| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Solve p(exponential) = 0 |
| Difficulty | Standard +0.3 This is a standard C2 factor theorem question with routine steps: substitute x=3 to find a constant, factorise a cubic (likely by inspection or division), then solve an exponential equation using substitution. Part (c) adds mild complexity by requiring the connection that g(y) = f(3^y), but this is a common exam pattern. Slightly above average due to the three-part structure and the exponential twist, but all techniques are standard C2 material with no novel insight required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks |
|---|---|
| \(3\) | \(-3.2376\) |
| \(3.1\) | \(-2.88739\) |
| \(3.2\) | \(-2.54427\) |
| \(3.3\) | \(-2.20771\) |
| \(3.4\) | \(-1.87722\) |
| \(3.5\) | \(-1.55236\) |
| \(3.6\) | \(-1.23274\) |
| \(3.7\) | \(-0.918\) |
| \(3.8\) | \(-0.60783\) |
| \(3.9\) | \(-0.30191\) |
Question 3:
$3$ | $-3.2376$
$3.1$ | $-2.88739$
$3.2$ | $-2.54427$
$3.3$ | $-2.20771$
$3.4$ | $-1.87722$
$3.5$ | $-1.55236$
$3.6$ | $-1.23274$
$3.7$ | $-0.918$
$3.8$ | $-0.60783$
$3.9$ | $-0.30191$3.
$$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + a x + 18$$
where $a$ is a constant.
Given that $( x - 3 )$ is a factor of $\mathrm { f } ( x )$,
\begin{enumerate}[label=(\alph*)]
\item show that $a = - 9$
\item factorise $\mathrm { f } ( x )$ completely.
Given that
$$\mathrm { g } ( y ) = 2 \left( 3 ^ { 3 y } \right) - 5 \left( 3 ^ { 2 y } \right) - 9 \left( 3 ^ { y } \right) + 18$$
\item find the values of $y$ that satisfy $\mathrm { g } ( y ) = 0$, giving your answers to 2 decimal places where appropriate.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2013 Q3 [9]}}