The polynomial p(x) is such that p(x) = ax³ + bx² +cx-5, where a, b and c are integers. It is given that p'(0) = 12. It is also given that p(x) has a factor of 3x-1 and a remainder of 95 when divided by x-2.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [7] (a) B1 c=12
2. [7] (a) M1 Allow one arithmetic or sign error, may substitute in their c.
3. [7] (a) M1 Allow one arithmetic or sign error, may substitute in their c.
4. [7] (a) A1 Allow multiples but c needs to have been eliminated and terms with powers evaluated.
5. [7] (a) A1 Allow multiples but c needs to have been eliminated
6. [7] (a) M1, dep on at least one previous M1, for attempt to solve their equations in a and b only, to find a or b.
7. [7] (a) A1 for both a and b.
8. [3] (b) M1 for attempt at 2 terms in their quadratic factor.
9. [3] (b) A1 for both factors.
10. [3] (b) B1 Allow other valid arguments, but must be using a correct quadratic factor and an attempt to evaluate the discriminant