| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find second derivative d²y/dx² |
| Difficulty | Standard +0.8 This Further Maths question requires implicit differentiation to find dy/dx, solving the resulting equation when dy/dx=0 (which involves substituting back into the original curve equation), then finding the second derivative using the quotient rule on an implicitly defined first derivative. The multi-step nature, algebraic complexity, and need to carefully handle implicit second derivatives places this above average difficulty. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(6x^2 + 6xy + 3x^2y' - 9y^2y' = 0 \Rightarrow 2x(x+y) = (3y^2 - x^2)y'\) | M1A1 | |
| \(y' = 0\) and \(x \neq 0 \Rightarrow x = -y\) | M1A1 | |
| \(\Rightarrow 2x^3 - 3x^3 + 3x^3 = 16 \Rightarrow A\) is \((2, -2)\) | A1 | |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(12x + 6xy' + 6y + 6xy' + 3x^2y'' - [18y(y')^2 + 9y^2y''] = 0\) | *M1 | |
| \(x=2,\ y=-2,\ y'=0 \Rightarrow 8 - 4 + 4y'' - 12y'' = 0\) | DM1 | |
| \(\Rightarrow y'' = \frac{1}{2}\) | A1 | |
| 3 |
## Question 5:
**5(i)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $6x^2 + 6xy + 3x^2y' - 9y^2y' = 0 \Rightarrow 2x(x+y) = (3y^2 - x^2)y'$ | M1A1 | |
| $y' = 0$ and $x \neq 0 \Rightarrow x = -y$ | M1A1 | |
| $\Rightarrow 2x^3 - 3x^3 + 3x^3 = 16 \Rightarrow A$ is $(2, -2)$ | A1 | |
| | **5** | |
**5(ii)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $12x + 6xy' + 6y + 6xy' + 3x^2y'' - [18y(y')^2 + 9y^2y''] = 0$ | *M1 | |
| $x=2,\ y=-2,\ y'=0 \Rightarrow 8 - 4 + 4y'' - 12y'' = 0$ | DM1 | |
| $\Rightarrow y'' = \frac{1}{2}$ | A1 | |
| | **3** | |
---5 The curve $C$ has equation $2 x ^ { 3 } + 3 x ^ { 2 } y - 3 y ^ { 3 } - 16 = 0$.\\
(i) Find the coordinates of the point $A$ on $C$ at which $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ and $x \neq 0$.\\
(ii) Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at $A$.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q5 [8]}}