Find the exact solutions of the equation 6p^(1/3) - 5p^(-1/3) - 13 = 0.

Mark scheme

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

1. [B1] May introduce their own variable e.g. x (6p^(2/3) - 13p^(1/3) - 5 = 0) soi
2. [M1] For attempt to solve quadratic equation to obtain p^(1/3) = .. or e.g. x = ...
3. [A1] Must be simplified and exact (p^(1/3) = 5/2 or p^(1/3) = -1/3)
4. [A1] Must be simplified and exact (p = 125/8 or p = -1/27)
5. [B1] lg(2x+5)²
6. [B1] Dep on first B1, must be a correct statement lg((2x+5)² / (x+2))
7. [B1] 1 = lg10 soi
8. [M1] Dep on second B mark For correct attempt to obtain a quadratic equation with no log terms, where k = 1 or 10 (2x+5)² / (x+2) = k oe
9. [M1] for attempt to solve their quadratic to obtain x = ..., implied by decimals of -1.8 or -0.69 or better
10. [A1] for both, A0 if one is discarded (-5 ± √5) / 4 or exact equivalent