A-LevelMathematicsComplex numbers9709/31 · May/June 2023 · Q1012 Marks
The polynomial x³ + 5x² + 31x + 75 is denoted by p(x).
- (a)Show that (x + 3) is a factor of p(x).[2]
- (b)Show that z = −1 + 2√6i is a root of p(z) = 0.[3]
- (c)Hence find the complex numbers z which are roots of p(z²) = 0.[7]
Mark scheme
Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.
- [M1] Substitute x = -3 to obtain value of p(-3)
- [A1] Obtain p(-3) = 0 and hence given result
- [M1] Substitute z = -1+ 2√6i and attempt expansions of z² and z³
- [M1] Use i² = -1
- [A1] Obtain p(z) = 0 and hence given result SC B1 if there is no evidence of working for the square or the cube. Total 1/3.
- [B1] State z1² = √3i and z2² = -√3i
- [M1] Expand (x + iy)² = -1+2√6i and compare real and imaginary parts Allow for use of z² = -1-2√6i.
- [A1] Obtain x² - y² = -1 and xy = √6
- [M1] Solve to obtain x and y
- [A1] Obtain z3 = √2 + √3i and z4 = -√2-√3i
- [M1] Use z² = -1-2√6i to obtain z5 and z6 Allow for use of z² = -1+2√6i.
- [A1] Obtain z5 = √2-√3i and z6 = -√2 + √3i
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