OCR Further Pure Core AS Specimen — Question 1 3 marks

Exam BoardOCR
ModuleFurther Pure Core AS (Further Pure Core AS)
SessionSpecimen
Marks3
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TopicRoots of polynomials
TypeQuadratic with transformed roots
DifficultyStandard +0.3 This is a standard Further Maths question on transformed roots requiring Vieta's formulas and algebraic manipulation. Students need to find α+β=-2 and αβ=5, then compute α²+β²=(α+β)²-2αβ=4-10=-6 and α²β²=(αβ)²=25, giving p=6 and q=25. While it requires multiple steps and is from Further Maths content, it follows a well-practiced technique with no novel insight needed, making it slightly above average difficulty.
Spec4.02i Quadratic equations: with complex roots4.05a Roots and coefficients: symmetric functions
1 In this question you must show detailed reasoning.
The equation \(x ^ { 2 } + 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\). The equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
Find the values of \(p\) and \(q\).
Question 1:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\alpha + \beta = -2\), \(\alpha\beta = 5\)B1 (AO 1.2) Allow alternative methods e.g. using complex roots or substituting \(x = \sqrt{y}\)
\(\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta\), \(\alpha^2\beta^2 = (\alpha\beta)^2\)M1 (AO 1.1a) Both used, must be seen
\(\Rightarrow p = 6\), \(q = 25\)A1 (AO 1.1)
[3] 
# Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\alpha + \beta = -2$, $\alpha\beta = 5$ | **B1** (AO 1.2) | Allow alternative methods e.g. using complex roots or substituting $x = \sqrt{y}$ |
| $\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$, $\alpha^2\beta^2 = (\alpha\beta)^2$ | **M1** (AO 1.1a) | Both used, must be seen |
| $\Rightarrow p = 6$, $q = 25$ | **A1** (AO 1.1) | |
| **[3]** | | |

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1 In this question you must show detailed reasoning.\\
The equation $x ^ { 2 } + 2 x + 5 = 0$ has roots $\alpha$ and $\beta$. The equation $x ^ { 2 } + p x + q = 0$ has roots $\alpha ^ { 2 }$ and $\beta ^ { 2 }$.\\
Find the values of $p$ and $q$.

\hfill \mbox{\textit{OCR Further Pure Core AS  Q1 [3]}}
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