| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2017 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Polynomial intersection with algebra |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard curve sketching techniques (finding intercepts), equating two functions algebraically, and solving a cubic equation with a given root. All steps are routine C1/C2 techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sketch showing a curve with maximum on \(y\)-axis, passing through \((-c, 0)\), \((c, 0)\) and \((0, c^2)\) | B1 | Shape must be smooth at maximum, not pointed. Branches must not turn back on themselves. Allow \((0,-c)\), \((0,c)\) and \((c^2, 0)\) if marked in correct places. Sketch takes precedence over labels. |
| Fully correct diagram: maximum on \(y\)-axis, branches extend below \(x\)-axis, correct intercepts | B1 | Intercepts must be correctly labelled and curve in correct position |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Positive cubic shape with one maximum and one minimum | B1 | Curve must be smooth at maximum and minimum (not pointed) |
| Curve touches or meets \(x\)-axis at origin and at \((3c, 0)\) with no other intersections | B1 | Origin need not be marked but \((3c,0)\) must be. Allow \(3c\) or \((0,3c)\) marked in correct place. Sketch takes precedence. |
| Maximum at the origin | B1 | Allow maximum to form a point or cusp |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^2(x-3c) = c^2 - x^2 \Rightarrow x^3 - 3cx^2 = c^2 - x^2\) | M1 | Sets equations equal and attempts to multiply out bracket or vice versa |
| \(x^3 + x^2 - 3cx^2 - c^2 = 0\) \(\Rightarrow x^3 + (1-3c)x^2 - c^2 = 0\)* | A1* | Collects to one side, factorises \(x^2\) terms. Must have intermediate line of working. Allow \(x^3 + x^2(1-3c) - c^2 = 0\) or \(0 = x^3 + (1-3c)x^2 - c^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(8 + 4(1-3c) - c^2 = 0\) | M1 | Substitutes \(x=2\) to give a correct unsimplified form |
| \(c^2 + 12c - 12 = 0\) | A1 | Correct 3-term quadratic. Allow equivalent forms (may be implied) |
| \((c+6)^2 - 36 - 12 = 0 \Rightarrow c = \ldots\) or \(c = \dfrac{-12 \pm \sqrt{12^2 - 4\times1\times(-12)}}{2}\) | M1 | Solves their 3TQ using formula or completing the square only. Implied by correct exact answer for their 3TQ |
| \(4\sqrt{3} - 6\) | A1 | \(c = 4\sqrt{3}-6\) or \(c = -6+4\sqrt{3}\) only |
## Question 13:
### Part (a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sketch showing a curve with maximum on $y$-axis, passing through $(-c, 0)$, $(c, 0)$ and $(0, c^2)$ | B1 | Shape must be smooth at maximum, not pointed. Branches must not turn back on themselves. Allow $(0,-c)$, $(0,c)$ and $(c^2, 0)$ if marked in correct places. Sketch takes precedence over labels. |
| Fully correct diagram: maximum on $y$-axis, branches extend below $x$-axis, correct intercepts | B1 | Intercepts must be correctly labelled and curve in correct position |
### Part (a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Positive cubic shape with one maximum and one minimum | B1 | Curve must be smooth at maximum and minimum (not pointed) |
| Curve touches or meets $x$-axis at origin and at $(3c, 0)$ with no other intersections | B1 | Origin need not be marked but $(3c,0)$ must be. Allow $3c$ or $(0,3c)$ marked in correct place. Sketch takes precedence. |
| Maximum at the origin | B1 | Allow maximum to form a point or cusp |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2(x-3c) = c^2 - x^2 \Rightarrow x^3 - 3cx^2 = c^2 - x^2$ | M1 | Sets equations equal and attempts to multiply out bracket or vice versa |
| $x^3 + x^2 - 3cx^2 - c^2 = 0$ $\Rightarrow x^3 + (1-3c)x^2 - c^2 = 0$* | A1* | Collects to one side, factorises $x^2$ terms. **Must have intermediate line of working.** Allow $x^3 + x^2(1-3c) - c^2 = 0$ or $0 = x^3 + (1-3c)x^2 - c^2$ |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $8 + 4(1-3c) - c^2 = 0$ | M1 | Substitutes $x=2$ to give a **correct** unsimplified form |
| $c^2 + 12c - 12 = 0$ | A1 | Correct 3-term quadratic. Allow equivalent forms (may be implied) |
| $(c+6)^2 - 36 - 12 = 0 \Rightarrow c = \ldots$ or $c = \dfrac{-12 \pm \sqrt{12^2 - 4\times1\times(-12)}}{2}$ | M1 | Solves their 3TQ using formula or completing the square **only**. Implied by correct exact answer for their 3TQ |
| $4\sqrt{3} - 6$ | A1 | $c = 4\sqrt{3}-6$ or $c = -6+4\sqrt{3}$ **only** |
---13. (a) On separate axes sketch the graphs of
\begin{enumerate}[label=(\roman*)]
\item $y = c ^ { 2 } - x ^ { 2 }$
\item $y = x ^ { 2 } ( x - 3 c )$\\
where $c$ is a positive constant.\\
Show clearly the coordinates of the points where each graph crosses or meets the $x$-axis and the $y$-axis.\\
(b) Prove that the $x$ coordinate of any point of intersection of
$$y = c ^ { 2 } - x ^ { 2 } \text { and } y = x ^ { 2 } ( x - 3 c )$$
where $c$ is a positive constant, is given by a solution of the equation
$$x ^ { 3 } + ( 1 - 3 c ) x ^ { 2 } - c ^ { 2 } = 0$$
Given that the graphs meet when $x = 2$\\
(c) find the exact value of $c$, writing your answer as a fully simplified surd.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2017 Q13 [11]}}