| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Decreasing or increasing function |
| Difficulty | Moderate -0.8 This is a straightforward application of standard calculus techniques: (i) requires factorising a quadratic and solving f'(x) > 0, (ii) requires integrating a polynomial and finding the constant using a boundary condition. Both parts are routine A-level procedures with no problem-solving insight needed, making this easier than average. |
| Spec | 1.07o Increasing/decreasing: functions using sign of dy/dx1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((3x+5)(x-1)(>0)\) | M1 | Attempt at factorisation |
| \(-5/3,\ 1\) | A1 | Both required |
| \(x < -5/3,\ x > 1\) | A1 | Ignore any words between answers; Condone \(<>\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) = x^3 + x^2 - 5x\ (+c)\) | M1, A1 | Attempt at integration; any unsimplified expression ok |
| \(3 = 1 + 1 - 5 + c\) | M1 | Sub. \((1, 3)\) |
| \(f(x) = x^3 + x^2 - 5x + 6\) | A1 | Accept \(c = 6\) |
| [4] |
## Question 6:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(3x+5)(x-1)(>0)$ | M1 | Attempt at factorisation |
| $-5/3,\ 1$ | A1 | Both required |
| $x < -5/3,\ x > 1$ | A1 | Ignore any words between answers; Condone $<>$ |
| **[3]** | | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = x^3 + x^2 - 5x\ (+c)$ | M1, A1 | Attempt at integration; any unsimplified expression ok |
| $3 = 1 + 1 - 5 + c$ | M1 | Sub. $(1, 3)$ |
| $f(x) = x^3 + x^2 - 5x + 6$ | A1 | Accept $c = 6$ |
| **[4]** | | |
---6 A curve has equation $y = \mathrm { f } ( x )$. It is given that $\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } + 2 x - 5$.\\
(i) Find the set of values of $x$ for which f is an increasing function.\\
(ii) Given that the curve passes through $( 1,3 )$, find $\mathrm { f } ( x )$.
\hfill \mbox{\textit{CAIE P1 2010 Q6 [7]}}