4MA0 Higher June 2014 Paper 4 Q18

EdexcelHigherOld spec7 marksQuadratic Inequalities Solving Quadratics

(a) Solve \(5x^2 - 6x - 2 = 0\)
Give your solutions correct to 3 significant figures.
Show your working clearly. (3)

(b) Solve the inequality \(\dfrac{m^2 + 3}{4} \gt 21\)
Show clear algebraic working. (4)

Mark scheme (a)

Scheme	Marks
\(\dfrac{--6 \pm \sqrt{(-6)^2 - 4 \times 5 \times -2}}{2 \times 5}\)	M1
\(\sqrt{76}\) or \(\sqrt{36 + 40}\) or \(2\sqrt{19}\) or 8.71.....	M1
1.47, −0.272	A1
	(3)

Notes

M1: for correct substitution; condone one sign error ; condone missing brackets around \((-6)^2\); accept 6 and \(6^2\) in place of \(--6\) and \((-6)^2\) There may be partial evaluation – if so, this must be correct

M1: (independent) for correct simplification of discriminant (if evaluated, at least 3sf rounded or truncated)

A1: for −0.27 to −0.272 and 1.47 to 1.472
Award 3 marks if first M1 scored and answer correct

Alternative

Scheme	Marks
\(x^2 - \dfrac{6}{5}x - \dfrac{2}{5} = 0\) \(\left(x - \dfrac{3}{5}\right)^2 - \dfrac{9}{25} - \dfrac{2}{5} = 0\)	M1
\(\left(x - \dfrac{3}{5}\right) = \pm\sqrt{\dfrac{19}{25}}\)	M1
1.47, −0.272	A1

Notes

M1: for \(\left(x - \dfrac{3}{5}\right)^2\) oe

M1: for \(\left(x - \dfrac{3}{5}\right) = \pm\sqrt{\dfrac{19}{25}}\) oe

A1: for −0.27 to −0.272 and 1.47 to 1.472
Award 3 marks if first M1 scored and answer correct

Mark scheme (b)

Scheme	Marks
\(m^2 \gt 81\) or \(m^2 - 81 \gt 0\)	M1
\(\pm\sqrt{\text{“}81\text{”}}\) or ±9 or \((m + 9)(m - 9)\)	B1
\(m \gt 9\) ; \(m \lt -9\)	A2
	(4)
	(7 marks)

Notes

M1: Allow \(m^2 = 81\) or \(m^2 - 81 = 0\)

A2: A1 for \(m \gt 9\);
A1 for \(m \lt -9\)
dep on at least M1 scored