AS October 2021 Paper 1 Q2
2 The equation \(2x^3 + 3x^2 - 2x + 5 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).
Use a substitution to find a cubic equation with integer coefficients whose roots are \(\alpha + 1\), \(\beta + 1\) and \(\gamma + 1\). [4]
| Scheme | Marks | AO |
|---|---|---|
| \(u = x + 1\) | B1 | 3.1a |
| \((u - 1)^3 = u^3 - 3u^2 + 3u - 1\) used in solution | M1 | 1.1 |
| \(2x^3 + 3x^2 - 2x + 5 = 0 \Rightarrow 2(u^3 - 3u^2 + 3u - 1) + 3(u^2 - 2u + 1) - 2(u - 1) + 5 = 0\) | M1 | 1.1 |
| \(2u^3 - 3u^2 - 2u + 8 = 0\) | A1 | 2.5 |
| [4] |
Notes
M1: (1st) Attempt to expand using binomial. 4 terms.
Follow through on their \(u = x + 1\)
M1: (2nd) Substituting into equation. Allow if no “= 0” here. Must have an attempt at expanding \((u - 1)^3\) and \((u - 1)^2\)
Follow through on their \(u = x + 1\)
A1: Must be an equation
For correct equation found using sums and products of roots allow SC2 (Method required was dictated in question)
Only allocate marks using main scheme, or SC method