Higher June 2023 Paper 1 Q24
24 Find the set of possible values of \(x\) for which
\[4x^2 - 25 \lt 0 \qquad \textbf{and} \qquad 12 - 5x - 3x^2 \gt 0\]You must show all your working. (5)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-2.5 \lt x \lt \dfrac{4}{3}\) | M1 | for method to find the critical values of \(4x^2 - 25 \lt 0\) eg \((2x + 5)(2x - 5)\) or critical values \(-2.5\) and 2.5 oe |
| M1 | (dep on M1) for \(x \gt -2.5\) and \(x \lt 2.5\) or \(x \gt a\) and \(x \lt b\) where \(a\) and \(b\) are their critical values and \(a \lt b\) | |
| M1 | for method to find the critical values of \(12 - 5x - 3x^2 \gt 0\) or \(3x^2 + 5x - 12 \lt 0\) eg \((4 - 3x)(x + 3)\) or \((3x - 4)(x + 3)\) \(\dfrac{--5 \pm \sqrt{(-5)^2 - 4 \times (-3) \times 12}}{2 \times (-3)}\) or \(\dfrac{-5 \pm \sqrt{5^2 - 4 \times 3 \times (-12)}}{2 \times 3}\) \(3\left[\left(x + \dfrac{5}{6}\right)^2 - \left(\dfrac{5}{6}\right)^2\right] - 12 = 0\) oe or critical values \(-3\) and \(\dfrac{4}{3}\) oe | |
| M1 | (dep on previous M1) for \(x \gt -3\) and \(x \lt \dfrac{4}{3}\) or \(x \gt c\) and \(x \lt d\) where \(c\) and \(d\) are their critical values and \(c \lt d\) | |
| A1 | for \(-2.5 \lt x \lt \dfrac{4}{3}\) oe eg \(x \lt \dfrac{4}{3}\), \(x \gt -2.5\) |
Additional guidance
accept use of = or incorrect inequality symbol for 1st and 3rd M marks
This may be implied by a suitable diagram
A correct answer with no supportive working gets 0 marks