| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Topic | Stationary points and optimisation |
| Type | Find stationary point then sketch curve |
| Difficulty | Moderate -0.3 This is a straightforward stationary points question requiring standard differentiation, solving a cubic equation that factors nicely, second derivative test, and basic curve sketching. The factorization (dy/dx = 12x²(x-3)(x-2)) is clean, and determining the range for four roots from the sketch is routine. Slightly easier than average due to the nice algebraic structure and standard techniques throughout. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
**(i)**
Attempt to differentiate by reducing powers by one **M1**
Obtain $12x^3 - 60x^2 + 72x = 0$ **A1**
Factorise $x$ and attempt to solve a 3 term quadratic (but condone cancellation of $x$) **M1**
Obtain $(0, 0)$, $(2, 32)$, $(3, 27)$ **A1**
Obtain the second derivative or compare gradients or $y$ values either side of each point **M1**
$36x^2 - 120x + 72$ must be used with either substitution of the relevant $x$ values, or the final values $72, -24$ and $36$ must be shown and similarly for comparison of gradients.
Conclude $(0, 0)$ min, $(2, 32)$ max, $(3, 27)$ min (condone incorrect or no $y$ values for this mark) **A1**
**[6]**
**(ii)**
Generally correct shape of a quartic, two min and one max **M1**
Stationary points marked OR correct $y = 27$ and $y = 32$ shown clearly **A1**
$27 < k < 32$ **A1**
**[3]**6 (i) Find the coordinates of the stationary points of the curve with equation
$$y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }$$
and determine their nature.\\
(ii) Sketch the graph of $y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }$ and hence state the set of values of $k$ for which the equation $3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 } = k$ has exactly four distinct real roots.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2016 Q6 [9]}}