A-LevelMathematicsComplex numbers9709/31 · May/June 2021 · Q57 Marks
Solve the equation z² – 2piz – q = 0, where p and q are real constants.
- (a)Solve the equation z² – 2piz – q = 0, where p and q are real constants.[2]
- (b)In an Argand diagram with origin O, the roots of this equation are represented by the distinct points A and B. Given that A and B lie on the imaginary axis, find a relation between p and q.[2]
- (c)Given instead that triangle OAB is equilateral, express q in terms of p.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.
- [M1] Use quadratic formula and i² =−1
- [A1] Obtain answers pi + √(q-p²) and pi - √(q-p²)
- [M1] State or imply that the discriminant must be negative
- [A1] State condition q<p²
- [M1] Carry out a correct method for finding a relation, e.g. use the fact that the argument of one of the roots is (±)60°
- [A1] State a correct relation in any form, e.g. p / √(q-p²) = (±)√3
- [A1] Simplify to q = (4/3)p²
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