| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Standard +0.8 This question requires knowledge of symmetric functions and Newton's identities to express sum of cubes in terms of elementary symmetric polynomials, then applying Vieta's formulas. Part (ii)(b) adds a non-routine twist requiring insight to scale the result appropriately. While the techniques are standard for Further Maths, the multi-step reasoning and the creative final part elevate it above average difficulty. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
**(i)** $x^3 + y^3 = (x+y)^3 - 3xy(x+y)$ or equivalent **M1 A1**
**[2]**
**(ii)(a)** $\alpha + \beta\ (=3)$ and $\alpha\beta\ (=\frac{8}{9})$ substituted into (i)'s result **ft** $\Rightarrow \alpha^3 + \beta^3 = 19$ **M1 A1**
**[2]**
**(b)** $9t^2 - 27t + 8 = 0 \Rightarrow (3t-1)(3t-8) = 0 \Rightarrow \alpha, \beta = \frac{1}{3}, \frac{8}{3}$ **M1 A1**
Then $\alpha^3 + \beta^3 = 19 = \left(\frac{1}{3}\right)^3 + \left(\frac{8}{3}\right)^3$ Explicit statement required **A1**
**[3]**7 (i) Express $x ^ { 3 } + y ^ { 3 }$ in terms of $( x + y )$ and $x y$.\\
(ii) The equation $t ^ { 2 } - 3 t + \frac { 8 } { 9 } = 0$ has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $\alpha ^ { 3 } + \beta ^ { 3 }$.
\item Hence express 19 as the sum of the cubes of two positive rational numbers.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q7 [7]}}