 2    nn . [5] (2003/FM/TPJC/P1/Q6) www.KiasuExamPaper.com 795

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Assignment 1) Page 2 of 2 4. Use de Moivre's theorem to prove that cos 5 =  cos5cos20cos16 35  . [3] Show that πcos10 is a root of the equation 52016 24  xx = 0, and obtain the other three roots in trigonometric form. [3] Hence show that π 3πcos cos10 10 = 4 5 . [3] (2002/FM/TJC/P1/Q10) www.KiasuExamPaper.com 796