| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2022 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with nonlinearly transformed roots |
| Difficulty | Challenging +1.2 This is a standard Further Maths question on transformed roots requiring systematic application of Vieta's formulas and algebraic manipulation. Part (a) uses the substitution y=1/x² to find the new equation; part (b) applies Vieta's formulas to recognize the sum of products of roots taken three at a time; part (c) uses Newton's identities or sum-of-squares techniques. While requiring multiple steps and careful algebra, these are well-practiced techniques in Further Maths syllabi with no novel insight needed. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = x^{-2}\) leading to \(x = y^{-\frac{1}{2}}\) | B1 | Correct substitution. |
| \(y^{-2} + 3y^{-1} + 2y^{-\frac{1}{2}} + 6 = 0\) leading to \(1 + 3y + 2y^{\frac{3}{2}} + 6y^2 = 0\) | M1 | Obtains an equation not involving radicals. |
| \(36y^4 + 32y^3 + 21y^2 + 6y + 1 = 0\) | A1 | |
| \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} + \frac{1}{\delta^2} = -\frac{32}{36} = -\frac{8}{9}\) | B1 FT |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha\beta\gamma\delta = 6\) | B1 | SOI |
| \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} + \frac{1}{\delta^2} = \frac{\alpha^2\beta^2\delta^2 + \alpha^2\beta^2\gamma^2 + \beta^2\gamma^2\delta^2 + \alpha^2\gamma^2\delta^2}{\alpha^2\beta^2\gamma^2\delta^2}\) | M1 | Relates to coefficients. |
| \(\beta^2\gamma^2\delta^2 + \alpha^2\gamma^2\delta^2\alpha^2\beta^2\delta^2 + \alpha^2\beta^2\gamma^2 = -32\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{\alpha^4} + \frac{1}{\beta^4} + \frac{1}{\gamma^4} + \frac{1}{\delta^4} = \left(\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} + \frac{1}{\delta^2}\right)^2 - 2\left(\alpha^{-2}\beta^{-2} + \alpha^{-2}\gamma^{-2} + \alpha^{-2}\delta^{-2} + \beta^{-2}\gamma^{-2} + \beta^{-2}\delta^{-2} + \gamma^{-2}\delta^{-2}\right) = \left(-\frac{8}{9}\right)^2 - 2\left(\frac{21}{36}\right)\) | M1 | Uses formula for sum of squares. |
| \(-\frac{61}{162}\) | A1 |
## Question 2(a):
$y = x^{-2}$ leading to $x = y^{-\frac{1}{2}}$ | B1 | Correct substitution.
$y^{-2} + 3y^{-1} + 2y^{-\frac{1}{2}} + 6 = 0$ leading to $1 + 3y + 2y^{\frac{3}{2}} + 6y^2 = 0$ | M1 | Obtains an equation not involving radicals.
$36y^4 + 32y^3 + 21y^2 + 6y + 1 = 0$ | A1 |
$\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} + \frac{1}{\delta^2} = -\frac{32}{36} = -\frac{8}{9}$ | B1 FT |
**Total: 4**
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## Question 2(b):
$\alpha\beta\gamma\delta = 6$ | B1 | SOI
$\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} + \frac{1}{\delta^2} = \frac{\alpha^2\beta^2\delta^2 + \alpha^2\beta^2\gamma^2 + \beta^2\gamma^2\delta^2 + \alpha^2\gamma^2\delta^2}{\alpha^2\beta^2\gamma^2\delta^2}$ | M1 | Relates to coefficients.
$\beta^2\gamma^2\delta^2 + \alpha^2\gamma^2\delta^2\alpha^2\beta^2\delta^2 + \alpha^2\beta^2\gamma^2 = -32$ | A1 |
**Total: 3**
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## Question 2(c):
$\frac{1}{\alpha^4} + \frac{1}{\beta^4} + \frac{1}{\gamma^4} + \frac{1}{\delta^4} = \left(\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} + \frac{1}{\delta^2}\right)^2 - 2\left(\alpha^{-2}\beta^{-2} + \alpha^{-2}\gamma^{-2} + \alpha^{-2}\delta^{-2} + \beta^{-2}\gamma^{-2} + \beta^{-2}\delta^{-2} + \gamma^{-2}\delta^{-2}\right) = \left(-\frac{8}{9}\right)^2 - 2\left(\frac{21}{36}\right)$ | M1 | Uses formula for sum of squares.
$-\frac{61}{162}$ | A1 |
**Total: 2**
---2 The equation $x ^ { 4 } + 3 x ^ { 2 } + 2 x + 6 = 0$ has roots $\alpha , \beta , \gamma , \delta$.
\begin{enumerate}[label=(\alph*)]
\item Find a quartic equation whose roots are $\frac { 1 } { \alpha ^ { 2 } } , \frac { 1 } { \beta ^ { 2 } } , \frac { 1 } { \gamma ^ { 2 } } , \frac { 1 } { \delta ^ { 2 } }$ and state the value of $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }$.
\item Find the value of $\beta ^ { 2 } \gamma ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \gamma ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }$.
\item Find the value of $\frac { 1 } { \alpha ^ { 4 } } + \frac { 1 } { \beta ^ { 4 } } + \frac { 1 } { \gamma ^ { 4 } } + \frac { 1 } { \delta ^ { 4 } }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q2 [9]}}