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Institution: University of the West Indies (St. Augustine) Course: Engineering Mathematics I (MATH 1180), Group I: Class Quiz #1 Lecturer: Mr. Oral Robertson Date: September 23, 2010 MATH 1180: Class Quiz #1 COMPLEX NUMBERS (Multiple Choice) Duration of Test: 50 mins. Instructions: This paper has twenty (20) questions. Circle the correct answer (a), (b), (c), or (d). The use of calculators is NOT ALLOWED. NAME: _____________________________________________ I.D.# : __________________________ 1. What is the value of i5 ? (a) i (b) –1 (c) –i (d) 1 2. If z = –3 – 4i, what is the value of Im(z) ? (a) –3 (b) –3i (c) –4 (d) –4i 3. Which of the following gives the correct centre and radius of the circle defined by: |z – 2 + 3i|= 4 ? (a) (–2, 3) and 4 units (b) (–2, 3) and 2 units (c) (2, –3) and 4 units (d) (2, –3) and 2 units 4. If z1 = a + bi and z 2 = c + di, then the product z 1 z 2 equals to? (a) (ac + bd) + i(–ad + bc) (b) (ac – bd) + i(ad + bc) (c) (ac + bd) + i(ad – bc) (d) (ac – bd) + i(–ad – bc) 5. Taking what you know of complex number conjugates (including their geometry), which of the following do you expect to be not true? (a) z 2 z z (b) Re(z) Re( z) 6. What is the value of arg(–1)? (a) 0 (b) /2 (c) z z 2 Im(z) i (d) z1 z 2 z1 z 2 (c) (d) 3/2 7. What expression gives the length of the line segment connecting two complex numbers z2 and z3 ? (a) |z3 – z2| (b) |z2 + z3| (c) |z3| – |z2| (d) |z2| + |z3| 8. When the complex number “ i tanh ix ” is simplified, you get ? (a) – tanh x (b) tanh x (c) – tan x (d) tan x 9. If z = cos + i sin, then which of the following is not true? (a) z = ei (b) |z| = 1 (c) 1/z = cos – i sin (d) arg(z) = tan 10. What expression below gives the complex number 1 i 3 in polar form? (a) 2 cos 2 i sin 2 3 3 (b) 4 cos i sin 3 3 (c) 2 cos 4 i sin 4 3 3 (d) 4 cos 5 i sin 5 3 3 1 Institution: University of the West Indies (St. Augustine) Course: Engineering Mathematics I (MATH 1180), Group I: Class Quiz #1 Lecturer: Mr. Oral Robertson Date: September 23, 2010 11. What statement describes the point P (representing a complex number z) such that its distance from the point (0, –1) is twice its distance from the point (3, 0) ? (a) z 1 2 z 3i (b) z 1 2 z 3i (c) z i 2 z 3 (d) z i 2 z 3 12. Which of the following is not true for two general complex numbers z1 and z 2 ? (a) z1 z 2 z1 z 2 (b) z1 z 2 z 1 z 2 13. If z = cos + i sin, then z 1 (a) 2 cos z (b) 2 sin (c) z1 z 2 z1 z 2 (d) z1 z 2 z1 z 2 = ? (c) 2i cos 14. What is the value of Im(w) given that w = cos(x – iy) ? (a) cos x cosh y (b) sin x cosh y (c) cos x sinh y (d) 2i sin (d) sin x sinh y z n 15. If two complex numbers are z1 = r1(cos1 + i sin1) and z2 = r2(cos2 + i sin2), then arg 2 equals ? z1 nθ 2 n n n (a) (b) θ 2 θ 1 (c) nθ 2 θ1 (d) θ 2 θ1 θ1 16. If z = cos + i sin, then z n 1 (a) 2 cos n zn (b) 2n cos = ? (c) 2i sin n (d) –2n sin n 17. Which of the following is not true for two complex numbers z1 and z 2 ? (a) z1 z 2 z1 z 2 (b) z1 z 2 z1 z 2 (c) z1 z 2 z1 z 2 (d) z1 z 2 z 2 z1 18. Graphically, which of the following is not true for a complex number z1 defined by arg z1 = (a) y > 0 (b) x < 0 (c) y = –x 3 ? 4 (d) y = x 19. On an Argand diagram, two complex numbers z1 and z2 are such that arg(z2) > arg(z1). What expression can accurately give the angle between the two lines Oz1 and Oz2 ? (a) arg(z1 + z2) (b) arg (z2 – z1) (c) arg(z2/z1) (d) arg(z1 z2) 20. If w = cos2 + i sin2, then which of the following is NOT an expression equivalent to Re(w)? (a) cos2 + sin2 (b) cos2 – sin2 (c) 2cos2 – 1 (d) 1 – 2sin2 2