Complex Numbers Exam Questions

2(3 – i) + i(4 + 5i) = 6 – 2i + 4i + 5(-1) = 6 – 2i + 4i - 5 = 1 + 2i

z = 5 + 4i z = 5 – 4i z = 5 + 4i z = 5 – 4i

z = 5 + 4i z = 5 – 4i

Imaginary parts are equal p(2 + i) + 8 – ki = 5k - 3 - i 2p + pi + 8 – ki = 5k - 3 - i 2p + 8 + pi – ki = 5k - 3 - i Real parts are equal Imaginary parts are equal 2p + 8 + i(p – k) = 5k - 3 - i 2p + 8 = 5k - 3 and p – k = - 1

k = 3 and p = 2 2p + 8 = 5k - 3 and p – k = - 1 Put 3 for k in (2) 2p – 5k = - 11 --- (1) p – (3) = - 1 p – k = - 1 --- (2) p – 3 = - 1 2p – 5k = - 11 --- (1) - p + k = 1 --- (2) x - 1 p = - 1 + 3 = 2 2p – 5k = - 11 --- (1) x 1 k = 3 and p = 2 - 2p + 2k = 2 --- (2) x 2 -3k = - 9 (1) + (2) k = 3

7(2 + i) + i(11 + 9i) = 14 + 7i + 11i + 9(-1) = 14 + 18i - 9 = 5 + 18i

w = 3 - i w + 6i = (3 – i) + 6i = 3 + 5i |w + 6i| = w + 6i = 3 + 5i

z = 2 + 4i z2 = (2 + 4i)2 = (2 + 4i)(2 + 4i) = 2(2 + 4i) + 4i(2 + 4i) = 4 + 8i + 8i + 16(-1) = 4 + 16i - 16 = -12 + 16i z2 + 28 = -12 + 16i + 28 = 16 + 16i

Imaginary Parts are Equal z = 2 + 4i |z| = |2 + 4i| = k(z2 + 28) =|z|(1 + i) Real Parts are Equal Imaginary Parts are Equal

z = 5 + 4i z – 4i = 5 + 4i - 41 = 5 + 0i = 5 z = 5 + 4i z – 4i = 5

u = 3 – 6i |u| =

u + 3i = 3 – 6i + 3i = 3 – 3i

w = i - 2 w2 = (-2 + i)2 = (-2 + i)(-2 + i) = -2(-2 + i) + i(-2 + i) = 4 - 2i - 2i -1 = 3 - 4i

kw2 = 2w + 1 + ti k(3 – 4i) = 2(-2 + i) + 1 + ti 3k – 4ki = -4 + 2i +1 + ti Real Parts are Equal Imaginary Parts are Equal 3k – 4ki = -3 + i(2 + t) -4k = 2 + t 3k = -3 -4(-1) = 2 + t k = -1 2 + t = 4 t = 2