June 2024 Paper 2 Q3

OCR ACurrent spec6 marksPolynomials Sketching & Transforming Graphs

3 The function f is defined by \(\mathrm{f}(x) = x^3 - x^2 - 5x - 3\).

(a) Show that \((x - 3)\) is a factor of \(\mathrm{f}(x)\). [1]

(b) Factorise \(\mathrm{f}(x)\) completely. [2]

Three students attempted to draw the graph of \(y = (x - a)(x - 1)(x + 1)\), each using a different value of the constant \(a\). Not all of their graphs were correct. Their graphs are given in the diagrams below. Copies of the diagrams are provided in the Printed Answer Booklet.

either give the value of \(a\) for which this is the correct graph of \(y = \mathrm{f}(x)\),
or, if there is no value of \(a\) for which this graph is correct, write “No value of \(a\)”. [3]

Fig. 1.1: cubic curve on a grid from x = -2 to 3, crossing the x-axis at 0, 1 and 2, with a maximum between 0 and 1 and a minimum between 1 and 2 — Fig. 1.1

Fig. 1.2: cubic curve on a grid crossing the x-axis at -1, with a maximum near x = 0, crossing at 1, a minimum between 1 and 2, and crossing at 2 — Fig. 1.2

Fig. 1.3: cubic curve on a grid crossing the x-axis at -1, with a maximum just left of the y-axis, touching the x-axis at 1 and then rising — Fig. 1.3

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{f}(3) = 3^3 - 3^2 - 5 \times 3 - 3 = 0\)	B1	1.1
	[1]

Notes

B1: Must see substitution; or factorise showing factor \((x - 3)\)
No conclusion required beyond =0 (so e.g. accept long division leading to remainder 0).

Mark scheme (b)

Scheme	Marks	AO
\((x - 3)(px^2 + qx + r)\)	M1	3.1a
\((x - 3)(x^2 + 2x + 1)\) \((x - 3)(x + 1)(x + 1)\)	A1	1.1
	[2]

Notes

M1: Attempt quadratic factor by inspection or division.
Look for \((x - 3)\) and a quadratic or two linear factors.
NB this step may be seen in (a).

A1: or \((x - 3)(x + 1)^2\)

Mark scheme (c)

Scheme	Marks	AO
Fig 1.1: no value (of \(a\))	B1	2.2a
Fig 1.2: \((a =)\ 2\)	B1	2.2a
Fig 1.3: \((a =)\ 1\)	B1	2.2a
	[3]

Notes

Fig 1.2: Accept values between 1.9 and 2.1

Fig 1.3: Accept values between 0.9 and 1.1