A2 June 2021 Paper 2 Q5
5 The equation
\[z^3 + 2z^2 - 5z - 3 = 0\]has roots \(\alpha\), \(\beta\) and \(\gamma\)
Find a cubic equation with roots
\[\frac{1}{2}\alpha - 1,\ \frac{1}{2}\beta - 1 \ \text{ and } \ \frac{1}{2}\gamma - 1\][5 marks]
| Scheme | Marks | AO |
|---|---|---|
| Expresses \(w\) in terms of \(z\) or States the values of the sum, product and pairwise sum of roots of original equation. Condone sign errors. | M1 | 3.1a |
| Expresses \(z\) correctly in terms of \(w\) or obtains correct value \((-4)\) of sum of roots of new equation | A1 | 1.1b |
| Substitutes their expression for \(z\) into original equation to form an equation in \(w\) or Expresses the pairwise sum of roots of new equation in terms \(\sum\alpha\) and \(\sum\alpha\beta\) | M1 | 1.1a |
| Simplifies their equation in \(w\) or Expresses product of roots of new equation in terms \(\sum\alpha\), \(\sum\alpha\beta\) and \(\alpha\beta\gamma\) | M1 | 1.1a |
| Obtains correct equation (any correct form) | A1 | 1.1b |
| (5 marks) |
Typical solution
\[w = \tfrac{1}{2}z - 1 \text{ so } z = 2w + 2\]Substituting in the original equation
\[\begin{aligned} (2w + 2)^3 + 2(2w + 2)^2 - 5(2w + 2) - 3 &= 0 \\ 8w^3 + 24w^2 + 24w + 8 + 8w^2 + 16w + 8 - 10w - 10 - 3 &= 0 \\ 8w^3 + 32w^2 + 30w + 3 &= 0 \end{aligned}\]