Higher November 2023 Paper 2 Q12
12. The equation \(2x^3 - 3x^2 - 2 = 0\) has only one positive solution.
Akriti says,
“The positive solution lies between 1·81 and 1·82.”
Is Akriti correct?
Yes ☐ No ☐
Show calculations and explain your decision. [2]
| Scheme | Marks | Comments |
|---|---|---|
| No indicated or implied with \((2 \times 1.81^3 - 3 \times 1.81^2 - 2 =)\ 0.03(1….)\) AND \((2 \times 1.82^3 - 3 \times 1.82^2 - 2 =)\ 0.1(19…)\) AND a correct explanation e.g ‘one should be positive, and one should be negative’. ‘0 is not between 0.119.. and 0.03..’ ‘0.119… and 0.03.. are both positive’ | B2 | Allow for 0.03 seen and an explanation saying that 1.82 will lead to a greater answer that is not negative. Award B1 for one of the following:
|
Alternative method
| Scheme | Marks | Comments |
|---|---|---|
| No indicated or implied with \((2 \times 1.81^3 - 3 \times 1.81^2 =)\ 2.03(1….)\) AND \((2 \times 1.82^3 - 3 \times 1.82^2 =)\ 2.1(19…)\) AND a correct explanation e.g ‘one should be more than 2, and one should be less than 2’. ‘2 is not between 2.119.. and 2.03..’ ‘2.119… and 2.03.. are both more than 2’ | B2 | Allow for 2.03 seen and an explanation saying that 1.82 will lead to a greater answer that is not negative. Award B1 for one of the following:
|
| (2) |
Corrected by Westie's Workshop
- The board's mark scheme printed “‘0 is not between 0.119.. and 0.03..’ and ‘0.119… and 0.03.. are both positive’”. Changed to “‘2 is not between 2.119.. and 2.03..’ and ‘2.119… and 2.03.. are both more than 2’” because in this method the values are 2.03… and 2.119…, so they are compared with 2, not 0.
Original: see the board's mark scheme PDF (link above).