| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Curve from derivative information |
| Difficulty | Moderate -0.8 This is a straightforward integration question requiring students to find a stationary point by setting f'(x)=0, then integrate to find f(x) and use the range condition to determine the constant. All steps are routine calculus techniques with no problem-solving insight needed, making it easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(3\) | B1 | [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) = x^2 - 6x(+c)\) | M1A1 | Dependent on \(c\) present |
| Subst \((3, -4)\) | M1 | cao |
| \(c = 5 \to f(x) = x^2 - 6x + 5\) | A1 | [4] |
## Question 4:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $3$ | B1 | [1] |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = x^2 - 6x(+c)$ | M1A1 | Dependent on $c$ present |
| Subst $(3, -4)$ | M1 | cao |
| $c = 5 \to f(x) = x^2 - 6x + 5$ | A1 | [4] |
---4 A function f is defined for $x \in \mathbb { R }$ and is such that $\mathrm { f } ^ { \prime } ( x ) = 2 x - 6$. The range of the function is given by $\mathrm { f } ( x ) \geqslant - 4$.\\
(i) State the value of $x$ for which $\mathrm { f } ( x )$ has a stationary value.\\
(ii) Find an expression for $\mathrm { f } ( x )$ in terms of $x$.
\hfill \mbox{\textit{CAIE P1 2011 Q4 [5]}}