2e Correct expression Correct solution. Arg(z) = - 450 or -1350 z = (a + bi) (3 i) = 3a + b + i(3b a) – – interpreted to give a without i2 (3 + i) (3 i) 10 10 – correct relationship for u such as 3b a = -1 if angle = -450 – im(z) = -1 for r 3a + b 3b-a re(z) -45 So 3b a = - 3a b 3a-b – – 4b = -2a so a = -2b Evidence Achievement (u) Merit (r) Excellence (t) Question THREE Apply the algebra Apply the algebra of Apply the algebra of of complex numbers complex numbers, complex numbers, in solving using relational using extended problems. thinking, in solving abstract thinking, in problems. solving problems. (1 + + - √2)(3 – √2)2 =(1 √2)(11 6√2) 3a Correct answer = -1 + 5√2 required for u and u = 8cis(5) v = 4cis(3) 3b 8cis (5) = 2cis(2) Correct solution. 4cis(3) required for u )9 3c Change to polar 1 form required for Correct solution = ( 1cis(-45) )9 u required for r = 1 cis(-405) = cis (-45) ) = 3d Multiplied by w = 1 = 1 × (x iy)2 – conjugate in some z2 (x + iy)2 (x iy)2 – form for u Could have = x2 + y2 2xyi – expanded 1 Correct exact (x + iy)(x iy)(x + iy)(x + iy) – . solution for r. = x2 + y2 2xyi = x2 + y2 2xyi . (x + iy)2 – – (x2 + y2) (x2 + y2) (x2 + y2)2 at start but method shown is better.
2e z = (a + bi) (3 – i) = 3a + b + i(3b – a) (3 + i) (3 – i) 10 10 3b – a = -1 if angle = -450 3a + b 3b-a -45 So 3b – a = - 3a – b 3a-b 4b = -2a so a = -2b Correct expression without i2 for u Arg(z) = - 450 or -1350 interpreted to give a correct relationship such as im(z) = -1 for r re(z) Correct solution. Question THREE Evidence Achievement (u) Merit (r) Excellence (t) Apply the algebra of complex numbers in solving problems. Apply the algebra of complex numbers, using relational thinking, in solving problems. Apply the algebra of complex numbers, using extended abstract thinking, in solving problems. 3a (1 + √2)(3 – √2)2 =(1 + √2)(11 - 6√2) = -1 + 5√2 Correct answer required for u 3b u = 8cis(5) and v = 4cis(3) 8cis (5) = 2cis(2) 4cis(3) Correct solution. required for u 3c )9 1 = ( 1cis(-45) )9 = 1 cis(-405) = cis (-45) = ) Change to polar form required for u Correct solution required for r 3d w = 1 = 1 × (x – iy)2 z2 (x + iy)2 (x – iy)2 = x2 + y2 – 2xyi (x + iy)(x – iy)(x + iy)(x + iy) = x2 + y2 – 2xyi = x2 + y2 – 2xyi (x2 + y2) (x2 + y2) (x2 + y2)2 Multiplied by conjugate in some form for u Could have expanded 1 . . (x + iy)2 at start but method shown is better. Correct exact solution for r.