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 significant figures. It is given that P and Q lie on the locus z a b  where a and b are real, and 0b . (iii) Give a geometrical description of this locus, and hence find the least possible value of b and the corresponding value of a. 4. The complex number z satisfies 2 5i 3z   . (i) On an Argand diagram, sketch the region in which the point representing z can lie. [3] (ii) Find exactly the maximum and minimum possible values of |z|. [2] (iii) It is given that 0 arg π 4z  . With this extra information, find the maximum value of 6 iz  . Label the point(s) that correspond to this maximum value on your diagram with the letter P. [3] (GCE 2011/ P2/Q1) www.KiasuExamPaper.com 815

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Tutorial Set 2) Page 3 of 4 5. Sketch in an Argand diagram, the set of points representing all complex numbers satisfying both the inequalities: 1 i 2z   and  π πarg 12 2z    . Find the greatest and least values of  tan arg z . 6. The complex number z satisfies the relations 6z  and 8 6i1 1 z   . (a) Illustrate both of these relations on a single Argand diagram. [3] (b) Find the greatest and least possible values of arg z, giving your answers in radians correct to 3 decimal places. [4] (GCE 2008/P2/Q3 (Modified)) 7. A fixed complex number a is such that  0 arg π 2a  . On a single Argand dia gram, sketch the loci given by 7z a z a   and 4 3z a a  . The two complex numbers that satisfy the above equations are represented by the complex numbers p and q. Find the possible values of arg p q      . Find p q in terms of a. (HCI/2009/ P1/4) 8. (i) Solve the equation 7 (1 i) 0z    , giving the roots in the form ier  , where 0r and π π   . [3] (ii) Show the roots on an Argand diagram. [4] (iii) The roots represented by 1z and 2z are such that 1 2 10 arg( ) arg( ) π2z z   . Explain why the locus of all points z such that 1 2z z z z   passes through the origin. Draw this locus on your Argand diagram and find its exact Cartesian equation. [5] (GCE 2009 / P1 / Q9) 9. Given that  2 1 cos i sinz     , where π π   , show that the locus of the points representing z , as θ varies, is a circle. Sketch the locus. Hence, or otherwise, find the greatest and the least value of iz , and find z in the exact form ix y that gives these values of iz . 10. Two loci in the Argand diagram are given by the equations * 1 i 2z    and   πarg 2i 5z  . Draw an Argand diagram to show both loci and find the value of z that corresponds to the point of intersection of these loci. [5] (HCI/2015/P1/Q10b modified) www.KiasuExamPaper.com 816

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Tutorial Set 2) Page 4 of 4 11. Solve the equation 5 32 0z   , expressing your answers in the form ier  , where 0r  and π π   . [2] 1z , 2z and 3z are three of the roots of 5 32 0z   such that 1 2 30 arg arg argz z z     . (i) Find the smallest positive integer n such that 1 2 n z z        is real and positive. [3] (ii) The points A and B represent the roots 1z and 3z respectively in the Argand diagram. The line segment 'BA is obtained by rotating the line segment BA through 2  clockwise about the point B. Find the real part of the complex number represented by point 'A , giving your answer in exact trigonometric form. [4] (AJC/2015/P2/Q2) 12. (i) Sketch, on an Argand diagram, the set of points representing the complex number z which satisfy both conditions: 2 i 4 2 iz z    and   1arg 2i ta0 2nz    . Hence, find the greatest value of 4iz which satisfy the above conditions, giving your answer in exact form. (ii) With the help of your sketch in (i), sketch, on another Argand diagram, the set of points representing the complex number z which satisfy both conditions: 2 i 4 2 iz z     and   1arg 2i ta0 2nz    Numerical Answers to Practice Questions 1. (i) π π 3π 3π2 cos isin , 2 cos isin3 3 4 4                       (ii) 11π 11π2 cos isin12 12     (iv) 2x 2. (ii) 2 < k < 4 3. (ii) 1.91 radians (iii) least 2 2, 1b a  4. (ii) 29 3 (iii) 17 5. Least 1 , greatest 3 6. (b) Least 0.058, greatest 1.229 7. 1.29 or −1.29, 8 a 8. (i) 8 11 i 2814 π 2 e , 0, 1, 2, 3 k z k       (ii) 5πtan 28y x 9. 5 2 with 4 5 2 52 i5 5z                , 5 2 with z = 4 5 2 52 i5 5                10. 2.71 3.97iz   11. 3π π π 3πi i i i πi5 5 5 5 2e , 2e , 2e , 2e , 2ez    (i) 5 (ii) π2 2sin 5  12. (i) 2 13