RVHS H2 Math Complex Numbers Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Complex Numbers 1 Complex Numbers 1. ACJC/2016/I/8 (part of) The complex number z is given by izk=+ where k is a non-zero real number. (i) Find the possible values of k if izk=+ satisfies the equation 32 i 2 4i 0z z z− − − = . [3] 2. ACJC Prelim 9758/2023/01/Q2 Do not use a calculator in answering this question. The complex numbers z and w satisfy the following equations. * 2 4i 2 3i wz zw + =− + += Find z and w, giving your answers in the form iab+ , where a and b are real numbers. [4] 3. CJC Prelim 9758/2023/01/Q1 The complex numbers z and w satisfy the following equations. 21zw+= 2 4 24iwz− = + Find z and w , giving your answers in the form iab+ where a and b are real numbers. [4] 4. CJC/2021/II/3 (part of) Do not use a calculator in answering this question. It is given that 1z and 2z are the roots of the equation 2 2i 2 0zz− − = , where ( ) ( )12arg argzz  . (i) Show that 1 1iz =+ . [3] (ii) Given that 1xz= , find 2x , 3x and 4x in cartesian form. Given also that 1xz= is a root of the equation 4 3 26 18 10 0x x sx x− + − + = , where s is real, find the value of s and the other roots of the equation. [6] 5. CJC/2016/I/3 The cubic equation 32 0,x ax bx c+ + + = where a, b and c are constants, has roots 3i+ and 2. (i) One JC2 student remarked that the third root is 3 − i. State a necessary assumption the student made in order that the remark is true. [1] (ii) Given that the assumption in part (i) holds, find the values of a, b and c. [4]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Complex Numbers 2 6. EJC Prelim 9758/2025/P1/Q8 Do not use a calculator in answering this question. The complex number w is such that 2 2iw = and 0 arg( ) 2w  . (a) Find w. [3] (b) Given that one of the roots of the equation 32 (1 i) 0z z z s− + − + = is w, find the other roots of the equation and the value of s. [5] Hence, find the roots of the equation 32i (1 i) 0v v v s− + + + + = . [2] 7. DHS/2016/I/2 The complex number w is such that 2kw + ** i i 1 0kww w w+ − − = , where *w is the complex conjugate of w and k is a real and non-zero constant. (i) For iw a b=+ where a and b are real numbers, obtain an expression for b in terms of a and k. Explain why w is either purely real or purely imaginary. [4] (ii) Using your result in part (i), or otherwise, find the real roots of the equation 22 2 * i i * 1 0w ww w w+ + − − = [2] 8. CJC/2009/I/8(a) (a) One of the roots of the equation 034 234 =+++− pzzzz is 1 – 2i, where p is a constant. A student says that “One of the other roots must be 1 + 2i.” Explain why this statement is not entirely correct. [1] (i) Determine the value of p. [1] (ii) Find the exact values of all the other roots of the equation. [3] 9. IJC/2016/I/8(b) (modified) (b) (i) If cos isin , where 0 2z   = +   , show that ( ) 21 2sin sin i cosz   − = − . [2] (ii) Hence find 21 z− in terms of  . [3]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Complex Numbers 3 10. The complex numbers a and b are given by ( )1 3ia=− + and ( )1 1i2b=− . (i) Without using a calculator, find the value of 2ab in the form ixy+ . [2] (ii) Find the modulus and argument of 2 .ab [3] (ii) The diagram below shows an isosceles right triangle ABC, where the points A, B and C represent the complex numbers a, b and c respectively. Find the exact value of c. [2] 11. MI/2016/I/12 (modified) (i) For real a and b, one root of the equation 32 9 5 0az z bz− + − = is 2iz=− . Find the values of a and b . [4] (ii) Hence find all the roots of the equation in part (i) in exact form. [4] 12. AJC/2011/I/13 (i) The complex numbers p and q satisfy the simultaneous equations * 10i i 5pq+ = + 2 5 2i 0pq− − + = . Given that Im(p) < 0, find p in the Cartesian form. [3] Hence find the values of n for which 2np is purely imaginary. [2] (ii) The complex number w is such that *arg 2 w pp  − =− and *2ww+ =− . Find w. [4] A C B y x O

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Complex Numbers 4 13. MJC/2016/I/10 (a) Solve the simultaneous equations 2i 1zw= + − and 2 5i0 2zw− + = , giving z and w in the form ixy+ where x and y are real. [5] (b) Given that 1zw w=− where ( )2 cos isinw =+ ,   −   , express the real and imaginary parts of z in terms of  . [3] 14. RI Promo 9758/2025/Q3 The point A on the Argand diagram below represents the complex number 1w with modulus r and argument  . It is given that the complex numbers 2w and 3w satisfy the equations 21ww=− and 32 iww= . Let B, C and D represent the complex numbers 2w , 3w and 23ww+ respectively. (a) On the copy of the Argand diagram in the Printed Answer Book, plot the points B, C and D, indicating clearly the modulus and argument of 2w and 3w . [3] (b) State, in radians, the angle BOC. [1] (c) By considering the quadrilateral OBDC, find 23ww+ in terms of r and 23arg( )ww+ in terms of  . [2] 15. NYJC/2011/I/8 A complex number i=+z x y is represented by the point P in an Argand diagram. If the complex number w, where 8i 6 zw z −= + , ( 6−z ), has its real part zero, show that the locus of P in the Argand diagram is a circle, and find the radius and t he coordinates of the centre of this circle. If, however, w is real, find the locus of P in this case. [6] Im Re r

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Complex Numbers 5 16. PJC/2011/I/9(modified) The polynomial ( )P z has real coefficients. The equation ( )P0z = has a root ixy+ , where x and y are real numbers. (i) Write down a second root in terms of x and y , and hence show that a quadratic factor of ( )P z is 2 2 22z xz x y− + + . [3] (ii) 1z and 2z are roots of the equation ( )P0z = , where 1 1 3iz =+ and 21 izz= . Write down the exact modulus and argument of 2z . State the geometrical relationship between 1z and 2z and illustrate this relationship clearly on an Argand diagram. [3] (iii) Given further that ( )P z is of degree four, express ( )P z as a product of two quadratic factors with real coefficients, giving each factor in exact form. [3] 17. ACJC/2014/I/4 (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. [4] (ii) For these values of a and b, solve the equation in part (ii), without the use of a calculator. [3] 18. SRJC/2016/II/3(a) (a) The complex number w is such that w = a + ib, where a and b are non-zero real numbers. The complex conjugate of w is denoted by w*. Given that *2() 3iw bw =− , solve for a and b and hence write down the possible values of w. [3] 19. TJC/2016/I/12(a) (a) The complex numbers 1z and 2z satisfy the following simultaneous equations 12 *2 i 7 6izz+ = − , 12 i 6 6izz− = − . Find 1z and 2z in the form ixy+ , where x and y are real. [4] 20. VJC/2016/II/3 The points A, B, C and D represent the complex numbers 2 5i−+ , 1z , 4i+ and 2z respectively. Given that ABCD is a square, labelled in an anti-clockwise direction, show that 1 1z =− . Find 2z . [4]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Complex Numbers 6 21. VJC Prelim 9758/2025/P2/Q4 Do not use a calculator in answering this question. The complex number z has modulus 1 and argume