| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Topic | Parametric curves and Cartesian conversion |
| Type | Convert to Cartesian (polynomial/rational) |
| Difficulty | Standard +0.3 Part (i) is straightforward algebraic manipulation (eliminate t by cubing x=3t). Part (ii) requires substituting the Cartesian equation into the second curve and solving a cubic, but knowing one root (x=3) allows factorization. This is a standard multi-step question requiring routine techniques with no novel insight, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02q Use intersection points: of graphs to solve equations1.03g Parametric equations: of curves and conversion to cartesian |
**Question 7:**
**(i)** $x^3 = 27t^3$, $y = 1+\frac{1}{27}x^3$ AG
- **M1**: Attempt to eliminate $t$
- **A1**: Obtain given answer convincingly. M1A0 for $y=1+\left(\frac{x}{3}\right)^3 = 1+\frac{1}{27}x^3$
**Total for (i): [2]**
**(ii)**
$$1+\frac{1}{27}x^3 = x^2+4x-19$$
$$x^3-27x^2-108x+540=0$$
$$(x-3)(x^2-24x-180)=0$$
$$(x-30)(x+6)=0$$
$x=30$ or $-6$; points $(30, 1001)$ and $(-6,-7)$
OR working in $t$:
$$1+t^3=9t^2+12t-19$$
$$t^3-9t^2-12t+20=0$$
$$(t-1)(t^2-8t-20)=0$$
$$(t-1)(t-10)(t+2)=0$$
$t=1, 10$ or $-2$; points $(30,1001)$ and $(-6,-7)$
- **M1**: Reduce to equation in one variable
- **M1***: Attempt division by $(x-3)$ [or $(t-1)$]
- **A1**: Obtain correct quotient
- **M1d***: Attempt to solve quadratic quotient
- **A1**: Obtain correct roots
- **A1**: Obtain coordinates of both points
**Total for (ii): [6]**7 A curve is given parametrically by $x = 3 t , y = 1 + t ^ { 3 }$ where $t$ is any real number.\\
(i) Show that a cartesian equation for this curve is given by $y = 1 + \frac { 1 } { 27 } x ^ { 3 }$.
A second curve is given by $y = x ^ { 2 } + 4 x - 19$.\\
(ii) Given that the curves intersect at the point $( 3,2 )$, find the coordinates of all the other points of intersection between the two curves.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2015 Q7 [6]}}