Higher June 2025 Paper 1 Q16

16 \(\mathrm{f}(x) = 9 - \sqrt{x}\) where \(x \geqslant 0\)
\(\mathrm{g}(x) = 4x^2\)

(a) Find \(\mathrm{f}(9)\) (1)

(b) Solve \(\mathrm{fg}(x) \lt 0\)
Show clear algebraic working. (3)

Scheme	Marks
6	B1
	(1)

Mark scheme (b)

Scheme	Marks
\(9 - \sqrt{4x^2}\;(\lt 0)\)	M1
eg \(9 - 2x\;(\lt 0)\) or \(9 \lt 2x\) or \(81 \lt 4x^2\) or \(9^2 \lt 4x^2\)	M1
Working required Answer: \(x \gt \dfrac{9}{2}\)	A1
	(3)
	(4 marks)

M1: for substituting \(\mathrm{g}(x)\) in \(\mathrm{f}(x)\), allow incorrect inequality sign or = sign

M1: for removing the square root, allow incorrect inequality sign or = sign

A1: (dep on M1)

oe eg \(x \gt 4.5\), \(\dfrac{9}{2} \lt x\), \(4.5 \lt x\)