| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2003 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (extended problem with normals, stationary points, or further geometry) |
| Difficulty | Easy -1.2 This is a straightforward integration question requiring only basic polynomial integration and using an initial condition to find the constant. Part (ii) involves solving a simple quadratic inequality. Both parts are routine textbook exercises with no problem-solving insight required, making this easier than average. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| \((1,5)\) used to give \(c=5\) | B2, 1, 0; B1, ∇ [3] | Co - unsimplified ok. Must have integrated + use of \(x=1\) and \(y=5\) for \(c\). |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow x<\frac{1}{3}\) and \(x>1\) | M1, A1, A1 [3] | Set to 0 and attempt to solve. Co for end values – even if "\(>\)" etc. Co (allow \(\leq\) and \(\geq\)). Allow \(1 |
**(i)** $y = x^3 - 2x^2 + x$ $(+c)$
$(1,5)$ used to give $c=5$ | B2, 1, 0; B1, ∇ [3] | Co - unsimplified ok. Must have integrated + use of $x=1$ and $y=5$ for $c$.
**(ii)** $3x^2-4x+1>0$
$\rightarrow$ end values of $1$ and $\frac{1}{3}$
$\rightarrow x<\frac{1}{3}$ and $x>1$ | M1, A1, A1 [3] | Set to 0 and attempt to solve. Co for end values – even if "$>$" etc. Co (allow $\leq$ and $\geq$). Allow $1<x<\frac{1}{3}$.4 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 4 x + 1$. The curve passes through the point $( 1,5 )$.\\
(i) Find the equation of the curve.\\
(ii) Find the set of values of $x$ for which the gradient of the curve is positive.
\hfill \mbox{\textit{CAIE P1 2003 Q4 [6]}}