A-LevelMathematicsComplex numbers9709/31 · May/June 2017 · Q78 Marks
Throughout this question the use of a calculator is not permitted. The complex numbers u and w are defined by u = −1 + 7i and w = 3 + 4i.
- (i)Showing all your working, find in the form x + iy, where x and y are real, the complex numbers u - 2w and u/w.[4]
- (ii)In an Argand diagram with origin O, the points A, B and C represent the complex numbers u, w and u - 2w respectively. Prove that angle AOB = ¼π.[2]
- (iii)State fully the geometrical relation between the line segments OB and CA.[2]
Mark scheme
Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.
- State that u – 2w = - 7 - i B1
- EITHER: Multiply numerator and denominator of u/w by 3 – 4i, or equivalent (M1
- Simplify the numerator to 25 + 25i or denominator to 25 A1
- Obtain final answer 1 + i A1)
- OR: Obtain two equations in x and y and solve for x or for y (M1
- Obtain x = 1 or y = 1 A1
- Obtain final answer 1 + i A1)
- Find the argument of u/w M1
- Obtain the given answer A1
- State that OB and CA are parallel B1
- State that CA = 2OB, or equivalent B1
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