| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Session | Specimen |
| Marks | 5 |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Standard +0.3 This is a straightforward application of Newton's identities and Vieta's formulas for a cubic polynomial. The first sum is immediate from coefficients, the second requires one algebraic manipulation (squaring the sum), and the third uses a standard identity. While it's a Further Maths topic, the execution is mechanical with no problem-solving insight required, making it slightly easier than average. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
$\alpha + \beta + \gamma = 14$: B1
$\alpha^2 + \beta^2 + \gamma^2 = (\alpha+\beta+\gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)$: M1
$= 14^2 - 2 \times 16 = 164$ **ft**: A1
$\alpha^3 + \beta^3 + \gamma^3 = 14\sum\alpha^2 - 16\sum\alpha - 21\sum 1$
For substituting roots into equation & adding: M1
$= 2009$ **cao**: A1
**Total: 5 marks**2 The equation $x ^ { 3 } - 14 x ^ { 2 } + 16 x + 21 = 0$ has roots $\alpha , \beta , \gamma$. Determine the values of $\alpha + \beta + \gamma$, $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$ and $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 Q2 [5]}}