Standard +0.3 This is a multiple-choice question testing understanding of relationships between coefficients and roots (Vieta's formulas). While it requires checking four statements systematically, each verification is straightforward using α+β=-b/a and αβ=c/a. The incorrect statement (third one) can be identified by counterexample without deep insight, making this easier than average for Further Maths.
3 The quadratic equation \(a x ^ { 2 } + b x + c = 0 ( a , b , c \in \mathbb { R } )\) has real roots \(\alpha\) and \(\beta\).
One of the four statements below is incorrect.
Which statement is incorrect?
Tick ( \(\checkmark\) ) one box.
\(c = 0 \Rightarrow \alpha = 0\) or \(\beta = 0\) □
\(c = a \Rightarrow \alpha\) is the reciprocal of \(\beta\) □
\(b < 0\) and \(c < 0 \Rightarrow \alpha > 0\) and \(\beta > 0\) □
\(b = 0 \Rightarrow \alpha = - \beta\) □
3 The quadratic equation $a x ^ { 2 } + b x + c = 0 ( a , b , c \in \mathbb { R } )$ has real roots $\alpha$ and $\beta$.
One of the four statements below is incorrect.
Which statement is incorrect?
Tick ( $\checkmark$ ) one box.\\
$c = 0 \Rightarrow \alpha = 0$ or $\beta = 0$ □\\
$c = a \Rightarrow \alpha$ is the reciprocal of $\beta$ □\\
$b < 0$ and $c < 0 \Rightarrow \alpha > 0$ and $\beta > 0$ □\\
$b = 0 \Rightarrow \alpha = - \beta$ □
\hfill \mbox{\textit{AQA Further Paper 1 2020 Q3 [1]}}