FP1 COMPLEX NUMBERS PAST EXAM QUESTIONS Nos.1-7 are standard plain-vanilla questions. Questions 8, 9, 10, 11, 13, 19, 26, 27, 30 demand a bit more thought. Q.21 is very easy if you use the sum-of-roots/ product-of-roots method, but a bit tricky otherwise. For Q.10 and Q.30, remember that if 64 is thought of as a complex number, then its argument is 0 or 2π or 4π or... When you multiply complex numbers, you add arguments. Therefore, when you take powers of complex numbers, you multiply arguments. When you take roots of complex numbers, you divide arguments. Therefore, the cube roots of 64 all have modulus 4, and they have arguments 0, 2π/3, 4π/3. 1. Given that z = –3 + 4i, (a) find the modulus of z, (2) (b) the argument of z in radians to 2 decimal places. (2) – 14 2i Given also that w = z (c) use algebra to find w, giving your answers in the form a + ib, where a and b are real. (4) The complex numbers z and w are represented by points A and B on an Argand diagram. (d) Show the points A and B on an Argand diagram. (2) (Total 10 marks) 2. f(x) = 2x3 – 5x2 + px – 5, p Given that 1 – 2i is a complex solution of f (x) = 0, (a) write down the other complex solution of f(x) = 0, (1) (b) solve the equation f(x) = 0, (6) (c) find the value of p. (2) (Total 9 marks) 3. z = 2 – 3i (a) Show that z2 = –5 – 12i. (2) Find, showing your working, (b) the value of z 2 , (2) City of London Academy 1 (c) the value of arg(z2), giving your answer in radians to 2 decimal places. (2) (d) Show z and z2 on a single Argand diagram. (1) (Total 7 marks) 4. f(x) = x3 + x2 + 44x + 150 Given that f(x) = (x + 3)(x2 + ax + b), where a and b are real constants, (a) find the value of a and the value of b. (2) (b) Find the three roots of f(x) = 0. (4) (c) Find the sum of the three roots of f(x) = 0. (1) (Total 7 marks) 5. The complex numbers z1 and z2 are given by z1= 2 + 8i and z2 = 1 – i Find, showing your working, z (a) 1 in the form a + bi, where a and b are real, (3) z 2 z (b) the value of 1 , z 2 (2) z1 (c) the value of arg , giving your answer in radians to 2 decimal places. z 2 (2) (Total 7 marks) City of London Academy 2 6. Given that 2 and 5 + 2i are roots of the equation x3 – 12x3 + cx + d = 0, c, d , (a) write down the other complex root of the equation. (1) (b) Find the value of c and the value of d. (5) (c) Show the three roots of this equation on a single Argand diagram. (2) (Total 8 marks) 7. The complex numbers z1and z2 are given by z1 = 2 – i and z2 = – 8 + 9i (a) Show z1 and z2 on a single Argand diagram. (1) Find, showing your working, (b) the value of z1 , (2) (c) the value of arg z1, giving your answer in radians to 2 decimal places, (2) z (d) 2 in the form a + bi, where a and b are real. z1 (3) (Total 8 marks) 8. f (x) = (x2 + 4)(x2 + 8x + 25) (a) Find the four roots of f (x) = 0. (5) (b) Find the sum of these four roots. (2) (Total 7 marks) City of London Academy 3 12 5i 9. Given that z1 = 3 + 2i and z2 , z1 (a) find z2 in the form a + ib, where a and b are real. (2) (b) Show on an Argand diagram the point P representing z1 and the point Q representing z2. (2) (c) Given that O is the origin, show that POQ = . 2 (2) The circle passing through the points O, P and Q has centre C. Find (d) the complex number represented by C, (2) (e) the exact value of the radius of the circle. (2) (Total 10 marks) 10. (a) Write down the value of the real root of the equation x3 – 64 = 0. (1) (b) Find the complex roots of x3 – 64 = 0, giving your answers in the form a + ib, where a and b are real. (4) (c) Show the three roots of x3 – 64 = 0 on an Argand diagram. (2) (Total 7 marks) 11. The complex number z is defined by City of London Academy 4 a 2i z , a , a 0 a i 1 Given that the real part of z is , find 2 (a) the value