Moderate -0.5 This is a standard C1 simultaneous equations question requiring substitution of a linear equation into a quadratic, then solving the resulting quadratic. It's slightly easier than average because the linear equation rearranges cleanly (y = 2x + 1) and the arithmetic is straightforward, but it still requires multiple steps and careful algebraic manipulation.
Either \(y = 2x + 1\) or \(y = \frac{x^2 + 11}{3}\)
M1
Substitute for \(x/y\) or attempt to get an equation in 1 variable only
\(x^2 - 6x + 8 = 0\)
A1
Obtain correct 3 term quadratic
\((x-2)(x-4) = 0\)
M1
Correct method to solve 3 term quadratic
\(x = 2, x = 4\)
A1
or one correct pair of values B1
\(y = 5, y = 9\)
A1
second correct pair of values B1 c.a.o
OR
\(x = \frac{y-1}{2}\)
\(\frac{(y-1)^2}{4} - 3y + 11 = 0\)
\(y^2 - 14y + 45 = 0\)
\((y-5)(y-9) = 0\)
\(y = 5, y = 9\)
\(x = 2, x = 4\)
SR If solution by graphical methods: setting out to draw a parabola and a line both correct
M1
reading off of coordinates at intersection point(s)
M1
one correct pair
A1
second correct pair
A1
OR No working shown:
one correct pair
B1
second correct pair
B1
full justification that these are the only solutions
B3
5
**Answer/Working** | **Marks** | **Guidance**
---|---|---
Either $y = 2x + 1$ or $y = \frac{x^2 + 11}{3}$ | M1 | Substitute for $x/y$ or attempt to get an equation in 1 variable only
$x^2 - 6x + 8 = 0$ | A1 | Obtain correct 3 term quadratic
$(x-2)(x-4) = 0$ | M1 | Correct method to solve 3 term quadratic
$x = 2, x = 4$ | A1 | or one correct pair of values B1
$y = 5, y = 9$ | A1 | second correct pair of values B1 c.a.o
**OR** | |
$x = \frac{y-1}{2}$ | |
$\frac{(y-1)^2}{4} - 3y + 11 = 0$ | |
$y^2 - 14y + 45 = 0$ | |
$(y-5)(y-9) = 0$ | |
$y = 5, y = 9$ | |
$x = 2, x = 4$ | |
**SR** If solution by graphical methods: setting out to draw a parabola and a line both correct | M1 |
reading off of coordinates at intersection point(s) | M1 |
one correct pair | A1 |
second correct pair | A1 |
**OR** No working shown: | |
one correct pair | B1 |
second correct pair | B1 |
full justification that these are the only solutions | B3 |
| **5** |