| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Find unknown coefficients from roots |
| Difficulty | Standard +0.3 This is a straightforward C2 question involving factor theorem and solving a cubic equation. Part (a) requires simple substitution into the equation (routine 2-mark work). Part (b) involves factoring out (x+4) and solving the resulting quadratic, which are standard techniques. The question is slightly easier than average because the method is clear and the algebra is manageable. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks |
|---|---|
| (a) \((-4, 0) \therefore 0 = 4 - 20 + 16k + 128\) | M1 |
| \(16k = -112, k = -7\) | A1 |
| Answer | Marks |
|---|---|
| \(x = -4\) is a solution \(\therefore (x+4)\) is a factor | B1 |
| Answer | Marks |
|---|---|
| \[\begin{array}{c | cc} |
| Answer | Marks |
|---|---|
| \end{array}\] | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((x+4)(1+2x)(1-x) = 0\) | M1 | |
| \(x = -4\) (at A), \(-\frac{1}{2}, 1\) | A1 |
**(a)** $(-4, 0) \therefore 0 = 4 - 20 + 16k + 128$ | M1 |
$16k = -112, k = -7$ | A1 |
**(b)** $4 + 5x - 7x^2 - 2x^3 = 0$
$x = -4$ is a solution $\therefore (x+4)$ is a factor | B1 |
$x + 4 \mid 2x^3 - 7x^2 + 5x + 4$
$$\begin{array}{c|cc}
& -2x^2 & +x & +1 \\
\hline
-2x^3 & -8x^2 & & \\
& x^2 & +5x & \\
& & x^2 & +4x \\
& & & x+4 \\
& & & x+4
\end{array}$$ | M1 A1 |
$\therefore (x+4)(1+x-2x^2) = 0$
$(x+4)(1+2x)(1-x) = 0$ | M1 |
$x = -4$ (at A), $-\frac{1}{2}, 1$ | A1 | | (7 marks)\includegraphics{figure_1}
Figure 1 shows the curve $y = f(x)$ where
$$f(x) = 4 + 5x + kx^2 - 2x^3,$$
and $k$ is a constant.
The curve crosses the $x$-axis at the points $A$, $B$ and $C$.
Given that $A$ has coordinates $(-4, 0)$,
\begin{enumerate}[label=(\alph*)]
\item show that $k = -7$, [2]
\item find the coordinates of $B$ and $C$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q3 [7]}}