| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Quadratic in exponential form |
| Difficulty | Moderate -0.3 This is a standard C2 exponential equation requiring substitution y=3^x to form a quadratic, then solving and converting back. The algebraic manipulation is straightforward with clear guidance from part (a), though students must handle the quadratic and logarithms correctly. Slightly easier than average due to the scaffolding provided. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks |
|---|---|
| (i) \(= 3^y \times 3^y = 3y\) | M1 A1 |
| (ii) \(= 3^{-y} \times (3^y)^2 = \frac{1}{3}y^2\) | M1 A1 |
| Answer | Marks |
|---|---|
| \(3y - \frac{1}{3}y^2 = 6\) | |
| \(y^2 - 9y + 18 = 0\) | M1 |
| \((y - 3)(y - 6) = 0\) | |
| \(y = 3, 6\) | A1 |
| \(3^x = 3, 6\) | |
| \(x = 1, \frac{\lg 6}{\lg 3}\) | B1 M1 |
| \(x = 1, 1.63 \text{ (2dp)}\) | A1 |
**(a)**
(i) $= 3^y \times 3^y = 3y$ | M1 A1 |
(ii) $= 3^{-y} \times (3^y)^2 = \frac{1}{3}y^2$ | M1 A1 |
**(b)**
$3y - \frac{1}{3}y^2 = 6$ | |
$y^2 - 9y + 18 = 0$ | M1 |
$(y - 3)(y - 6) = 0$ | |
$y = 3, 6$ | A1 |
$3^x = 3, 6$ | |
$x = 1, \frac{\lg 6}{\lg 3}$ | B1 M1 |
$x = 1, 1.63 \text{ (2dp)}$ | A1 |\begin{enumerate}
\item (a) Given that $y = 3 ^ { x }$, find expressions in terms of $y$ for\\
(i) $3 ^ { x + 1 }$,\\
(ii) $3 ^ { 2 x - 1 }$.\\
(b) Hence, or otherwise, solve the equation
\end{enumerate}
$$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$
giving non-exact answers to 2 decimal places.\\
\hfill \mbox{\textit{Edexcel C2 Q6 [9]}}