A-LevelMathematicsComplex numbers9709/32 · Feb/Mar 2019 · Q710 Marks
Showing all working and without using a calculator, solve the equation (1 + i)z² – (4 + 3i)z + 5 + i = 0. Give your answers in the form x + iy, where x and y are real.
- (a)Showing all working and without using a calculator, solve the equation (1 + i)z² – (4 + 3i)z + 5 + i = 0. Give your answers in the form x + iy, where x and y are real.[6]
- (b)The complex number u is given by u = -1 - i. On a sketch of an Argand diagram show the point representing u. Shade the region whose points represent complex numbers satisfying the inequalities |z| < |z − 2i| and ¼π < arg(z – u) < ½π.[4]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2019). The mark or mark type is shown in brackets.
- [M1] Use quadratic formula to solve for z
- [M1] Use i² =-1 throughout
- [A1] Obtain correct answer in any form
- [M1] Multiply numerator and denominator by 1 – i, or equivalent
- [A1] Obtain final answer, e.g. 1 - i
- [A1] Obtain second final answer, e.g. 5/2 + 1/2 i
- [B1] Show the point representing u in relatively correct position
- [B1] Show the horizontal line through z = i
- [B1ft] Show correct half-lines from u, one of gradient 1 and the other vertical
- [B1] Shade the correct region
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