| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Determine if function is increasing/decreasing |
| Difficulty | Standard +0.3 This is a straightforward multi-part calculus question requiring standard techniques: differentiation to find stationary points, second derivative for the gradient's stationary value, and basic integration for area. All parts follow routine procedures with no novel insight required, making it slightly easier than average despite being multi-step. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = 3x^2 - 4x + 5\) | B1 | CAO |
| Using \(b^2 - 4ac \rightarrow 16 - 60 \rightarrow\) negative \(\rightarrow\) some explanation or completed square and explanation | M1 A1 | Uses discriminant on equation (set to 0). CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(m = 3x^2 - 4x + 5\); \(\frac{dm}{dx} = 6x - 4\ (= 0)\) (must identify as \(\frac{dm}{dx}\)) | B1FT | FT providing differentiation is equivalent |
| \(x = \frac{2}{3}\), \(m = \frac{11}{3}\) or \(\frac{dy}{dx} = \frac{11}{3}\); Alt1: \(m = 3\left(x - \frac{2}{3}\right)^2 + \frac{11}{3}\); Alt2: \(3x^2 - 4x + 5 - m = 0\), \(b^2 - 4ac = 0\), \(m = \frac{11}{3}\) | M1 A1 | Sets to 0 and solves; A1 for correct \(m\). Alt1: B1 completing square, M1A1 ans. Alt2: B1 coefficients, M1A1 ans |
| \(\frac{d^2m}{dx^2} = 6\) +ve \(\rightarrow\) Minimum value or check values of \(m\) either side of \(x = \frac{2}{3}\) | M1 A1 | M1 correct method. A1 (no errors anywhere) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate \(\rightarrow \frac{x^4}{4} - \frac{2x^3}{3} + \frac{5x^2}{2}\) | B2,1 | Loses a mark for each incorrect term |
| Uses limits \(0\) to \(6\ \rightarrow 270\) (may not see use of lower limit) | M1 A1 | Use of limits on an integral. CAO. Answer only 0/4 |
## Question 10(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 3x^2 - 4x + 5$ | **B1** | CAO |
| Using $b^2 - 4ac \rightarrow 16 - 60 \rightarrow$ negative $\rightarrow$ some explanation or completed square and explanation | **M1 A1** | Uses discriminant on equation (set to 0). CAO |
## Question 10(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $m = 3x^2 - 4x + 5$; $\frac{dm}{dx} = 6x - 4\ (= 0)$ (must identify as $\frac{dm}{dx}$) | **B1FT** | FT providing differentiation is equivalent |
| $x = \frac{2}{3}$, $m = \frac{11}{3}$ or $\frac{dy}{dx} = \frac{11}{3}$; Alt1: $m = 3\left(x - \frac{2}{3}\right)^2 + \frac{11}{3}$; Alt2: $3x^2 - 4x + 5 - m = 0$, $b^2 - 4ac = 0$, $m = \frac{11}{3}$ | **M1 A1** | Sets to 0 and solves; A1 for correct $m$. Alt1: B1 completing square, M1A1 ans. Alt2: B1 coefficients, M1A1 ans |
| $\frac{d^2m}{dx^2} = 6$ +ve $\rightarrow$ Minimum value or check values of $m$ either side of $x = \frac{2}{3}$ | **M1 A1** | M1 correct method. A1 (no errors anywhere) |
## Question 10(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate $\rightarrow \frac{x^4}{4} - \frac{2x^3}{3} + \frac{5x^2}{2}$ | **B2,1** | Loses a mark for each incorrect term |
| Uses limits $0$ to $6\ \rightarrow 270$ (may not see use of lower limit) | **M1 A1** | Use of limits on an integral. CAO. Answer only 0/4 |10 The curve with equation $y = x ^ { 3 } - 2 x ^ { 2 } + 5 x$ passes through the origin.\\
(i) Show that the curve has no stationary points.\\
(ii) Denoting the gradient of the curve by $m$, find the stationary value of $m$ and determine its nature.\\
(iii) Showing all necessary working, find the area of the region enclosed by the curve, the $x$-axis and the line $x = 6$.\\
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 2018 Q10 [12]}}