| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve from second derivative |
| Difficulty | Moderate -0.8 This question involves straightforward double integration with constants of integration determined by given conditions, finding stationary points by setting dy/dx=0, and using the second derivative test. All techniques are standard P1/C1 procedures requiring no problem-solving insight, though the multi-part structure and inclusion of a second unrelated problem (normal/area) adds some length. Slightly easier than average due to the routine nature of all steps. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07e Second derivative: as rate of change of gradient1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrating \(\rightarrow \frac{dy}{dx} = x^2 - 5x \ (+c)\) | B1 | |
| \(= 0\) when \(x = 3\) | M1 | Uses the point to find \(c\) after \(\int = 0\) |
| \(c = 6\) | A1 | |
| Integrating again \(\rightarrow y = \frac{x^3}{3} - \frac{5x^2}{2} + 6x \ (+d)\) | B1 | FT Integration again FT if a numerical constant term is present |
| Use of \((3, 6)\) | M1 | Uses the point to find \(d\) after \(\int = 0\) |
| \(d = 1\frac{1}{2}\) | A1 | |
| Total: 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = x^2 - 5x + 6 = 0 \rightarrow x = 2\) | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = 3\), \(\frac{d^2y}{dx^2} = 1\) and/or \(+\)ve Minimum; \(x = 2\), \(\frac{d^2y}{dx^2} = -1\) and/or \(-\)ve Maximum | B1 | www |
| May use shape of '\(+x^3\)' curve or change in sign of \(\frac{dy}{dx}\) | B1 | www; SC: \(x=3\) minimum, \(x=2\) maximum, B1 |
| Total: 2 |
## Question 10(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrating $\rightarrow \frac{dy}{dx} = x^2 - 5x \ (+c)$ | B1 | |
| $= 0$ when $x = 3$ | M1 | Uses the point to find $c$ after $\int = 0$ |
| $c = 6$ | A1 | |
| Integrating again $\rightarrow y = \frac{x^3}{3} - \frac{5x^2}{2} + 6x \ (+d)$ | B1 | **FT** Integration again FT if a numerical constant term is present |
| Use of $(3, 6)$ | M1 | Uses the point to find $d$ after $\int = 0$ |
| $d = 1\frac{1}{2}$ | A1 | |
| **Total: 6** | | |
## Question 10(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = x^2 - 5x + 6 = 0 \rightarrow x = 2$ | B1 | |
| **Total: 1** | | |
## Question 10(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = 3$, $\frac{d^2y}{dx^2} = 1$ and/or $+$ve Minimum; $x = 2$, $\frac{d^2y}{dx^2} = -1$ and/or $-$ve Maximum | B1 | www |
| May use shape of '$+x^3$' curve or change in sign of $\frac{dy}{dx}$ | B1 | www; SC: $x=3$ minimum, $x=2$ maximum, B1 |
| **Total: 2** | | |10 A curve for which $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 5$ has a stationary point at $( 3,6 )$.\\
(i) Find the equation of the curve.\\
(ii) Find the $x$-coordinate of the other stationary point on the curve.\\
(iii) Determine the nature of each of the stationary points.\\
\includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-20_700_616_262_762}
The diagram shows part of the curve $y = \frac { 3 } { \sqrt { ( 1 + 4 x ) } }$ and a point $P ( 2,1 )$ lying on the curve. The normal to the curve at $P$ intersects the $x$-axis at $Q$.\\
(i) Show that the $x$-coordinate of $Q$ is $\frac { 16 } { 9 }$.\\
(ii) Find, showing all necessary working, the area of the shaded region.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P1 2019 Q10 [9]}}