Question 4 - A-Level Maths

Edexcel C1 2014 June — Question 4 5 marks

Exam Board	Edexcel
Module	C1 (Core Mathematics 1)
Year	2014
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Curve Sketching
Type	Polynomial with line intersection
Difficulty	Moderate -0.8 This is a straightforward C1 curve sketching question requiring basic skills: finding x-intercept of a reciprocal function, sketching a cubic with given roots, and counting intersections graphically. All parts involve routine procedures with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part nature.
Spec	1.02m Graphs of functions: difference between plotting and sketching 1.02n Sketch curves: simple equations including polynomials 1.02o Sketch reciprocal curves: y=a/x and y=a/x^2 1.02q Use intersection points: of graphs to solve equations

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-05_945_1026_269_466} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ with equation $$y = \frac { 1 } { x } + 1 , \quad x \neq 0$$ The curve $C$ crosses the $x$-axis at the point $A$.

State the $x$ coordinate of the point $A$. The curve $D$ has equation $y = x ^ { 2 } ( x - 2 )$, for all real values of $x$.
A copy of Figure 1 is shown on page 7. On this copy, sketch a graph of curve $D$.
Show on the sketch the coordinates of each point where the curve $D$ crosses the coordinate axes.
Using your sketch, state, giving a reason, the number of real solutions to the equation $$x ^ { 2 } ( x - 2 ) = \frac { 1 } { x } + 1$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-06_942_1026_516_466} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}

Show mark scheme Show mark scheme source

Answer	Marks	Guidance
(a) $-1$ accept $(-1, 0)$	B1
(1 mark)
(b)	Shape: The curve must have two clear turning points and be a $+ve$ $x^4$ curve (with a maximum and minimum)	B1
Touches at (0,0): The graph touches the origin. Accept touching as a maximum or minimum. There must be a sketch for this mark but sketch may be wrong and this mark is independent of previous marks. Origin is where axes cross and may not be labelled. This may be a quadratic or quartic curve for this mark.	B1
Crosses at (2,0) only: The graph crosses the $x$-axis at the point $(2,0)$ only. If it crosses at $(2,0)$ and $(0,0)$ this is B0. Accept $(0,2)$ or $2$ marked on the correct axis. Accept $(2, 0)$ in the text of the answer provided that the curve crosses the positive $x$ axis. There must be a sketch for this mark. Do not give credit if $(2,0)$ appears only in a table with no indication that this is the intersection point. (If in doubt send to review). Graph takes precedence over text for third B mark.	B1
(3 marks)
(c) $2$ solutions as curves cross twice	B1 ft
(1 mark)
Notes:
N.B.	Check original diagram as answer may appear there.
(a)	B1: The $x$ coordinate of $A$ is $-1$. Accept $-1$ or $(-1,0)$ on the diagram or stated with or without diagram. Allow $(0, -1)$ on the diagram if it is on the correct axis.
(b)	If no graph is drawn then no marks are available in part (b).
B1: Correct shape. The position is not important for this mark but the curve must have two clear turning points and be a $+ve$ $x^4$ curve (with a maximum and minimum). There must be a sketch for this mark but sketch may be wrong and this mark is independent of previous marks. Origin is where axes cross and may not be labelled. This may be a quadratic or quartic curve for this mark.
B1: The graph touches the origin. Accept touching as a maximum or minimum. There must be a sketch for this mark but sketch may be wrong and this mark is independent of previous marks. Origin is where axes cross and may not be labelled. This may be a quadratic or quartic curve for this mark.
B1: The graph crosses the $x$-axis at the point $(2,0)$ only. If it crosses at $(2,0)$ and $(0,0)$ this is B0. Accept $(0,2)$ or $2$ marked on the correct axis. Accept $(2, 0)$ in the text of the answer provided that the curve crosses the positive $x$ axis. There must be a sketch for this mark. Do not give credit if $(2,0)$ appears only in a table with no indication that this is the intersection point. (If in doubt send to review). Graph takes precedence over text for third B mark.
(c)	B1ft: Two (solutions) as there are two intersections (of the curves). N.B. Just states $2$ with no reason is B0. If the answer states $2$ roots and two intersections – or crosses twice this is enough for B1. BUT B0 if there is any wrong reason given – e.g. crosses $x$ axis twice, or crosses asymptote twice. Isw – is not used for this mark so any wrong statement listed to follow a correct statement will result in B0. Allow ft – so if their graph crosses the hyperbola once – allow "one solution as there is one intersection". And if it crosses three times – allow "three solutions as there are three intersections" or four etc. However in (c) if they have sketched a curve (even a fully correct one) but not extended it to intersect the hyperbola and they put "no points of intersection so no solutions" then this scores B0. Allow "lines or curves cross over twice, or touch twice, or meet twice…etc as explanation, but need some form of words).

