| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 9 |
| 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 integration problem with standard techniques. Part (i) requires simple substitution into the second derivative to determine concavity. Parts (ii) and (iii) involve routine integration of power functions and using given conditions to find constants—all standard AS-level calculus with no novel problem-solving required. Slightly easier than average due to the mechanical nature of the steps. |
| Spec | 1.07e Second derivative: as rate of change of gradient1.07f Convexity/concavity: points of inflection1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) (If \(x = 2\)) it's negative \(\rightarrow\) Max | B1 [1] | www |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow A = 11\) | B2,1,0, M1, A1 [4] | oe one per term; Attempt at the constant \(A\) after \(\int n\); co |
| Answer | Marks | Guidance |
|---|---|---|
| (If \(x = 2\)) \(y = 12\) | B2,1,0, ✓, M1, A1 [4] | oe Doesn't need '+c', but does need a term \(A\) to give "\(4x\)".; Attempt at \(c\) after \(\int n\); co |
$\frac{d^2y}{dx^2} = \frac{24}{x^3} - 4$
**(i)** (If $x = 2$) it's negative $\rightarrow$ Max | B1 [1] | www
**(ii)** $\left(\frac{dy}{dx}\right) = -12x^{-2} - 4x + (A)$
$= 0$ when $x = 2$
$\rightarrow A = 11$ | B2,1,0, M1, A1 [4] | oe one per term; Attempt at the constant $A$ after $\int n$; co
**(iii)** $(y =) 12x^{-1} - 2x^2 + Ax + (c)$
$y = 13$ when $x = 1 \rightarrow c = -8$
(If $x = 2$) $y = 12$ | B2,1,0, ✓, M1, A1 [4] | oe Doesn't need '+c', but does need a term $A$ to give "$4x$".; Attempt at $c$ after $\int n$; coA curve is such that $\frac{d^2y}{dx^2} = \frac{24}{x^3} - 4$. The curve has a stationary point at $P$ where $x = 2$.
\begin{enumerate}[label=(\roman*)]
\item State, with a reason, the nature of this stationary point. [1]
\item Find an expression for $\frac{dy}{dx}$. [4]
\item Given that the curve passes through the point $(1, 13)$, find the coordinates of the stationary point $P$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q10 [9]}}