| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Classify nature of stationary points |
| Difficulty | Moderate -0.3 This is a straightforward stationary points question requiring standard differentiation (chain rule for the first term), solving a quadratic equation, and using the second derivative test. All techniques are routine for A-level, though the algebra requires care. Slightly easier than average due to being a textbook-style exercise with no novel problem-solving required. |
| Spec | 1.07e Second derivative: as rate of change of gradient1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative |
| Answer | Marks | Guidance |
|---|---|---|
| 4(i) | dy/dx=−2 ( 2x−1 )−2 +2 | B2,1,0 |
| Answer | Marks | Guidance |
|---|---|---|
| d2y/dx2 =8 ( 2x−1 )−3 | B1 | Unsimplified form ok |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 4(ii) | Set dy/dx to zero and attempt to solve – at least one correct step | M1 |
| x = 0, 1 | A1 | Expect ( 2x−1 )2 =1 |
| When x = 0, d2y/dx2 =−8 (or<0). Hence MAX | B1 | |
| When x = 1, d2y/dx2 =8 (or>0). Hence MIN | B1 | Both final marks dependent on correct x and correct |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(i) ---
4(i) | dy/dx=−2 ( 2x−1 )−2 +2 | B2,1,0 | Unsimplified form ok (–1 for each error in ‘–2’,
( )−2
‘ 2x−1 ’ and ‘2’)
d2y/dx2 =8 ( 2x−1 )−3 | B1 | Unsimplified form ok
3
Question | Answer | Marks | Guidance
--- 4(ii) ---
4(ii) | Set dy/dx to zero and attempt to solve – at least one correct step | M1
x = 0, 1 | A1 | Expect ( 2x−1 )2 =1
When x = 0, d2y/dx2 =−8 (or<0). Hence MAX | B1
When x = 1, d2y/dx2 =8 (or>0). Hence MIN | B1 | Both final marks dependent on correct x and correct
d2y/dx2 and no errors
May use change of sign of dy/dx but not at x=1/2
4
Question | Answer | Marks | GuidanceA curve has equation $y = (2x - 1)^{-1} + 2x$.
\begin{enumerate}[label=(\roman*)]
\item Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$. [3]
\item Find the $x$-coordinates of the stationary points and, showing all necessary working, determine the nature of each stationary point. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2019 Q4 [7]}}