RI Complex numbers C5 Add Prac (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________________ Additional Practice C5 Complex Numbers Page 1 of 4 Additional Practice Questions for Chapter 5: Complex Numbers 1 Given that 2(2 3 i ) (2 3 i ) 0     , find the real numbers  and  . [6] [ 4, 13 ] 2 The complex number ixy is such that 2(i )ixy . Find the possible values of , xy   , giving your answers in exact form. [4] Hence find the possible values of the complex number w such that 2 iw  . [2] [ 11, 22 xy , 11, 22 xy  ; 11 11i o r i 22 22   ] 3 The complex number is given by , where a is a non-zero real number. Given that is real, find the possible values of z. [3] [ 13 iz or 13 iz ] 4 The complex number w is such that iwa b , where a and b are non-zero real numbers. The complex conjugate of w is denoted by *w . Given that 2 * w w is purely imaginary, find the possible values of w in terms of a. [5] [ i o r i 33 aaaa ] 5 (a) Solve the simultaneous equations i1 iwz   , 402( 1 i ) 62 izw   . [3] (b) Two complex numbers w and z are such that * 2iwz , 2 6wz  . Find w in the form iab , where a and b are real and positive. [4] [(a) 22 iw  , 1iz  (b) 22 iw ] z 12 iza 3z

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________________________ Additional Practice C5 Complex Numbers Page 2 of 4 6 The complex numbers s and w satisfy the equations 6isw and 10sw . Given that Re(s) > 0, solve the equations for s and w, giving all answers in the form ix y , where x and y are real. [4] Hence find a solution to the following equations 6uv  and 10uv . Give your answers for u and v in the form ix y , where x and y are real. [2] [ 13 i , 13 isw  ; 3iu  and 3iv ] 7 Do not use a calculator in answering this question. Given that 23 i is a root of the equation 43 2 10 48 122 143 0zzz z  , solve the equation, giving your answers in exact form. [4] [ and 32 i ] 8 It is given that 12 i satisfies the equation 2 (i )1 6i 0za z b , where ,ab  . Find the value of a and b and the other root. [5] [ 15a and 27b , 14 i ] 9 Do not use a calculator in answering this question. (a) Verify that 15 i is a root of the equation 2 ( 1 8i) ( 17 7i) 0ww    . Hence, or otherwise, find the second root of the equation in cartesian form, ip q , showing your working. [5] (b) The equation 32 51 6 0zz z k  , where k is a real constant, has a root 1iza , where a is a positive real constant. Find the values of a and k, showing your working. [5] [(a) 23 i (b) 3a , 30k  ] 23 i

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________________________ Additional Practice C5 Complex Numbers Page 3 of 4 10 Do not use a calculator in answering this question. (a) Find the roots of the equation    2 1i 2 55 i 0zz    , giving your answers in cartesian form iab . [3] (b) (i) Given that 1i , find 2 , 3 and 4 in cartesian form. Given also that 43 2 39 58 0pq     , where p and q are real, find p and q. [4] (ii) Using the values of p and q in part (b)(i), express 43 2 39 58pq    as the product of two quadratic factors. [3] [(a) 12 i or 2i (bi) 6, 66pq  (bii)   22 22 42 9   ] 11 A graphic calculator is not to be used in answering this question. (i) It is given that 1 13 iz  . Find the value of 3 1z , showing clearly how you obtain your answer. [3] (ii) Given that 13 i is a root of the equation 3224 0za zb z   , Find the values of the real numbers a and b . [4] (iii) For these values of a and b , solve the equation in part (ii), and show all the roots on an Argand diagram. [4] [(i) 8 (ii) 3a , 6b (iii) 1 13 iz  , 2 13 iz  and 3 1 2z  ] 12 Given that 1 12 iz  is a root of the quadratic equation 2 0za z b  where , ab   , find the values of a and b . Mark on an Argand diagram the points 1P and 2P , which represent the roots 1z and 2z of the quadratic equation 2 0za z b  . A third point 3P is on the real axis such that 1P , 2P and 3P form a triangle which encloses the origin wi th an area of 8 square units. Find the complex number represented by the point 3P . [5] [ 2, 5ab  ; 3 3z  ] 13 The complex number 𝑧 and 𝑤 are given by and 32 iz   and 54 iw . In an Argand diagram, the points Z and W represent the complex numbers z and w respectively, and 'Z represents *z , the complex conjugate of z. The point 𝑃 is such that 'ZWZ P is a parallelogram. Find the complex number p represented by P. [ 11 4i ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________________________ Additional Practice C5 Complex Numbers Page 4 of 4 14 The complex number 75 i is represented by the point A in an Argand diagram with origin O. Given that OABC is a