F08 - Further Complex Numbers - Assignment 2

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Assignment 2`) Page 1 of 2 National Junior College 2016 – 2017 H2 Further Mathematics Topic F7: Further Complex Numbers (Assignment 2) Name: _____________________________ Suggested Duration: 60 min 1. The complex number z satisfies 1 2z  and z+ i  z 2 i . Sketch the locus of points representing z on an Argand diagram. [3] Hence show that the maximum value of  arg 2 2iz+ + is 1 22sin . 13       Show your working clearly. [3] 2. (a) Given that iez  ,where 0  and π0 2  , and i 3w z , find (i) z w+ , in terms of  , [2] (ii)  arg z w+ , in terms of  . [2] (b) The complex number z satisfies the relations 1 i 3 3z +  and  arg 2 4i 3 3z +  . (i) Illustrate, on an Argand diagram, the locus of points representing the complex number z. [3] (ii) Find the exact value of z that gives the greatest possible value of arg( ) .z [3] 3. On an Argand diagram, the complex number a 3+ 4i is represented by A and i a is represented by B. On a single Argand diagram, sketch the following loci: (i) 5z a  , [2] (ii) i 5z a  . [2] (iii) State a single transformation that will map the locus of (i) to the locus of (ii). [1] The point C represents the complex number a+ ia on the Argand diagram. (iv) Explain why C lies on the loci of 5z a  and i 5z a  . [1] (v) Find the exact area of the region satisfying both (i) and (ii). [2] (vi) If z satisfies both (i) and (ii), find the maximum value of arg(z ia+a), leaving your answer in exact form. [2] www.KiasuExamPaper.com 802

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Assignment 2`) Page 2 of 2 4. The complex number z satisfies the relations   πarg 3 3i 4z+   and 3 3i ,z b +  where b is a constant and 1 3.b  (i) Illustrate each of the above relations on a single Argand diagram. [2] (ii) Find the exact least possible value of 5i .z+ [1] (iii) Given that the least possible value of z is 18 2 , (a) find the value of b, [1] (b) hence find an exact expression for z, in the form ix y+ . [2] (c) State the cartesian equation of the locus of the point representing complex variable w such that 1w w z  , where 1z is the complex number found in part (b). [1] (ACJC/2014/P2/Q2) www.KiasuExamPaper.com 803