Question 5 - A-Level Maths

CAIE FP1 2013 November — Question 5 8 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2013
Session	November
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Roots of polynomials
Type	Roots with special relationships
Difficulty	Standard +0.8 This is a Further Maths question requiring students to use the relationship between roots and coefficients (Vieta's formulas) with algebraically related roots. While the arithmetic progression structure simplifies the algebra considerably (the linear term in d vanishes when summing), students must still set up and solve a system using sum and product of roots, requiring systematic algebraic manipulation across multiple steps. This is more demanding than standard FP1 exercises but follows a recognizable pattern for this topic.
Spec	1.02j Manipulate polynomials: expanding, factorising, division, factor theorem 4.05a Roots and coefficients: symmetric functions

5 The equation $$8 x ^ { 3 } + 36 x ^ { 2 } + k x - 21 = 0$$ where $k$ is a constant, has roots $a - d , a , a + d$. Find the numerical values of the roots and determine the value of $k$.

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Question 5:

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$\sum\alpha = 3a = -\frac{36}{8} = -\frac{9}{2} \Rightarrow a = -\frac{3}{2}$	M1A1	Uses sum of roots
$\alpha\beta\gamma = a(a^2 - d^2) = \frac{21}{8}$	M1	Uses product of roots
$\frac{9}{4} - d^2 = -\frac{2}{3} \times \frac{21}{8} = -\frac{7}{4}$	M1	Substitutes for $a$
$\Rightarrow d^2 = 4 \Rightarrow d = \pm 2$	A1
Roots are $-\frac{7}{2}, -\frac{3}{2}, \frac{1}{2}$	A1
$\sum\alpha\beta = \frac{21}{4} - \frac{7}{4} - \frac{3}{4} = \frac{k}{8}$	M1	Uses sum of products in pairs
$\Rightarrow k = 22$	A1	(8)

Total: [8]

## Question 5:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\sum\alpha = 3a = -\frac{36}{8} = -\frac{9}{2} \Rightarrow a = -\frac{3}{2}$ | M1A1 | Uses sum of roots |
| $\alpha\beta\gamma = a(a^2 - d^2) = \frac{21}{8}$ | M1 | Uses product of roots |
| $\frac{9}{4} - d^2 = -\frac{2}{3} \times \frac{21}{8} = -\frac{7}{4}$ | M1 | Substitutes for $a$ |
| $\Rightarrow d^2 = 4 \Rightarrow d = \pm 2$ | A1 | |
| Roots are $-\frac{7}{2}, -\frac{3}{2}, \frac{1}{2}$ | A1 | |
| $\sum\alpha\beta = \frac{21}{4} - \frac{7}{4} - \frac{3}{4} = \frac{k}{8}$ | M1 | Uses sum of products in pairs |
| $\Rightarrow k = 22$ | A1 | **(8)** |

**Total: [8]**

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5 The equation

$$8 x ^ { 3 } + 36 x ^ { 2 } + k x - 21 = 0$$

where $k$ is a constant, has roots $a - d , a , a + d$. Find the numerical values of the roots and determine the value of $k$.

\hfill \mbox{\textit{CAIE FP1 2013 Q5 [8]}}

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