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