Question 3 - A-Level Maths

CAIE FP1 2011 June — Question 3 6 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2011
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Roots of polynomials
Type	Finding polynomial from root properties
Difficulty	Standard +0.3 This is a straightforward application of Newton's identities/Vieta's formulas for Further Maths students. Given sum of roots, sum of squares, and product, students need to find the sum of products of pairs using the identity (Σα)² = Σα² + 2Σαβ, then construct the cubic using Vieta's formulas. The final step of solving the resulting cubic is routine. While it requires multiple steps and knowledge of symmetric functions, it follows a standard algorithmic approach taught explicitly in FP1 with no novel insight required.
Spec	4.05a Roots and coefficients: symmetric functions

3 Find a cubic equation with roots $\alpha , \beta$ and $\gamma$, given that $$\alpha + \beta + \gamma = - 6 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 38 , \quad \alpha \beta \gamma = 30 .$$ Hence find the numerical values of the roots.

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Question 3:

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$36 = 38 + 2\sum\alpha\beta \Rightarrow \sum\alpha\beta = -1$	M1A1	Uses $(\sum\alpha)^2 = \sum\alpha^2 + 2\sum\alpha\beta$
$\therefore t^3 + 6t^2 - t - 30 = 0$ is the required equation	A1	States equation with required roots
$\Rightarrow (t-2)(t+3)(t+5) = 0$	M1A1	Factorises
Hence $\alpha, \beta, \gamma$ are $2, -3$ and $-5$ (in any order). N.B. Answers written down with no working get B1.	A1	Gives values of $\alpha, \beta, \gamma$

# Question 3:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $36 = 38 + 2\sum\alpha\beta \Rightarrow \sum\alpha\beta = -1$ | M1A1 | Uses $(\sum\alpha)^2 = \sum\alpha^2 + 2\sum\alpha\beta$ |
| $\therefore t^3 + 6t^2 - t - 30 = 0$ is the required equation | A1 | States equation with required roots |
| $\Rightarrow (t-2)(t+3)(t+5) = 0$ | M1A1 | Factorises |
| Hence $\alpha, \beta, \gamma$ are $2, -3$ and $-5$ (in any order). N.B. Answers written down with no working get B1. | A1 | Gives values of $\alpha, \beta, \gamma$ |

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3 Find a cubic equation with roots $\alpha , \beta$ and $\gamma$, given that

$$\alpha + \beta + \gamma = - 6 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 38 , \quad \alpha \beta \gamma = 30 .$$

Hence find the numerical values of the roots.

\hfill \mbox{\textit{CAIE FP1 2011 Q3 [6]}}

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