| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2019 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | State domain or range |
| Difficulty | Standard +0.8 This question requires completing the square to find the range of a quadratic with a restricted domain, then composing functions and solving a resulting quadratic equation. The multi-step nature, algebraic manipulation with parameters, and need to verify which solutions are valid (k positive) make this moderately challenging, though the techniques are standard C3/C4 material. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f\!\left(-\frac{3k}{4}\right) = ...\) or \(f(-4k) = ...\) | M1 | Attempts \(f\!\left(-\frac{3k}{4}\right)\) or \(f(-4k)\) |
| \(y_{\min} = -\frac{k^2}{8}\) or \(y > -\frac{k^2}{8}\) or \(y \geq -\frac{k^2}{8}\) or \(y_{\max} = 21k^2\) or \(y < 21k^2\) or \(y \leq 21k^2\) | A1 | One correct "end" of the range. May be implied by their final answer. Allow strict and non-strict inequality symbols |
| \(f\!\left(-\frac{3k}{4}\right) = ...\) and \(f(-4k) = ...\) | M1 | Attempts \(f\!\left(-\frac{3k}{4}\right)\) and \(f(-4k)\) |
| \(-\frac{k^2}{8} \leq f(x) \leq 21k^2\) | A1 | Correct range. Allow alternative notation. Allow \(y\) or "range" for \(f(x)\) but do not allow \(x\) for \(f(x)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{gf}(-2) = 2k - 3\!\left(2(-2)^2 + 3k(-2) + k^2\right)\) or \(\text{gf}(x) = 2k - 3\!\left(2x^2 + 3kx + k^2\right)\) | B1 | Correct expression for \(\text{gf}(-2)\) or \(\text{gf}(x)\). Award as soon as correct expression seen |
| \(2k - 3\!\left(2(-2)^2 + 3k(-2) + k^2\right) = -12\) | M1 | Puts their \(\text{gf}(-2) = \pm 12\) to obtain equation in \(k\) only. Must be using \(x = -2\) |
| \(3k^2 - 20k + 12 = 0\) | dM1 | Solves a 3TQ. Dependent on previous M |
| \(\Rightarrow (3k-2)(k-6) = 0 \Rightarrow k = \frac{2}{3},\ 6\) | A1 | Correct values. Allow equivalent fractions for \(\frac{2}{3}\) or \(0.\dot{6}\) |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f\!\left(-\frac{3k}{4}\right) = ...$ or $f(-4k) = ...$ | M1 | Attempts $f\!\left(-\frac{3k}{4}\right)$ **or** $f(-4k)$ |
| $y_{\min} = -\frac{k^2}{8}$ or $y > -\frac{k^2}{8}$ or $y \geq -\frac{k^2}{8}$ **or** $y_{\max} = 21k^2$ or $y < 21k^2$ or $y \leq 21k^2$ | A1 | One correct "end" of the range. May be implied by their final answer. Allow strict and non-strict inequality symbols |
| $f\!\left(-\frac{3k}{4}\right) = ...$ **and** $f(-4k) = ...$ | M1 | Attempts $f\!\left(-\frac{3k}{4}\right)$ **and** $f(-4k)$ |
| $-\frac{k^2}{8} \leq f(x) \leq 21k^2$ | A1 | Correct range. Allow alternative notation. Allow $y$ or "range" for $f(x)$ but do not allow $x$ for $f(x)$ |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{gf}(-2) = 2k - 3\!\left(2(-2)^2 + 3k(-2) + k^2\right)$ **or** $\text{gf}(x) = 2k - 3\!\left(2x^2 + 3kx + k^2\right)$ | B1 | Correct expression for $\text{gf}(-2)$ or $\text{gf}(x)$. Award as soon as correct expression seen |
| $2k - 3\!\left(2(-2)^2 + 3k(-2) + k^2\right) = -12$ | M1 | Puts their $\text{gf}(-2) = \pm 12$ to obtain equation in $k$ only. Must be using $x = -2$ |
| $3k^2 - 20k + 12 = 0$ | dM1 | Solves a 3TQ. Dependent on previous M |
| $\Rightarrow (3k-2)(k-6) = 0 \Rightarrow k = \frac{2}{3},\ 6$ | A1 | Correct values. Allow equivalent fractions for $\frac{2}{3}$ or $0.\dot{6}$ |
---3. The function f is defined by
$$f : x \mapsto 2 x ^ { 2 } + 3 k x + k ^ { 2 } \quad x \in \mathbb { R } , - 4 k \leqslant x \leqslant 0$$
where $k$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $k$, the range of f .
The function g is defined by
$$\mathrm { g } : x \mapsto 2 k - 3 x \quad x \in \mathbb { R }$$
Given that $\operatorname { gf } ( - 2 ) = - 12$
\item find the possible values of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2019 Q3 [8]}}