The paper was Edexcel Practice Paper A.
Question 1: roots of i
That, in the complex number system, every number has k k-th roots, is a basic result of the Fundamental Theorem of Algebra, which is, well… fundamental.
And that those roots are evenly spaced like the spokes of a wheel is a basic element of visualising how complex numbers work.
It’s worrying that some of you botched this question. Study
Question 2. Partial fractions and method of differences
You were all ok with the basic methods here. However, three of you made the exact same mistake from not reading the question carefully.
In part (a) you found the sum of 2/(4r2−1). In part (b) you’re asked for a sum of 1/(4r2−1). You can’t just plug in the formula you got from part (a). You must also divide by 2.
Question 3. Polar area
One of you had mixed up areas in polar coordinates – where an integral always gives you the area of a sector between two θ-values, a slice like a slice of pizza, not a vertical slice between two x-values.
You won’t lose marks by missing off the dθ (or dx, or whatever) from integrals – in fact Edexcel mark schemes sometimes miss them off – but it’s bad mathematical writing, and will cause you trouble when you come to do more complicated maths.
Question 4. Complex loci
This is a very standard plain-vanilla question, so it is worrying that some of you did badly.
Two of you made the same slip in part (c), writing 3 × 3 = 3 instead of 3 × 3 = 9. Everyone makes slips like that from time to time. However, the question had told you that the answer was a line, so when you got a circle for your answer you could know you’d made a slip and look for it.
Also, you could tell the answer must be a line without the question telling you that. Since the circle goes through the origin, a w=2i/z transformation of it must produce a locus which goes to infinity as z approaches the origin. That can’t be a circle. It must be a line.
Question 5. Maclaurin
You all did this efficiently and correctly.
Question 6. 2nd order differential equation
Your answers here showed you all know the general method, but some of you fumbled the details and lost marks. You need more practice to fix that.
Question 7. Inequalities
Question 8. 1st order differential equation
As in question 6, you all indicated you knew the general method, but there was more fumbling here. Sometimes a fatal amount of fumbling, as in making an understandable slip calculating the integrating factor and then ploughing ahead trying to make that wrong integrating factor work by way of writing nonsense.