| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Show derivative satisfies condition |
| Difficulty | Standard +0.3 This is a straightforward application of the product rule with standard derivatives (e^(2x) and tan x). Part (a) requires setting dy/dx = 0 and simple algebraic manipulation. Part (b) is routine tangent line calculation. Slightly above average difficulty due to the product rule with exponential and trig functions, but still a standard textbook exercise with no novel insight required. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07m Tangents and normals: gradient and equations1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = 2e^{2x}\tan x + e^{2x}\sec^2 x\) | M1 A1+A1 | |
| \(\frac{dy}{dx} = 0 \Rightarrow 2e^{2x}\tan x + e^{2x}\sec^2 x = 0\) | M1 | |
| \(2\tan x + 1 + \tan^2 x = 0\) | A1 | |
| \((\tan x + 1)^2 = 0\) | ||
| \(\tan x = -1\) | A1 | cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left(\frac{dy}{dx}\right)_0 = 1\) | M1 | |
| Equation of tangent at \((0,0)\) is \(y = x\) | A1 |
# Question 2:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 2e^{2x}\tan x + e^{2x}\sec^2 x$ | M1 A1+A1 | |
| $\frac{dy}{dx} = 0 \Rightarrow 2e^{2x}\tan x + e^{2x}\sec^2 x = 0$ | M1 | |
| $2\tan x + 1 + \tan^2 x = 0$ | A1 | |
| $(\tan x + 1)^2 = 0$ | | |
| $\tan x = -1$ | A1 | cso |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left(\frac{dy}{dx}\right)_0 = 1$ | M1 | |
| Equation of tangent at $(0,0)$ is $y = x$ | A1 | |
---2. A curve $C$ has equation
$$y = \mathrm { e } ^ { 2 x } \tan x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Show that the turning points on $C$ occur where $\tan x = - 1$.
\item Find an equation of the tangent to $C$ at the point where $x = 0$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2008 Q2 [8]}}