Excellence Questions: Complex Numbers

Excellence Questions: Complex Numbers Manipulate real and complex numbers, and solve equations 90638

1. Simplify i4n+ 3

2. Find the modulus and argument of the complex numbers 1+ i 2 z=, w = 1-i 1 - i Plot the points representing z, w, z + w on an Argand diagram and show from the 骣3p diagram that tan琪 = 1 + 2 桫8

3. If z= x + iy satisfies the equation (2-i) x -( 1 + 3 i) y - 7 = 0 , find x and y.

4. If z= a + ib , and z satisfies the equation (2+i)( z + 3 i) = 7 i - 6 , find a and b.

5. simplify each expression, giving the answer in the form a+ bi a) (cos3q+i sin 3 q)( cos5 q + i sin 5 q ) cos9q+ i sin 9 q b) cos 2q- i sin 2 q (cos 2q+ i sin 2 q )3 c) (cosq+ i sin q )2 (cos1p- i sin 1 p )3 d) 7 7 1 1 4 (cos7p+ i sin 7 p )

6. Use De Moivre’s Theorem to prove: a) cos 4q= 8cos4 q - 8cos 2 q + 1

b) sin 5q= 5sin q - 20sin3 q + 16sin 5 q 4 tanq- 4 tan3 q c) tan 4q = 1- 6 tan2q + tan 4 q

1 1 7. Prove that if z is a complex number, zn+ =2cos nq , z n - = 2 i sin n q zn z n

8. If w is a complex root of 1, show that 1+w + w2 = 1.

Complex numbers 90638 1 Excellence 9. If w is a complex root of 1, simplify the following; a) (1+ 3w)( 1 + 3 w2 ) 2 b) (1+ 3w + w2 ) 2 c) (1+w + 3 w2 )

骣2p 骣 2 p n2 n 3 n 10. If w=cos琪 + i sin 琪 , find all possible values of (w+ w + w ) where n is 桫3 桫 3 an integer.

11. Find the cube roots of -1 in the form a+ ib . Using this result, solve the equation ( z-1)3 + z3 = 0

20 3- 12i 12. Given that z2 = , find the two values of z in polar form. 3- 9i

13. Given that cosq+i sin q = eiq , show that cosq-i sin q = e-iq and that eiq+ e- i q = 2cosq

14. Express 3 - i in the form reiq where r >0, -p < q p and show that n n n+1 骣np 3-i + 3 + i = 2 cos琪 ( ) ( ) 桫6

15. Sketch on an Argand diagram the locus of the point which moves according to each equation. a) z -3 = 2 b) z+ i = 1 c) z-1 + 2 i = 3

16. Shade the areas represented by: a) z -2 < 3 b) z+1 - i 1 c) z- i > 2 d) z� z i

17. Find the locus of the point which moves so that z-1 = z + i

18. Find the locus of the point which moves so that z-2 = x + 2 , where z= x + iy . p 19. Sketch the locus of the point P( x , y ) represented by arg( z- a) = , where a=1 + i . 2 20. Sketch the locus represented by

Complex numbers 90638 2 Excellence p 骣z -1 p a) arg( z- i) = b) arg琪 = 4 桫z +1 2

骣z -1 p 21. If arg琪 = , show that the locus of P( x , y ) where z= x + iy lies on the arc of a 桫z +1 4 circle and find the coordinates of the centre and the radius.

22. Sketch the locus of P( x , y ) , which moves such that z-1 = z - 3 i

23. Find the ratio of the greatest value of z +1 to its least value when z- i = 1

24. Given that 2 + i is a root of the equation x4+ ax 3 + bx 2 +25 = 0 , find the other roots.

25. a) Find the complex numbers u and v which satisfy the simultaneous equations 3u+ v = 4 i u- iv =8 + 2 i b) Find the locus of P such that z- u = z - v

c) Find the two complex numbers z1, z 2 of this locus for which z-4 - 3 i = 5

26. Find the locus of z-1 = z + 1 + 1

27. Prove that the roots of z3+3 pz 2 + 3 qz + r = 0 form an equilateral triangle if and only if p2 = q .

28. If u= a + ib, v = c + id , prove that a) uv= u. v b) if u- 1� u 1 then a 0

29. a) Find all the solutions to z4 = 81 4 b) Find all the solutions to ( z-2) = 81 z4

骣2p 骣 2 p 30. If z=cos琪 + i sin 琪 , where k is a positive integer, show that: 桫k 桫 k 1+z + z2 + z 3 + ..... + z k - 1 = 0

31. For the complex equation z3 =cosq + i sin q , prove that the sum of the three solutions for z is always zero.

Complex numbers 90638 3 Excellence Answers 1. -i 1骣 1 3p 2. z+ w = +琪1 + i . The angle this makes with the x-axis is 2桫 2 8 3. x=3, y = - 1 4. a= -1, b = 1 5. a) cos8q+ i sin8 q b) cos11q+ i sin11 q c) cos 4q+ i sin 4 q d) -1 9. a) 7 b) 4w2 c) 4w 10. If n is a multiple of 3, the value is 1 + 1 + 1 = 3 If n is not a multiple of 3, the value is w+ w2 +1 = 0 1 1 1 11. , 1+i 3 , 1 - i 3 2 2( ) 2 ( ) 骣p p 骣 -5 p - 5 p 12. 2琪 cos+i sin , 2 琪 cos + i sin 桫6 6 桫 6 6 15. a) circle, centre (3, 0) radius 2 b) circle, centre (0, -1) radius 1 c) circle, centre (1, -2) radius 3 16. a) area inside circle, centre (2, 0) radius 3 b) area inside and including the circle, centre (-1, 1) radius 1 c) area outside circle, centre (0, 1) radius 2 d) y 0.5 17. The distance from the point to (1, 0) is the same as the distance of the point from (0, -1). So the perpendicular bisector of the line joining these points. ( y= - x ) 18. The distance between ( x, y) and (2, 0) is equal to x + 2 so ( x-2)2 + y2 = x + 2 This gives y2 = 8 x , a parabola. 19. This is the line PA where A is (1, 1). PA makes an angle of 90° with the x-axis. So the vertical line x =1, where y 1 20. a) y= x +1, x 0 b) x2+ y 2 =1, y 0 21. centre (0, 1), radius = 2 1 22. y=3 ( x + 4) 2+ 1 23. ratio is 2- 1 24. 2+i , 2 - i , - 2 + i , - 2 - i

25. a) u=1 - i , v = - 3 + 7 i . b) Locus is 2y= x + 7 c) z1=7 + 7 i , z 2 = - 1 + 3 i 26. Hyperbola, foci 1, -1 27. Put z+ p = Z 1 1 3i 29. a) 北3, 3i b) -1, , 2 5 5

Complex numbers 90638 4 Excellence