| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Substitution to find new equation |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on polynomial roots requiring standard substitution technique and application of Vieta's formulas. The substitution x=√u is given explicitly, making it mechanical algebra, and part (ii) follows directly from recognizing that the new equation's roots are α², β², γ² and applying the elementary symmetric polynomial formula for sum of products of pairs. No novel insight required beyond textbook methods. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
5 The cubic equation $x ^ { 3 } + 5 x ^ { 2 } + 7 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(i) Use the substitution $x = \sqrt { u }$ to find a cubic equation in $u$ with integer coefficients.\\
(ii) Hence find the value of $\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }$.
\hfill \mbox{\textit{OCR FP1 2009 Q5 [5]}}