13 2016 - 2017 H2 Maths Complex Numbers Tutorial

National Junior College Mathematics Department 2017 Complex Numbers (Tutorial) Page 1 of 4 National Junior College 2016 – 2017 H2 Mathematics Complex Numbers Tutorial Basic Mastery Questions 1. Simplify the following complex numbers in the Cartesian form, yx i : (a)   3 8i 5 7i (b) 7 5i 4 3i   2. Find the modulus and argument of the following complex numbers, leaving the arguments in exact radians. Hence express the complex numbers in polar form and exponential form. (a) 2i (b) 1 3i (c)   1 i 1 3i    (d) 1 3i 1i   (e) πi 31e (f) πi 33e (g) 100 (h)* 5πi 63e 3. Convert the following complex numbers into Cartesian form. (a) 5π 5π2 cos isin 66               (b) πi 32e  4. See the following proof:    21 i i i 1 1 1 1        11 ??? What went wrong? Practice Questions 1. Find the exact values of the modulus and argument of 3iz  and 4 4iw . Hence evaluate (i) 1 z , (ii) 1 *z , (iii)   3 *w , (iv) *z w , (v) 23zw , leaving your answers in exact polar form. 2. The complex numbers p and w are such that    7π 5π3, arg , and 2, arg .88p p w w    (i) Find the exact values of the modulus and argument of 2 2 p w . (ii) Hence find the smallest positive value of n such that 2 2 n p w   is purely imaginary. (2011/MJC/P1/Q11a modified)

National Junior College Mathematics Department 2017 Complex Numbers (Tutorial) Page 2 of 4 3. (i) The complex number w has modulus r and argument θ, where 10 π2 , and w* denotes the conjugate of w. State the modulus and argument of p, where * wp w . (ii) Given p5 is real and positive, find the possible values of θ. (GCE 2008/P2/Q3 modified) 4. The complex number a has modulus r and argument  , where 01 r and π0 2 . The complex number b is such that 1b a  and arg( ) arg( ) πab . Let the points A, B, C, D and E represent the complex numbers a, b, a + b, * b a and ia respectively, where *a denotes the conjugate of a. On a single clearly labelled Argand diagram, illustrate these five points. 5. Show that 2 i 2 iee  is a real number for all α. The complex number w is given by 4i 2 1ew   , where 0 π . Show that Re(w) = 1. 6. The complex number w is such that * 2 3 4ww w i   , where *w is the complex conjugate of w. Find w in the form iab , where a and b are real. (GCE 2007/P1/Q3b) 7. Solve the simultaneous equations i 2 1zw and  4 3 i * 6zw   , giving z and w in the form a + bi, where a and b are real. 8. Given that   5i12i 2  yx , where x and y are real numbers, find the set of possible values of yx i . Hence solve the equation 9i1242  zz . 9. (i) Given that 1i is a root of the equation 322 2 0,w aw bw    find the values of the real numbers a and b. (ii) For these values of a and b, solve the equation in part (ii) without the use of a calculator. 10. One of the roots of the equation 32 2 1 3i 0z z az     is iz . Find the complex number a and the other roots. (2014/HCI/P1/Q3) 11. The complex number z is defined by cos isinz  , where ππ    . Find, in Cartesian form, ixy , the complex number z such that    i π 3 i 3 1 ez     .

National Junior College Mathematics Department 2017 Complex Numbers (Tutorial) Page 3 of 4 Challenging Questions 1. The complex numbers 1 πi122e and 5 i122e  are represented by points A and B respectively in an Argand diagram with origin O. Show that triangle OAB is equilateral. 2. If z = i is a root of the equation     32 1 3i 2 3i 2 0z z z      , determine the other roots. Hence find the roots of the equation     32 1 3i 3i 2 2 0w w w      . 3. A complex number w is such that 16 3i 8i 0 and Im( ) 5ww w w    , where w is the conjugate of w . (i) Find w in the form i, where , .x y x y (ii) Find the integer values of n such that is realnw . (iii) Evaluate 3 6 9 21 1 ... 4 4 4 4 w w w w                          . Another complex number z has modulus 4 and satisfies i3arg π1 i 4 z   . (iv) Express z in the form of iab , where , ab  . (v) Find the area of the triangle ZWO where Z and W are points on the Argand diagram that represent the complex numbers z and w respectively, and O is the origin. (2009/HCI/P2/Q3) Numerical Answers to Basic Mastery Questions 1(a) 71–19i (b) i25 41 25 13  2(a) πi 2π π π2, ,2e ,2 cos +isin2 2 2                 (b) 2πi 32π 2π 2π2, ,2e , 2 cos isin3 3 3     (c) πi 12π π π2 2, ,2 2e ,2 2 cos + isin12 12 12                 (d) 7πi 1277 π 7π2, π, 2e , 2 cos + isin12 12 12                 (e) 3 , 6  , ππ3 cos isin66   , πi 63e   (f) πi 3π π π3, ,3e ,3 cos isin3 3 3     (g)     i π 3,π,3e ,3 cos π isin π (h) πi 6π π π3, ,3e ,3 cos isin6 6 6                  3(a) i3 (b) 1 i 3 Numerical Answers to Practice Questions 1.(i) 1 ππcos isin2 6 6   (ii) 15 π 5πcos isin2 6 6   (iii) 3π 3π128 2 cos isin 44               (iv) 2 11 π 11πcos isin4 12 12  

National Junior College Mathematics Department 2017 Complex Numbers (Tutorial) Page 4 of 4 (v) 5π 5π512 2 cos isin12 12   2. (i) 3/2, π 8 (ii) 4 3. (i) 1, 2θ (ii) π 2π,55 6. 1 2i 7. 1 i, 2 iwz    8. 2 3i or 2 3i, 3i, 4 3izz      9. (i) a = -5, b = 6 (ii) 11 i, 1 i, 2w   10. 2 3ia  ; 1 or 3 iz  11. 0.6 0.8iz  Numerical Answers to Challenging Questions 2(i) 1,2i ; i, 2i, 1    3(i) 2 3 2i 2( 3 i)w    (ii) 6 where n k k (iii) 0 (iv) 4i (v) 24 3 unit