Question 3 - A-Level Maths

OCR MEI FP1 2011 June — Question 3 5 marks

Exam Board	OCR MEI
Module	FP1 (Further Pure Mathematics 1)
Year	2011
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Roots of polynomials
Type	Roots with given sum conditions
Difficulty	Standard +0.8 This is a Further Maths question requiring systematic application of symmetric function relationships and algebraic manipulation. Students must use Vieta's formulas to connect the given conditions about sums of roots and their squares to the coefficients p and q, requiring the identity (α+β+γ)² = α²+β²+γ² + 2(αβ+βγ+γα). While methodical, it demands careful algebraic reasoning beyond standard A-level and is typical of FP1 difficulty.
Spec	4.05a Roots and coefficients: symmetric functions

3 The equation $x ^ { 3 } + p x ^ { 2 } + q x + 3 = 0$ has roots $\alpha , \beta$ and $\gamma$, where $$\begin{gathered} \alpha + \beta + \gamma = 4 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 6 \end{gathered}$$ Find $p$ and $q$.

Show LaTeX source

3 The equation $x ^ { 3 } + p x ^ { 2 } + q x + 3 = 0$ has roots $\alpha , \beta$ and $\gamma$, where

$$\begin{gathered}
\alpha + \beta + \gamma = 4 \\
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 6
\end{gathered}$$

Find $p$ and $q$.

\hfill \mbox{\textit{OCR MEI FP1 2011 Q3 [5]}}

This paper (9 questions)

View full paper

Q1 5 Q2 8 Q3 5 Q4 6 Q5 5 Q6 7 Q7 12 Q8 11 Q9 13