A2 June 2024 Paper 2 Q15

AQACurrent spec7 marksGraphs & Inequalities

15 The diagram shows the line \(y = 5 - x\)

Axes x and y through O with the straight line y = 5 − x, crossing the positive y-axis and the positive x-axis

(a) On the diagram above, sketch the graph of \(y = \left|x^2 - 4x\right|\), including all parts of the graph where it intersects the line \(y = 5 - x\)

(You do not need to show the coordinates of the points of intersection.) [3 marks]

(b) Find the solution of the inequality\[\left|x^2 - 4x\right| \gt 5 - x\]

Give your answer in an exact form. [4 marks]

Sketch of y = |x² − 4x|: touching the x-axis at O and at 4, with a hump between them, rising steeply outside them, and crossing the line y = 5 − x at four points — Mark scheme (a)

Scheme	Marks	AO
Draws curve with basically correct shape and no negative \(y\)-values.	B1	1.1b
Their graph intersects line at four distinct points.	B1	1.1b
Value of 4 shown at \(x\)-intercept	B1	1.1b
	(3)

Mark scheme (b)

Scheme	Marks	AO
Uses the modulus function to obtain two separate quadratic equations.	M1	3.1a
Obtains four correct \(x\)-values of points of intersection (condone decimal approximations).	A1	1.1a
Uses their graph to obtain at least one subset of the solution set (condone decimal approximations).	M1	2.2a
Deduces a completely correct solution set with exact values.	A1	2.2a
	(4)
	(7 marks)

Typical solution

\[x^2 - 4x = 5 - x\]\[x^2 - 3x - 5 = 0\]\[x = \frac{3 \pm \sqrt{29}}{2}\]\[4x - x^2 = 5 - x\]\[0 = x^2 - 5x + 5\]\[x = \frac{5 \pm \sqrt{5}}{2}\]\[x \lt \frac{3 - \sqrt{29}}{2},\ \frac{5 - \sqrt{5}}{2} \lt x \lt \frac{5 + \sqrt{5}}{2},\ x \gt \frac{3 + \sqrt{29}}{2}\]