| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with linearly transformed roots |
| Difficulty | Standard +0.8 This is a substantial Further Maths question requiring Vieta's formulas for symmetric functions (parts i-iv involve standard and moderately complex manipulations), followed by a substitution to transform and solve the quartic. While the techniques are systematic, the multi-part structure, the need to recognize how to simplify part (iv), and solving the transformed quartic make this more demanding than typical A-level questions but still within standard FP1 scope. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
The roots of the quartic equation $x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0$ are $\alpha , \beta , \gamma$ and $\delta$. Find the values of\\
(i) $\alpha + \beta + \gamma + \delta$,\\
(ii) $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }$,\\
(iii) $\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } + \frac { 1 } { \delta }$,\\
(iv) $\frac { \alpha } { \beta \gamma \delta } + \frac { \beta } { \alpha \gamma \delta } + \frac { \gamma } { \alpha \beta \delta } + \frac { \delta } { \alpha \beta \gamma }$.
Using the substitution $y = x + 1$, find a quartic equation in $y$. Solve this quartic equation and hence find the roots of the equation $x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0$.
\hfill \mbox{\textit{CAIE FP1 2014 Q11 EITHER}}