| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2002 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find stationary points and nature |
| Difficulty | Moderate -0.3 This is a straightforward stationary points question requiring basic differentiation of a polynomial, solving a quadratic equation, and substituting back into the original equation. All steps are routine A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(dy/dx = 3x^2 + 6x - 9\) | B2, | One off for each error including \(+k\) left |
| (ii) \(= 0\) when \((x+3)(x-1) = 0\). \(x = -3\) or \(x = 1\) | M1, A1 | Use of \(dy/dx = 0\); Both values somewhere |
| (iii) Subbing the values into \(y = 0\). \(k = -27\) or \(k = 5\) | M1, DM1, A1 | Using \(y = 0\) at least once; Subbing his values for \(x\) into \(y = 0 + \text{soln}\); Both correct |
**(i)** $dy/dx = 3x^2 + 6x - 9$ | B2, | One off for each error including $+k$ left
**(ii)** $= 0$ when $(x+3)(x-1) = 0$. $x = -3$ or $x = 1$ | M1, A1 | Use of $dy/dx = 0$; Both values somewhere
**(iii)** Subbing the values into $y = 0$. $k = -27$ or $k = 5$ | M1, DM1, A1 | Using $y = 0$ at least once; Subbing his values for $x$ into $y = 0 + \text{soln}$; Both correct8 A curve has equation $y = x ^ { 3 } + 3 x ^ { 2 } - 9 x + k$, where $k$ is a constant.\\
(i) Write down an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(ii) Find the $x$-coordinates of the two stationary points on the curve.\\
(iii) Hence find the two values of $k$ for which the curve has a stationary point on the $x$-axis.
\hfill \mbox{\textit{CAIE P1 2002 Q8 [7]}}