| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Session | Specimen |
| Marks | 7 |
| 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 straightforward product rule application with exponential and trigonometric functions, followed by routine algebraic manipulation to match a given form. The 'show that' structure provides the target expression, making it easier than open-ended problems. Parts (ii) and (iii) require minimal additional insight once part (i) is complete. Slightly above average due to the algebraic manipulation required, but still a standard textbook-style question. |
| Spec | 1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply that the derivative of \(e^{-2x}\) is \(-2e^{-2x}\) | B1 | |
| Use product or quotient rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Use Pythagoras | M1 | |
| Justify the given form | A1 | |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Fully justify the given statement | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State answer \(x = \frac{1}{4}\pi\) | B1 | |
| Total: 1 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply that the derivative of $e^{-2x}$ is $-2e^{-2x}$ | B1 | |
| Use product or quotient rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Use Pythagoras | M1 | |
| Justify the given form | A1 | |
| **Total: 5** | | |
## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Fully justify the given statement | B1 | |
| **Total: 1** | | |
## Question 5(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State answer $x = \frac{1}{4}\pi$ | B1 | |
| **Total: 1** | | |5 The equation of a curve is $y = \mathrm { e } ^ { - 2 x } \tan x$, for $0 \leqslant x < \frac { 1 } { 2 } \pi$.\\
(i) Obtain an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and show that it can be written in the form $\mathrm { e } ^ { - 2 x } ( a + b \tan x ) ^ { 2 }$, where $a$ and $b$ are constants.\\
(ii) Explain why the gradient of the curve is never negative.\\
(iii) Find the value of $x$ for which the gradient is least.\\
\hfill \mbox{\textit{CAIE P3 Q5 [7]}}