Moderate -0.8 This is a straightforward C1 question testing basic factorisation (taking out common factor x, then recognising a perfect square), standard cubic sketching, and applying a horizontal translation. All steps are routine procedures with no problem-solving required beyond recognising the translation pattern.
10. (a) Factorise completely \(x ^ { 3 } - 6 x ^ { 2 } + 9 x\)
(b) Sketch the curve with equation
$$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$$
showing the coordinates of the points at which the curve meets the \(x\)-axis.
Using your answer to part (b), or otherwise,
(c) sketch, on a separate diagram, the curve with equation
$$y = ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 9 ( x - 2 )$$
showing the coordinates of the points at which the curve meets the \(x\)-axis.
B1 for taking out factor \(x\); M1 for attempt to factorize TQ \((x+p)(x+q)\) where \(
Part (b)
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
Shape \(\wedge\vee\) (positive cubic)
B1
"Sharp points" lose this B1
Through origin (not touching)
B1
Curve must not start/terminate at \((0,0)\)
Touching \(x\)-axis only once
B1
Touching at \((3,0)\), or \(3\) on \(x\)-axis
B1ft
For curve touching (not crossing) at \((a,0)\) where \(y=x(x-a)^2\)
Part (c)
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
Graph moved horizontally with \((2,0)\) and \((5,0)\) marked, or \(2\) and \(5\) on \(x\)-axis
M1, A1
M1 for horizontal translation only; A1 for translation 2 to right crossing/touching at \(2\) and \(5\) only; fully correct graph from (b) scores M1A1 regardless
# Question 10:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x(x^2-6x+9) = x(x-3)(x-3)$ | B1, M1 A1 | B1 for taking out factor $x$; M1 for attempt to factorize TQ $(x+p)(x+q)$ where $|pq|=9$; A1 for fully correct: accept $x(x-3)^2$ |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Shape $\wedge\vee$ (positive cubic) | B1 | "Sharp points" lose this B1 |
| Through origin (not touching) | B1 | Curve must not start/terminate at $(0,0)$ |
| Touching $x$-axis only once | B1 | |
| Touching at $(3,0)$, or $3$ on $x$-axis | B1ft | For curve touching (not crossing) at $(a,0)$ where $y=x(x-a)^2$ |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Graph moved horizontally with $(2,0)$ and $(5,0)$ marked, or $2$ and $5$ on $x$-axis | M1, A1 | M1 for horizontal translation only; A1 for translation 2 to right crossing/touching at $2$ and $5$ only; fully correct graph from (b) scores M1A1 regardless |
10. (a) Factorise completely $x ^ { 3 } - 6 x ^ { 2 } + 9 x$\\
(b) Sketch the curve with equation
$$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$$
showing the coordinates of the points at which the curve meets the $x$-axis.
Using your answer to part (b), or otherwise,\\
(c) sketch, on a separate diagram, the curve with equation
$$y = ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 9 ( x - 2 )$$
showing the coordinates of the points at which the curve meets the $x$-axis.\\
\hfill \mbox{\textit{Edexcel C1 2009 Q10 [9]}}