of a, (4) (b) the argument of z, giving your answer in radians to 2 decimal places. (3) (Total 7 marks) 12. z = –2 + i (a) Express in the form a + ib 1 (i) , z (ii) z2 (4) (b) Show that z 2 z 5 2 (2) (c) Find arg (z2 – z) (2) (d) Display z and z2 – z on a single Argand diagram. (2) (Total 10 marks) 13. z = √3 – i. z* is the complex conjugate of z. z 1 3 (a) Show that i. z * 2 2 (3) City of London Academy 5 z (b) Find the value of . z * (2) z (c) Verify, for z = √3 – i, that arg = arg z – arg z*. z * (4) z (d) Display z, z* and on a single Argand diagram z * (2) (e) Find a quadratic equation with roots z and z* in the form ax2 + bx + c = 0, where a, b and c are real constants to be found. (2) (Total 13 marks) 14. (a) Find the roots of the equation z2 + 2z + 17 = 0, giving your answers in the form a + ib, where a and b are integers. (3) (b) Show these roots on an Argand diagram. (1) (Total 4 marks) 15. The complex numbers z1 and z2 are given by z1 = 5 + 3i, z2 = 1 + pi where p is an integer. z (a) Find 2 in the form a + ib, where a and b are expressed in terms of p. z1 City of London Academy 6 (3) z2 Given that arg , z1 4 (b) find the value of p. (2) (Total 5 marks) 16. The complex numbers z and w satisfy the simultaneous equations 2z + iw = –1, z – w = 3 + 3i. (a) Use algebra to find z, giving your answer in the form a + ib, where a and b are real. (4) (b) Calculate arg z, giving your answer in radians to 2 decimal places. (2) (Total 6 marks) 17. Given that 3 – 2i is a solution of the equation x4– 6x3 + 19x2– 36x + 78 = 0, (a) solve the equation completely, (7) (b) show on a single Argand diagram the four points that represent the roots of the equation. (2) (Total 9 marks) z 2i 18. Given that = i, where is a positive, real constant, z i (a) show that z 1 i 1 2 2 (5) City of London Academy 7 1 Given also that arg z = arctan , calculate 2 (b) the value of λ, (3) (c) the value of |z|2. (2) (Total 10 marks) 19. Given that 1 + 3i is a root of the equation z3 + 6z + 20 = 0, (a) find the other two roots of the equation, (3) (b) show, on a single Argand diagram, the three points representing the roots of the equation, (1) (c) prove that these three points are the vertices of a right-angled triangle. (2) (Total 6 marks) 20. z = –4 + 6i. (a) Calculate arg z, giving your answer in radians to 3 decimal places. (2) A The complex number w is given by w = , where A is a positive constant. Given that 2 i w = 20, (b) find w in the form a + ib, where a and b are constants, (4) w (c) calculate arg . z (3) (Total 9 marks) City of London Academy 8 21. Given that 3 + i is a root of the equation f(x) = 0, where f(x) = 2x3 + ax2 + bx – 10, a, b , (a) find the other two roots of the equation f(x) = 0, (4) (b) find the value of a and the value of b. (3) (Total 7 marks) 3 3 22. Given that z = 4 cos isin and w = 1 – i3, find 4 4 z (a) , w (3) z (b) arg , in radians as a multiple of . w (3) (c) On an Argand diagram, plot points A, B, C and D representing the complex numbers z, w, and 4, respectively. (3) (d) Show that AOC = DOB. (3) (e) Find the area of triangle AOC. (2) (Total 14 marks) 23. The complex number z = a + ib, where a and b are real numbers, satisfies the equation z2 + 16 – 30i = 0. City of London Academy 9 (a) Show that ab = 15. (2) (b) Write down a second equation in a and b and hence find the roots of z2 + 16 – 30i = 0. (4) (Total 6 marks) w 24. Given that z = 1 + 3i and that = 2 + 2i, find z (a) w in the form a + ib, where a, b , (3) (b) the argument of w, (2) (c) the exact value for the modulus of w. (2) On an Argand diagram, the point A represents z and the point B represents w. (d) Draw the Argand diagram, showing the points A and B. (2) (e) Find the distance AB, giving your answer as a simplified surd.