| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Tangent to exponential curve |
| Difficulty | Moderate -0.3 This is a standard C3 exponential function question covering routine techniques: range identification, finding inverse functions, solving exponential equations, and finding tangent equations. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average but still requiring multiple techniques and careful algebraic manipulation. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07m Tangents and normals: gradient and equations |
| Answer | Marks |
|---|---|
| \(f(x) > 5\) | B1 |
| Answer | Marks |
|---|---|
| \(y = 5 + e^{2x - 3}\) | M1 |
| \(2x - 3 = \ln(y - 5)\) | M1 |
| \(x = \frac{1}{2}[3 + \ln(y - 5)]\) | M1 |
| \(\therefore f^{-1}(x) = \frac{1}{2}[3 + \ln(x - 5)], \mathbb{R}, x > 5\) | A2 |
| Answer | Marks |
|---|---|
| \(x = f^{-1}(7) = \frac{1}{2}(3 + \ln 2)\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(x) = 2e^{2x - 3}\) | M1 | |
| grad \(= 4\) | A1 | |
| \(\therefore y - 7 = 4[x - \frac{1}{2}(3 + \ln 2)]\) | M1 A1 | |
| \([y = 4x + 1 - 2\ln 2]\) | M1 A1 | (11 marks) |
**(a)**
$f(x) > 5$ | B1 |
**(b)**
$y = 5 + e^{2x - 3}$ | M1 |
$2x - 3 = \ln(y - 5)$ | M1 |
$x = \frac{1}{2}[3 + \ln(y - 5)]$ | M1 |
$\therefore f^{-1}(x) = \frac{1}{2}[3 + \ln(x - 5)], \mathbb{R}, x > 5$ | A2 |
**(c)**
$x = f^{-1}(7) = \frac{1}{2}(3 + \ln 2)$ | M1 A1 |
**(d)**
$f'(x) = 2e^{2x - 3}$ | M1 |
grad $= 4$ | A1 |
$\therefore y - 7 = 4[x - \frac{1}{2}(3 + \ln 2)]$ | M1 A1 |
$[y = 4x + 1 - 2\ln 2]$ | M1 A1 | (11 marks)5.
$$\mathrm { f } ( x ) = 5 + \mathrm { e } ^ { 2 x - 3 } , \quad x \in \mathbb { R } .$$
(a) State the range of f .\\
(b) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state its domain.\\
(c) Solve the equation $\mathrm { f } ( x ) = 7$.\\
(d) Find an equation for the tangent to the curve $y = \mathrm { f } ( x )$ at the point where $y = 7$.\\
\hfill \mbox{\textit{Edexcel C3 Q5 [11]}}