| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2018 |
| Session | October |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Standard +0.3 This is a standard C3/C4 composite and rational function question with multiple routine parts: finding range/intercepts (basic substitution), computing gg(x) (algebraic manipulation of rational functions), sketching |g(x)| (reflection of negative parts), and solving |g(x)|=8 (two cases). While it has many parts, each requires only standard techniques without novel insight, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02l Modulus function: notation, relations, equations and inequalities1.02t Solve modulus equations: graphically with modulus function1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y < 3\) | B1 | Accept \(y<3\), \(g(x)<3\), \(g<3\), \(-\infty < y < 3\), \((-\infty, 3)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left(\frac{4}{3}, 0\right)\) | B1 | Allow \(x=\frac{4}{3}\), \(y=0\) or \(x=\frac{4}{3}\), \(g(x)=0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left(0, \frac{4}{3}\right)\) | B1 | Allow \(x=0\), \(y=\frac{4}{3}\) or \(x=0\), \(g(x)=\frac{4}{3}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts \(gg(x) = \frac{3\times\frac{3x-4}{x-3}-4}{\frac{3x-4}{x-3}-3}\) | M1 | Attempt to substitute \(g\) into \(g\) |
| Multiplies numerator and denominator by \((x-3)\) | dM1 | To form fraction \(\frac{ax+b}{cx+d}\); condone poor bracketing |
| \(gg(x) = \frac{3(3x-4)-4(x-3)}{3x-4-3(x-3)} = \frac{5x}{5} = x\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct shape with cusp (not minimum) at \(A\); same asymptotes as original curve | B1 | Curve must have correct curvature on rhs, not bend back on itself |
| Intersects \(y\)-axis at \(\left(0,\frac{4}{3}\right)\), meets \(x\)-axis at \(\left(\frac{4}{3},0\right)\) | B1ft | Follow through from (b) |
| Asymptotes at \(x=3\) and \(y=3\) | B1 | Both equations required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{3x-4}{x-3} = -8\) leading to solution of \(\ | g(x)\ | =8\) |
| Solves by cross-multiplying, collecting terms | dM1 | Do not allow from \(-\frac{3x-4}{x-3}=8\Rightarrow\frac{-3x+4}{-x+3}=8\) (scores M1 M0) |
| \(3x-4=-8x+24 \Rightarrow x=\frac{28}{11}\) | A1 | oee only; if both \(x=4\) and \(x=\frac{28}{11}\) found, mark scored only when \(x=4\) deleted or \(\frac{28}{11}\) chosen as answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left(\frac{3x-4}{x-3}\right)^2 = 64\) | M1 | |
| \(\Rightarrow Ax^2+Bx+c=0\); correct quadratic \(55x^2-360x+560=0\Rightarrow(11x-28)(x-4)=0\) | dM1 | Solves by usual methods |
| Selects \(x=\frac{28}{11}\) | A1 |
# Question 10:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y < 3$ | B1 | Accept $y<3$, $g(x)<3$, $g<3$, $-\infty < y < 3$, $(-\infty, 3)$ |
## Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(\frac{4}{3}, 0\right)$ | B1 | Allow $x=\frac{4}{3}$, $y=0$ or $x=\frac{4}{3}$, $g(x)=0$ |
## Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(0, \frac{4}{3}\right)$ | B1 | Allow $x=0$, $y=\frac{4}{3}$ or $x=0$, $g(x)=\frac{4}{3}$ |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts $gg(x) = \frac{3\times\frac{3x-4}{x-3}-4}{\frac{3x-4}{x-3}-3}$ | M1 | Attempt to substitute $g$ into $g$ |
| Multiplies numerator and denominator by $(x-3)$ | dM1 | To form fraction $\frac{ax+b}{cx+d}$; condone poor bracketing |
| $gg(x) = \frac{3(3x-4)-4(x-3)}{3x-4-3(x-3)} = \frac{5x}{5} = x$ | A1 | |
## Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct shape with cusp (not minimum) at $A$; same asymptotes as original curve | B1 | Curve must have correct curvature on rhs, not bend back on itself |
| Intersects $y$-axis at $\left(0,\frac{4}{3}\right)$, meets $x$-axis at $\left(\frac{4}{3},0\right)$ | B1ft | Follow through from (b) |
| Asymptotes at $x=3$ and $y=3$ | B1 | Both equations required |
## Part (e):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{3x-4}{x-3} = -8$ leading to solution of $\|g(x)\|=8$ | M1 | Allow $-\frac{3x-4}{x-3}=8$, $\frac{-3x+4}{x-3}=8$, $\frac{3x-4}{3-x}=8$, $\left(\frac{3x-4}{x-3}\right)^2=64$ |
| Solves by cross-multiplying, collecting terms | dM1 | Do not allow from $-\frac{3x-4}{x-3}=8\Rightarrow\frac{-3x+4}{-x+3}=8$ (scores M1 M0) |
| $3x-4=-8x+24 \Rightarrow x=\frac{28}{11}$ | A1 | **oee only**; if both $x=4$ and $x=\frac{28}{11}$ found, mark scored only when $x=4$ deleted or $\frac{28}{11}$ chosen as answer |
**Solution from squaring:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(\frac{3x-4}{x-3}\right)^2 = 64$ | M1 | |
| $\Rightarrow Ax^2+Bx+c=0$; correct quadratic $55x^2-360x+560=0\Rightarrow(11x-28)(x-4)=0$ | dM1 | Solves by usual methods |
| Selects $x=\frac{28}{11}$ | A1 | |
---10.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c6bde466-61ec-437d-a3b4-84511a98d788-32_492_636_260_660}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of part of the graph with equation $y = \mathrm { g } ( x )$, where
$$\mathrm { g } ( x ) = \frac { 3 x - 4 } { x - 3 } , \quad x \in \mathbb { R } , \quad x < 3$$
The graph cuts the $x$-axis at the point $A$ and the $y$-axis at the point $B$, as shown in Figure 2 .
\begin{enumerate}[label=(\alph*)]
\item State the range of g .
\item State the coordinates of
\begin{enumerate}[label=(\roman*)]
\item point $A$
\item point $B$
\end{enumerate}\item Find $\operatorname { gg } ( x )$ in its simplest form.
\item Sketch the graph with equation $y = | \mathrm { g } ( x ) |$
On your sketch, show the coordinates of each point at which the graph meets or cuts the axes and state the equation of each asymptote.
\item Find the exact solution of the equation $| \mathrm { g } ( x ) | = 8$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2018 Q10 [12]}}