| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Show derivative equals given algebraic form |
| Difficulty | Standard +0.3 This is a standard C3 quotient rule question with exponential functions. Part (i) requires product rule for xe^(2x) then quotient rule to reach the given form—routine algebraic manipulation. Part (ii) involves setting the derivative to zero and solving a quadratic discriminant condition, which is slightly above average but still follows a predictable pattern for stationary point questions. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt use of product rule for \(xe^{2x}\) | M1 | obtaining \(\ldots + \ldots\) |
| Obtain \(e^{2x} + 2xe^{2x}\) | A1 | or equiv; maybe within QR attempt |
| Attempt use of quotient rule | M1 | with or without product rule |
| Obtain unsimplified \(\frac{(x + k)(e^{2x} + 2xe^{2x}) - xe^{2x}}{(x + k)^2}\) | A1 | |
| Obtain \(\frac{e^{2x}(2x^2 + 2kx + k)}{(x + k)^2}\) | A1 5 | AG; necessary detail required |
| (ii) Attempt use of discriminant | M1 | or equiv |
| Obtain \(4k^2 - 8k = 0\) or equiv and hence \(k = 2\) | A1 | |
| Attempt solution of \(2x^2 + 2kx + k = 0\) | M1 | using their numerical value of \(k\) or solving in terms of \(k\) using correct formula |
| Obtain \(x = -1\) | A1 | |
| Obtain \(-e^{-2}\) | A1 5 | or exact equiv |
**(i)** Attempt use of product rule for $xe^{2x}$ | M1 | obtaining $\ldots + \ldots$
Obtain $e^{2x} + 2xe^{2x}$ | A1 | or equiv; maybe within QR attempt
Attempt use of quotient rule | M1 | with or without product rule
Obtain unsimplified $\frac{(x + k)(e^{2x} + 2xe^{2x}) - xe^{2x}}{(x + k)^2}$ | A1 |
Obtain $\frac{e^{2x}(2x^2 + 2kx + k)}{(x + k)^2}$ | A1 5 | AG; necessary detail required
**(ii)** Attempt use of discriminant | M1 | or equiv
Obtain $4k^2 - 8k = 0$ or equiv and hence $k = 2$ | A1 |
Attempt solution of $2x^2 + 2kx + k = 0$ | M1 | using their numerical value of $k$ or solving in terms of $k$ using correct formula
Obtain $x = -1$ | A1 |
Obtain $-e^{-2}$ | A1 5 | or exact equiv
---7 A curve has equation $y = \frac { x \mathrm { e } ^ { 2 x } } { x + k }$, where $k$ is a non-zero constant.\\
(i) Differentiate $x \mathrm { e } ^ { 2 x }$, and show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { 2 x } \left( 2 x ^ { 2 } + 2 k x + k \right) } { ( x + k ) ^ { 2 } }$.\\
(ii) Given that the curve has exactly one stationary point, find the value of $k$, and determine the exact coordinates of the stationary point.
\hfill \mbox{\textit{OCR C3 2008 Q7 [10]}}