**(a)** $-1$ accept $(-1, 0)$ | B1 |

| | **(1 mark)** |

**(b)** | **Shape:** The curve must have two clear turning points and be a $+ve$ $x^4$ curve (with a maximum and minimum) | B1 |
| | **Touches at (0,0):** The graph touches the origin. Accept touching as a maximum or minimum. There must be a sketch for this mark but sketch may be wrong and this mark is independent of previous marks. Origin is where axes cross and may not be labelled. This may be a quadratic or quartic curve for this mark. | B1 |
| | **Crosses at (2,0) only:** The graph crosses the $x$-axis at the point $(2,0)$ only. If it crosses at $(2,0)$ and $(0,0)$ this is B0. Accept $(0,2)$ or $2$ marked on the correct axis. Accept $(2, 0)$ in the text of the answer provided that the curve crosses the positive $x$ axis. There must be a sketch for this mark. Do not give credit if $(2,0)$ appears only in a table with no indication that this is the intersection point. (If in doubt send to review). Graph takes precedence over text for third B mark. | B1 |

| | **(3 marks)** |

**(c)** $2$ solutions as **curves cross twice** | B1 ft |

| | **(1 mark)** |

| **Notes:** | |
|---|---|
| **N.B.** | Check original diagram as answer may appear there. |
| **(a)** | B1: The $x$ coordinate of $A$ is $-1$. Accept $-1$ or $(-1,0)$ on the diagram or stated with or without diagram. Allow $(0, -1)$ on the diagram if it is on the correct axis. |
| **(b)** | **If no graph is drawn then no marks are available in part (b).** |
| | B1: Correct shape. The position is not important for this mark but the curve must have two clear turning points and be a $+ve$ $x^4$ curve (with a maximum and minimum). There must be a sketch for this mark but sketch may be wrong and this mark is independent of previous marks. Origin is where axes cross and may not be labelled. This may be a quadratic or quartic curve for this mark. |
| | B1: The graph touches the origin. Accept touching as a maximum or minimum. There must be a sketch for this mark but sketch may be wrong and this mark is independent of previous marks. Origin is where axes cross and may not be labelled. This may be a quadratic or quartic curve for this mark. |
| | B1: The graph crosses the $x$-axis at the point $(2,0)$ only. If it crosses at $(2,0)$ and $(0,0)$ this is B0. Accept $(0,2)$ or $2$ marked on the correct axis. Accept $(2, 0)$ in the text of the answer provided that the curve crosses the positive $x$ axis. There must be a sketch for this mark. Do not give credit if $(2,0)$ appears only in a table with no indication that this is the intersection point. (If in doubt send to review). Graph takes precedence over text for third B mark. |
| **(c)** | B1ft: Two (solutions) as there are two intersections (of the curves). N.B. Just states $2$ with no reason is B0. If the answer states $2$ roots and two intersections – or crosses twice this is enough for B1. BUT B0 if there is any wrong reason given – e.g. crosses $x$ axis twice, or crosses asymptote twice. Isw – is not used for this mark so any wrong statement listed to follow a correct statement will result in B0. Allow ft – so if their graph crosses the hyperbola once – allow "one solution as there is one intersection". And if it crosses three times – allow "three solutions as there are three intersections" or four etc. However in (c) if they have sketched a curve (even a fully correct one) but not extended it to intersect the hyperbola and they put "no points of intersection so no solutions" then this scores B0. Allow "lines or curves cross over twice, or touch twice, or meet twice…etc as explanation, but need some form of words). |

---

Show LaTeX source

4.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-05_945_1026_269_466}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}

Figure 1 shows a sketch of the curve $C$ with equation

$$y = \frac { 1 } { x } + 1 , \quad x \neq 0$$

The curve $C$ crosses the $x$-axis at the point $A$.
\begin{enumerate}[label=(\alph*)]
\item State the $x$ coordinate of the point $A$.

