| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Topic | Tangents, normals and gradients |
| Type | Find stationary points |
| Difficulty | Moderate -0.8 This is a straightforward calculus question requiring basic differentiation of a polynomial and solving a quadratic equation to find stationary points. Both parts are standard textbook exercises with no problem-solving insight needed, making it easier than average but not trivial since it requires correct application of the quadratic formula or factorisation. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
**Part (i)**
- Attempt differentiation of at least one term: M1
- Obtain $3x^2 + 2x - 1$: A1 **[2]**
**Part (ii)**
- State or imply their derivative equal to 0: B1
- Attempt to solve quadratic: M1
- Obtain $x = -1$ and $\frac{1}{3}$: A1
- Obtain $y = 4$ and $\frac{76}{27}$ ($= 2.81$) NIS: A1 **[4]**
**[Total: 6]**3 The equation of a curve is $y = x ^ { 3 } + x ^ { 2 } - x + 3$.\\
(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(ii) Hence find the coordinates of the stationary points on the curve.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q3 [6]}}