| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with nonlinearly transformed roots |
| Difficulty | Standard +0.3 This is a standard Further Maths question on transformed roots requiring systematic application of Vieta's formulas and algebraic manipulation. Part (a) uses the reciprocal root transformation (reverse coefficients), while parts (b) and (c) involve expressing power sums in terms of elementary symmetric functions—techniques that are well-practiced in Further Pure syllabi. The calculations are straightforward with no conceptual surprises, making this slightly easier than average for Further Maths content. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = x^{-1}\) | B1 | |
| \(7y^{-3} + 3y^{-2} + 5y^{-1} + 1 = 0 \Rightarrow y^3 + 5y^2 + 3y + 7 = 0\) | M1 A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\alpha^{-2} + \beta^{-2} + \gamma^{-2} = (-5)^2 - 2(3) = 19\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\alpha^{-3} + \beta^{-3} + \gamma^{-3} = -5(19) - 3(-5) - 21 = -101\) | M1 A1 | |
| Total | 4 |
## Question 1:
**Part (a)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = x^{-1}$ | B1 | |
| $7y^{-3} + 3y^{-2} + 5y^{-1} + 1 = 0 \Rightarrow y^3 + 5y^2 + 3y + 7 = 0$ | M1 A1 | |
| **Total** | **3** | |
**Part (b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\alpha^{-2} + \beta^{-2} + \gamma^{-2} = (-5)^2 - 2(3) = 19$ | M1 A1 | |
**Part (c)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\alpha^{-3} + \beta^{-3} + \gamma^{-3} = -5(19) - 3(-5) - 21 = -101$ | M1 A1 | |
| **Total** | **4** | |
---1 The cubic equation $7 x ^ { 3 } + 3 x ^ { 2 } + 5 x + 1 = 0$ has roots $\alpha , \beta , \gamma$.
\begin{enumerate}[label=(\alph*)]
\item Find a cubic equation whose roots are $\alpha ^ { - 1 } , \beta ^ { - 1 } , \gamma ^ { - 1 }$.
\item Find the value of $\alpha ^ { - 2 } + \beta ^ { - 2 } + \gamma ^ { - 2 }$.
\item Find the value of $\alpha ^ { - 3 } + \beta ^ { - 3 } + \gamma ^ { - 3 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q1 [7]}}