The polynomials f(x) and g(x) are defined by f(x) = 4x³ + ax² + 8x + 15 and g(x) = x² + bx + 18, where a and b are constants.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [M1] (a) Substitute x=-3, equate to zero and attempt solution for a (Allow attempt at synthetic division, must be a complete method. Allow one sign error carried through. Allow attempt at algebraic long division, must be complete with the remainder equated to zero.)
2. [A1] (a) Obtain a=13
3. [M1] (b) Substitute x=2, equate to 40 and attempt solution for b
4. [A1] (b) Obtain b=9
5. [B1] (c) Identify x+3 as factor of f(x) - g(x) (May be implied by synthetic division. If working backwards from solutions from a calculator then B0 M0.)
6. [M1] (c) Attempt, by division or equivalent, to find quadratic factor
7. [A1] (c) Obtain (x+3)(2x-1)(2x+1)
8. [M1] (d) Attempt correct process to find at least 1 value from cosecθ=k where k<-1
9. [A1] (d) Obtain 3.48 or 5.94 (Allow for 199.5° or -19.5°.)
10. [A1] (d) Obtain a second correct solution (And no others within the range.)