RI Complex numbers C5 Tut (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ___________________________ Tutorial 5: Complex Numbers Page 1 of 4 Tutorial 5 : Complex Numbers Section A (Basic Questions) 1 It is given that 12 i .z (i) Without using a calculator, find the values of 2z and 3 1 z in Cartesian form ixy , showing your working. [4] (ii) The real numbers p and q are such that 2 3 qpz z is real. Find, in terms of p, the value of q and the value of 2 3 qpz z . [3] [(i) 2 34 iz   , 3 11 1 2 i125 125z   (ii) 250qp , 19 p ] 2 Solve the following simultaneous equations: (a) 62 iwz , 203 2iwz  ; (b) 3i 2zw , 2 i5 2 i 0zw  . [(a) 11 i22z   , 13 5 i22w (b) 2i, 2 izw   or i, 2 4izw   ] 3 The complex number w is such that *2 3 4 iww w  , where *w is the complex conjugate of w. Find w in the form iab , where a and b are real. [4] [ 12 iw  ] 4 For each of the following complex number, represent it on an Argand diagram and find its modulus and argument. (a) 3i (b) 1i3 (c) 2i( 1 i ) (d) i(1 i) 5 The complex number z is such that 1z  and arg z  , where 0 4  . (i) Mark a possible point A representing z on an Argand diagram . Given that  2arg 2argzz  , mark the points B and C representing 2z and 2zz respectively on the same Argand diagram corresponding to point A. [2] (ii) State the geometrical shape of OACB. [1]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ________________________ Tutorial 5: Complex Numbers Page 2 of 4 6 (i) Find the roots of the equation   2i5 i 2 6 i 0zz   , giving your answers in cartesian form iab , where ,ab   . [2] (ii) Hence find the roots of the equation  2i1 5 i 2 6 i 0ww     , giving your answers in cartesian form iab , where ,ab   . [2] (iii) Given that the roots found in part (i) are also roots of the equation () 0Pz  , where ()Pz is a polynomial of degree 4 with real coefficients, find ()Pz . [3] [(i) 13 i o r 2 i (ii) 3 i or 2w (iii) 43 2 21 48 4 0zz zz   ] 7 The cubic equation 32 31 212 0,az z z b  where a and b are real numbers, has a complex root 13 i .z (i) Explain why the equation must have a real root. [2] (ii) Find the values of a and b and the real root, showing your working clearly. [5] [(ii) 25, 190ab , 19 25 ] Section B (Discussion Questions) 1 Solve for iza b , , ab   if *(i )2 iizz . [ 42 i33z  ] 2 Do not use a calculator in answering this question. (i) Given that i8 6 ixy  , determine the possible values of x and y, where x and y are real numbers. [2] (ii) Hence solve 2 2i 9 6i 0ww   , giving your answer in the form iab , where a and b are real numbers. [2] [(i) 3, 1 or 3, 1x yx y    (ii) 3 2i or 3ww   ] 3 It is given that 1 *1 zw z   , where iza b , , ab   and 0b . By expressing w in the form iuv , , uv   , find the conditions under which (a) w is real, (b) w is purely imaginary. [(a) 0a (b) 22 1ab ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ________________________ Tutorial 5: Complex Numbers Page 3 of 4 4 Given that 2*2i 1iz zz     , find z in the form x + iy. State the exact values of |𝑧| and arg ሺ𝑧ሻ. [5] [ 4 and 4 ixz   , 17z  , 1 1arg tan 4z    ] 5 Do not use a calculator in answering this question. It is given that 2i is a root of the equation 3222 0 .za z z b   (a) Find the values of the real numbers a and b and the remaining roots of the equation. [4] (b) Using these values of a and b, deduce the roots of the equation 32 22 0 .bz z az   [2] [(a) 2iz or 3 2z  (b) 21 21 2i, i or 55 55 3z    ] 6 Do not use a calculator in answering this question. Given that one of the roots of the equation 43 10 25 0za z z  is 12 i where a is real, show that 2.a Find the other roots of the equation. Hence find the roots of the equation 43( 1) 2( 1) 10( 1) 25 0ww w    . [ 12 i o r 5z   , 2iw or 15 ] 7 Do not use a calculator in answering this question (i) Given that i is a root of the equation   32 82 2 i 8 i 0zk z z    , where k is a constant to be determined, find the other roots, leaving your answers in exact cartesian form ix y , showing your working. [3] (ii) Hence solve the equation   32i2 2 8 i 8 i 0zk z z    , leaving your answers in exact cartesian form. [2] [(i) 22 ik  , 26 i and 26 i (ii) 1, 6 2i, or 6 2iz     ] 8 Given that iza b and arg( )z  , where 0a , 0b , find, in terms of  and , the values of (a) *arg( )z , (b) arg( )z , (c) arg( ib)a , (d) arg( ia)b [(a)  , (b)   , (c)  , (d) 2   ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ________________________ Tutorial 5: Complex Numbers Page 4 of 4 9 In an Argand diagram, the points ,,,A BCD represent the complex numbers ,,,abcd respectively. Given that ABCD is a square described in an anticlockwise sense, with 1ia and 73 ic , find b and d . [ 5ib , 35 i ]d  10 The complex numbers z and w are given by 3iz  and 1i3w  . Sketch an Argand diagram, with origin O, showing the points Z, W and P representing the complex numbers z, w and zw respectively. Show that OWPZ is a square. [3] By co