| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Solve exponential equation by substitution |
| Difficulty | Moderate -0.8 This is a structured, multi-part C1 question that guides students through each step of solving an exponential equation via substitution. Part (a) tests basic index law manipulation, part (b) is a 'show that' requiring straightforward algebraic substitution, and part (c) involves solving a quadratic then taking logarithms. The scaffolding makes this easier than average, requiring only routine application of standard techniques with no problem-solving insight needed. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) \(2^{x+2} = 2^x \times 2^1 = 4y\) | M1 A1 | |
| (ii) \(2^{1-x} = \frac{2^1}{2^x} = \frac{8}{y}\) | M1 A1 | |
| (b) \(2^{x+2} + 2^{3-x} = 33 \Rightarrow 4y + \frac{8}{y} = 33\) | ||
| \(4y^2 + 8 = 33y\) | M1 | |
| \(4y^2 - 33y + 8 = 0\) | A1 | |
| (c) \((4y - 1)(y - 8) = 0\) | M1 | |
| \(y = \frac{1}{4}, 8\) | A1 | |
| \(2^x = \frac{1}{4}, 8\) | ||
| \(x = -2, 3\) | A2 | (10 marks) |
**(a)** (i) $2^{x+2} = 2^x \times 2^1 = 4y$ | M1 A1 |
(ii) $2^{1-x} = \frac{2^1}{2^x} = \frac{8}{y}$ | M1 A1 |
**(b)** $2^{x+2} + 2^{3-x} = 33 \Rightarrow 4y + \frac{8}{y} = 33$ | |
$4y^2 + 8 = 33y$ | M1 |
$4y^2 - 33y + 8 = 0$ | A1 |
**(c)** $(4y - 1)(y - 8) = 0$ | M1 |
$y = \frac{1}{4}, 8$ | A1 |
$2^x = \frac{1}{4}, 8$ | |
$x = -2, 3$ | A2 | (10 marks)7. (a) Given that $y = 2 ^ { x }$, find expressions in terms of $y$ for
\begin{enumerate}[label=(\roman*)]
\item $2 ^ { x + 2 }$,
\item $2 ^ { 3 - x }$.\\
(b) Show that using the substitution $y = 2 ^ { x }$, the equation
$$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$
can be rewritten as
$$4 y ^ { 2 } - 33 y + 8 = 0$$
(c) Hence solve the equation
$$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q7 [10]}}