F08 - Further Complex Numbers - Tutorial Set 2

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Tutorial Set 2) Page 1 of 4 National Junior College 2016 – 2017 H2 Further Mathematics Topic F7: Further Complex Numbers (Tutorial Set 2) Basic Mastery Questions 1. Describe and sketch the locus of z in each of the following cases: (a) 3z (b) 21 z (c) 4 2i 6z  (d) 3 4i 3 4iz    (e) i312  zz (f) ii  zz (g) 2 1 2i 1 2z z    (h)   2 5 1 150z  (i)   πarg 4 2z  (j)   πarg 3 i 3z   (k)   3πarg 1 i 4z  (l) 1 πarg 1 i 6 z      (m) * 6zz z  (n) 2 i 12i z z   2. Shade, in separate diagrams, the region represented by each of the following inequalities: (a)  Im 2z  (b) 2 36z  (c) 1 2 1 15i 1z    (d) 2 1 3iz z    (e)  π 2πarg 2 22 3z   (f)   2 πarg 1 i 3z   and * 1z z  (g) 1z z  and π πarg4 4 z   (h)  Im 3z  and   1π arg 1 i tan 24 z     (i) i 2z  and 1 1 iz z    (j)  π πarg i4 2z   and   π0 arg 2i 4z   3. The point P in an Argand diagram represents the variable complex number z, and the point A in the first quadrant represents the fixed complex number a. Sketch, on separate diagrams, the locus of P in the following cases, making clear the relationship between the locus and A. (i) z a (ii) 2z a a  (iii) z a z  (iv)    arg argz a a  www.KiasuExamPaper.com 814

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Tutorial Set 2) Page 2 of 4 Practice Questions 1. The complex numbers 1z and 2z are given by 1 i 3 and 1 i  respectively. (i) Express each of 1z and 2z in polar form  cos isinr   where 0r and π π   . Give r and θ in exact form. (ii) Find the complex conjugate of 1 2 z z in exact polar form. (iii) On a single Argand diagram, sketch the loci (a) 1 2z z  , (b)  2arg 4z z   (iv) Find where the locus 1 2z z  meets the positive real axis. (GCE 2010/P1/Q8) 2. The point P in an Argand diagram represents the variable complex z where   πarg 2 3i . 3z   Sketch the locus of P. Give a geometrical description of the locus given by 2 i ,z k   where k  . (i) If the locus 2 iz k   just touches the locus of P, show that 2k  . (ii) Find the set of values of k for which the two loci intersect at two points exactly. 3. The complex number z satisfies the equation 2z z  . (i) Show that the real part of z is –1. The complex number z also satisfies the equation 3z  . The two possible values of z are represented by the points P and Q in an Argand diagram. (ii) Draw a sketch showing the positions of P and Q, and calculate the two possible values of arg z, leaving the answers in radians correct to 3 signifi