Question 1 - A-Level Maths

CAIE Further Paper 1 2021 November — Question 1 6 marks

Exam Board	CAIE
Module	Further Paper 1 (Further Paper 1)
Year	2021
Session	November
Marks	6
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 question on symmetric functions requiring knowledge of Newton's identities or algebraic manipulation to find elementary symmetric functions from power sums. While it requires multiple steps and careful algebra (finding αβ+βγ+γα from the given sums, then finding αβγ), the technique is well-practiced in Further Maths courses and follows a predictable pattern without requiring novel insight.
Spec	4.05a Roots and coefficients: symmetric functions

1 It is given that $$\alpha + \beta + \gamma = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 5 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 6 .$$ The cubic equation $\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0$ has roots $\alpha , \beta , \gamma$.
Find the values of $b , c$ and $d$.

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Question 1:

Answer	Marks	Guidance
Answer	Marks	Guidance
$b = -(\alpha + \beta + \gamma) = -3$	B1
$5 = 3^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)$	M1 A1	Uses formula for sum of squares.
$c = 2$	A1
$6 - 3(5) + 2(3) + 3d = 0$	M1	Uses original equation or formula for sum of cubes.
[Equation is $x^3 - 3x^2 + 2x + 1 = 0$] $d = 1$	A1
Total	6

**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $b = -(\alpha + \beta + \gamma) = -3$ | B1 | |
| $5 = 3^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)$ | M1 A1 | Uses formula for sum of squares. |
| $c = 2$ | A1 | |
| $6 - 3(5) + 2(3) + 3d = 0$ | M1 | Uses original equation or formula for sum of cubes. |
| [Equation is $x^3 - 3x^2 + 2x + 1 = 0$] $d = 1$ | A1 | |
| **Total** | **6** | |

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1 It is given that

$$\alpha + \beta + \gamma = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 5 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 6 .$$

The cubic equation $\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0$ has roots $\alpha , \beta , \gamma$.\\
Find the values of $b , c$ and $d$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2021 Q1 [6]}}

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Q1 6 Q2 9 Q3 8 Q4 11 Q5 13 Q6 13 Q7 15