Standard +0.3 This is a straightforward parametric-to-Cartesian conversion requiring algebraic manipulation. Students rearrange x = 2t - 1 to find t = (x+1)/2, substitute into y, and simplify the resulting rational expression. The question guides students to the final form and only requires finding two integer constants, making it slightly easier than average.
5. A curve \(C\) has parametric equations
$$x = 2 t - 1 , \quad y = 4 t - 7 + \frac { 3 } { t } , \quad t \neq 0$$
Show that the Cartesian equation of the curve \(C\) can be written in the form
$$y = \frac { 2 x ^ { 2 } + a x + b } { x + 1 } , \quad x \neq - 1$$
where \(a\) and \(b\) are integers to be found.
Substitutes \(t=\frac{x+1}{2}\) into \(y \Rightarrow y = 4\!\left(\frac{x+1}{2}\right)-7+\frac{6}{(x+1)}\)
M1
Score for attempt at substituting \(t=\frac{x+1}{2}\) or equivalent into \(y=4t-7+\frac{3}{t}\)
Attempts single fraction: \(y = \frac{(2x-5)(x+1)+6}{(x+1)}\)
M1
Award for attempt at single fraction with correct common denominator; \(4\!\left(\frac{x+1}{2}\right)-7\) term may be simplified first
\(y = \frac{2x^2-3x+1}{x+1}, \quad a=-3,\ b=1\)
A1
Correct answer only
## Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes $t=\frac{x+1}{2}$ into $y \Rightarrow y = 4\!\left(\frac{x+1}{2}\right)-7+\frac{6}{(x+1)}$ | M1 | Score for attempt at substituting $t=\frac{x+1}{2}$ or equivalent into $y=4t-7+\frac{3}{t}$ |
| Attempts single fraction: $y = \frac{(2x-5)(x+1)+6}{(x+1)}$ | M1 | Award for attempt at single fraction with correct common denominator; $4\!\left(\frac{x+1}{2}\right)-7$ term may be simplified first |
| $y = \frac{2x^2-3x+1}{x+1}, \quad a=-3,\ b=1$ | A1 | Correct answer only |
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5. A curve $C$ has parametric equations
$$x = 2 t - 1 , \quad y = 4 t - 7 + \frac { 3 } { t } , \quad t \neq 0$$
Show that the Cartesian equation of the curve $C$ can be written in the form
$$y = \frac { 2 x ^ { 2 } + a x + b } { x + 1 } , \quad x \neq - 1$$
where $a$ and $b$ are integers to be found.\\
\hfill \mbox{\textit{Edexcel Paper 1 Q5 [3]}}