| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find gradient at point |
| Difficulty | Standard +0.3 Part (i) requires straightforward application of the quotient rule and substituting x=0. Part (ii) involves implicit differentiation with a cubic term, then solving for dy/dx and substituting x=0, y=2. Both are standard techniques with minimal problem-solving required, making this slightly easier than average. |
| Spec | 1.07q Product and quotient rules: differentiation1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use correct quotient rule or equivalent | M1 | |
| Obtain \(\frac{(1+e^{2x})2x - (1+x^2)2e^{2x}}{(1+e^{2x})^2}\) or equivalent | A1 | |
| Substitute \(x=0\) and obtain \(-\frac{1}{2}\) or equivalent | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiate \(y^3\) and obtain \(3y^2\frac{dy}{dx}\) | B1 | |
| Differentiate \(5xy\) and obtain \(5y + 5x\frac{dy}{dx}\) | B1 | |
| Obtain \(6x^2 + 5y + 5x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0\) | B1 | |
| Substitute \(x=0\), \(y=2\) to obtain \(-\frac{5}{6}\) or equivalent following correct work | B1 | [4] |
## Question 5(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct quotient rule or equivalent | M1 | |
| Obtain $\frac{(1+e^{2x})2x - (1+x^2)2e^{2x}}{(1+e^{2x})^2}$ or equivalent | A1 | |
| Substitute $x=0$ and obtain $-\frac{1}{2}$ or equivalent | A1 | [3] |
## Question 5(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiate $y^3$ and obtain $3y^2\frac{dy}{dx}$ | B1 | |
| Differentiate $5xy$ and obtain $5y + 5x\frac{dy}{dx}$ | B1 | |
| Obtain $6x^2 + 5y + 5x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0$ | B1 | |
| Substitute $x=0$, $y=2$ to obtain $-\frac{5}{6}$ or equivalent following correct work | B1 | [4] |
---5 For each of the following curves, find the gradient at the point where the curve crosses the $y$-axis:\\
(i) $y = \frac { 1 + x ^ { 2 } } { 1 + \mathrm { e } ^ { 2 x } }$;\\
(ii) $2 x ^ { 3 } + 5 x y + y ^ { 3 } = 8$.
\hfill \mbox{\textit{CAIE P3 2013 Q5 [7]}}