Question 7 - A-Level Maths

CAIE FP1 2017 June — Question 7 8 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2017
Session	June
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Roots of polynomials
Type	Finding polynomial from root properties
Difficulty	Challenging +1.2 This is a standard Further Maths technique of converting symmetric function conditions into a polynomial via Vieta's formulas. Students must recognize that the three given equations translate to elementary symmetric polynomials (sum of roots = -1, sum of products of pairs can be derived from the second equation using the identity, and sum of reciprocals gives product of roots), then construct and solve the cubic. While it requires multiple steps and is beyond Core A-level, it's a textbook FP1 exercise with a well-established method.
Spec	4.05a Roots and coefficients: symmetric functions

7 By finding a cubic equation whose roots are $\alpha , \beta$ and $\gamma$, solve the set of simultaneous equations $$\begin{aligned} \alpha + \beta + \gamma & = - 1 , \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 29 , \\ \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - 1 . \end{aligned}$$

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Question 7:

Answer	Marks	Guidance
Answer	Marks	Guidance
$2\sum\alpha\beta = 1 - 29 \Rightarrow \sum\alpha\beta = -14$	M1A1
$\frac{\sum\alpha\beta}{\alpha\beta\gamma} = \frac{-14}{\alpha\beta\gamma} = -1 \Rightarrow \alpha\beta\gamma = 14$	M1A1 FT
$\Rightarrow x^3 + x^2 - 14x - 14 = 0$	A1
$\Rightarrow (x+1)(x^2-14)$	M1A1	Attempt to factorise cubic
Solution is $-1$, $\pm\sqrt{14}$, any order. Accept $\pm3.74$ (awrt). SR B1 for correct roots without working	A1

## Question 7:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $2\sum\alpha\beta = 1 - 29 \Rightarrow \sum\alpha\beta = -14$ | M1A1 | |
| $\frac{\sum\alpha\beta}{\alpha\beta\gamma} = \frac{-14}{\alpha\beta\gamma} = -1 \Rightarrow \alpha\beta\gamma = 14$ | M1A1 FT | |
| $\Rightarrow x^3 + x^2 - 14x - 14 = 0$ | A1 | |
| $\Rightarrow (x+1)(x^2-14)$ | M1A1 | Attempt to factorise cubic |
| Solution is $-1$, $\pm\sqrt{14}$, any order. Accept $\pm3.74$ (awrt). SR **B1** for correct roots without working | A1 | |

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7 By finding a cubic equation whose roots are $\alpha , \beta$ and $\gamma$, solve the set of simultaneous equations

$$\begin{aligned}
\alpha + \beta + \gamma & = - 1 , \\
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 29 , \\
\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - 1 .
\end{aligned}$$

\hfill \mbox{\textit{CAIE FP1 2017 Q7 [8]}}

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