The curve $D$ has equation $y = x ^ { 2 } ( x - 2 )$, for all real values of $x$.
\item A copy of Figure 1 is shown on page 7.

On this copy, sketch a graph of curve $D$.\\
Show on the sketch the coordinates of each point where the curve $D$ crosses the coordinate axes.
\item Using your sketch, state, giving a reason, the number of real solutions to the equation

$$x ^ { 2 } ( x - 2 ) = \frac { 1 } { x } + 1$$

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-06_942_1026_516_466}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\end{enumerate}

\hfill \mbox{\textit{Edexcel C1 2014 Q4 [5]}}

This paper (11 questions)

View full paper

Q1 3 Q2 4 Q3 7 Q4 5 Q5 5 Q6 5 Q7 7 Q8 9 Q9 10 Q10 10 Q11 10

Answer	Marks	Guidance
(a) \(-1\) accept \((-1, 0)\)	B1
(1 mark)
(b)	Shape: The curve must have two clear turning points and be a \(+ve\) \(x^4\) curve (with a maximum and minimum)	B1
Touches at (0,0): The graph touches the origin. Accept touching as a maximum or minimum. There must be a sketch for this mark but sketch may be wrong and this mark is independent of previous marks. Origin is where axes cross and may not be labelled. This may be a quadratic or quartic curve for this mark.	B1
Crosses at (2,0) only: The graph crosses the \(x\)-axis at the point \((2,0)\) only. If it crosses at \((2,0)\) and \((0,0)\) this is B0. Accept \((0,2)\) or \(2\) marked on the correct axis. Accept \((2, 0)\) in the text of the answer provided that the curve crosses the positive \(x\) axis. There must be a sketch for this mark. Do not give credit if \((2,0)\) appears only in a table with no indication that this is the intersection point. (If in doubt send to review). Graph takes precedence over text for third B mark.	B1
(3 marks)
(c) \(2\) solutions as curves cross twice	B1 ft
(1 mark)
Notes:
N.B.	Check original diagram as answer may appear there.
(a)	B1: The \(x\) coordinate of \(A\) is \(-1\). Accept \(-1\) or \((-1,0)\) on the diagram or stated with or without diagram. Allow \((0, -1)\) on the diagram if it is on the correct axis.
(b)	If no graph is drawn then no marks are available in part (b).
B1: Correct shape. The position is not important for this mark but the curve must have two clear turning points and be a \(+ve\) \(x^4\) curve (with a maximum and minimum). There must be a sketch for this mark but sketch may be wrong and this mark is independent of previous marks. Origin is where axes cross and may not be labelled. This may be a quadratic or quartic curve for this mark.
B1: The graph touches the origin. Accept touching as a maximum or minimum. There must be a sketch for this mark but sketch may be wrong and this mark is independent of previous marks. Origin is where axes cross and may not be labelled. This may be a quadratic or quartic curve for this mark.
B1: The graph crosses the \(x\)-axis at the point \((2,0)\) only. If it crosses at \((2,0)\) and \((0,0)\) this is B0. Accept \((0,2)\) or \(2\) marked on the correct axis. Accept \((2, 0)\) in the text of the answer provided that the curve crosses the positive \(x\) axis. There must be a sketch for this mark. Do not give credit if \((2,0)\) appears only in a table with no indication that this is the intersection point. (If in doubt send to review). Graph takes precedence over text for third B mark.
(c)	B1ft: Two (solutions) as there are two intersections (of the curves). N.B. Just states \(2\) with no reason is B0. If the answer states \(2\) roots and two intersections – or crosses twice this is enough for B1. BUT B0 if there is any wrong reason given – e.g. crosses \(x\) axis twice, or crosses asymptote twice. Isw – is not used for this mark so any wrong statement listed to follow a correct statement will result in B0. Allow ft – so if their graph crosses the hyperbola once – allow "one solution as there is one intersection". And if it crosses three times – allow "three solutions as there are three intersections" or four etc. However in (c) if they have sketched a curve (even a fully correct one) but not extended it to intersect the hyperbola and they put "no points of intersection so no solutions" then this scores B0. Allow "lines or curves cross over twice, or touch twice, or meet twice…etc as explanation, but need some form of words).