The complex number −1 + √7i is denoted by u. It is given that u is a root of the equation 2x³ + 3x² + 14x + k = 0, where k is a real constant.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. *M1 Substitute x = −1+√7i in the equation and attempt expansions of x² and x³
2. DM1 Use i² = −1 correctly at least once and solve for k 2(20-4√7i)+3(-6-2√7i) +14(-1+√7i) + k = 0
3. A1 Obtain answer k = - 8 SC B1 only for those who show no working for the cube and square and obtain answer k = – 8.
4. B1 State answer -1-√7i Can be seen simply stated on its own, or in a list of roots. Allow if stated clearly in part 10(a).
5. M1 Carry out a method for finding a quadratic factor with zeros −1+√7i and -1-√7i Or state (x-(-1+√7i))(x-(-1-√7i))(2x-p)
6. A1 Obtain x² + 2x+8 Or obtain (-1+√7i)(-1-√7i) p = -8 Or obtain (-1+√7i)+(-1−√7i)+3/2 = −p/2
7. A1 Obtain root x = 1/2, or equivalent, via division or inspection Needs to follow from the working.
8. B1 Show a circle with centre -1+√7i [Figure X.X]
9. B1 Show circle with radius 2 and centre not at the origin There needs to be some evidence of scale e.g. radius marked or a scale on the axes If the scales are very different from each other then B1 for centre in the correct position and B1 for an ellipse. If there is more than one circle the max score is B1.
10. M1 Carry out a complete method for calculating the maximum value of arg z for correct circle e.g. ½π+tan⁻¹(2/√7) + ¼π Can be implied by 155.7° .
11. A1 Obtain answer 2.72 radians CAO. The question requires radians.