Question 6 - A-Level Maths

WJEC Further Unit 1 2022 June — Question 6 8 marks

Exam Board	WJEC
Module	Further Unit 1 (Further Unit 1)
Year	2022
Session	June
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Roots of polynomials
Type	Roots with special relationships
Difficulty	Challenging +1.2 This is a Further Maths question requiring knowledge of Vieta's formulas and geometric progressions. Students must set up roots as α, -3α, 9α, apply sum/product of roots systematically, and solve for α before finding p and q. While methodical, it's a standard Further Maths exercise with clear structure and no novel insight required—moderately above average difficulty.
Spec	4.05a Roots and coefficients: symmetric functions 4.05b Transform equations: substitution for new roots

6. The roots of the cubic equation $$2 x ^ { 3 } + p x ^ { 2 } - 126 x + q = 0$$ form a geometric progression with common ratio - 3 .
Find the possible values of $p$ and $q$, giving your answers in surd form.

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Question 6:

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$\alpha + \beta + \gamma = -\frac{p}{2}$	B1	May be seen later in working
$\alpha\beta + \beta\gamma + \gamma\alpha = -63$	B1
$\alpha\beta\gamma = -\frac{q}{2}$	B1
Let initial root be $\alpha$ AND use of g.p. property	M1	Accept solutions where $\alpha, \beta, \gamma$ interchanged
Then other roots are $-3\alpha$ and $9\alpha$	A1	oe (e.g. $-3\alpha, 9\alpha, -27\alpha$)
$7\alpha = -\frac{p}{2}$, $-21\alpha^2 = -63$, $-27\alpha^3 = -\frac{q}{2}$	A1	provided M1 awarded
$\therefore \alpha^2 = 3 \Rightarrow \alpha = \pm\sqrt{3}$	A1	cao
If $\alpha = +\sqrt{3}$: $p = -14\sqrt{3}$ and $q = 162\sqrt{3}$
If $\alpha = -\sqrt{3}$: $p = 14\sqrt{3}$ and $q = -162\sqrt{3}$	A1

## Question 6:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\alpha + \beta + \gamma = -\frac{p}{2}$ | B1 | May be seen later in working |
| $\alpha\beta + \beta\gamma + \gamma\alpha = -63$ | B1 | |
| $\alpha\beta\gamma = -\frac{q}{2}$ | B1 | |
| Let initial root be $\alpha$ AND use of g.p. property | M1 | Accept solutions where $\alpha, \beta, \gamma$ interchanged |
| Then other roots are $-3\alpha$ and $9\alpha$ | A1 | oe (e.g. $-3\alpha, 9\alpha, -27\alpha$) |
| $7\alpha = -\frac{p}{2}$, $-21\alpha^2 = -63$, $-27\alpha^3 = -\frac{q}{2}$ | A1 | provided M1 awarded |
| $\therefore \alpha^2 = 3 \Rightarrow \alpha = \pm\sqrt{3}$ | A1 | cao |
| If $\alpha = +\sqrt{3}$: $p = -14\sqrt{3}$ and $q = 162\sqrt{3}$ | | |
| If $\alpha = -\sqrt{3}$: $p = 14\sqrt{3}$ and $q = -162\sqrt{3}$ | A1 | |

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6. The roots of the cubic equation

$$2 x ^ { 3 } + p x ^ { 2 } - 126 x + q = 0$$

form a geometric progression with common ratio - 3 .\\
Find the possible values of $p$ and $q$, giving your answers in surd form.\\

\hfill \mbox{\textit{WJEC Further Unit 1 2022 Q6 [8]}}

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