Higher November 2024 Paper 2 Q22
22 Solve the simultaneous equations
\(x^2 + y^2 + y = 3\)
\(x + 2 = y\)
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
| \((y - 2)^2 + y^2 + y = 3\) or \(x^2 + (x + 2)^2 + x + 2 = 3\) | M1 |
| \(2y^2 - 3y + 1\;[= 0]\) oe (any form with 3 terms) or \(2x^2 + 5x + 3\;[= 0]\) oe (any form with 3 terms) | M1 |
\((2y - 1)(y - 1)\;[= 0]\) \(\dfrac{-(-3) \pm \sqrt{(-3)^2 - 4 \times 2 \times 1}}{2 \times 2}\) \(2\left[\left(y - \dfrac{3}{4}\right)^2 - \dfrac{9}{16}\right] + 1 = 0\) oe (leading to \(y\) values of \(\dfrac{1}{2}\) and 1) (allow \(x\) used for \(y\) here) or \((2x + 3)(x + 1)\;[= 0]\) \(\dfrac{-5 \pm \sqrt{5^2 - 4 \times 2 \times 3}}{2 \times 2}\) \(2\left[\left(x + \dfrac{5}{4}\right)^2 - \dfrac{25}{16}\right] + 3 = 0\) oe (leading to \(x\) values of \(-\dfrac{3}{2}\) and \(-1\)) (allow \(y\) used for \(x\)) | M1ft |
| eg \((x =)\;\text{``}{\tfrac{1}{2}}\text{''} - 2\) oe \(\text{``}{1}\text{''} - 2\) oe or \((y =)\;\text{``}{-\tfrac{3}{2}}\text{''} + 2\) oe, \(\text{``}{-1}\text{''} + 2\) oe | M1 |
Working required Answer: \(x = -\dfrac{3}{2},\; y = \dfrac{1}{2}\) \(x = -1,\; y = 1\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: substitution of linear equation into quadratic – allow one sign error in substituted expression
M1: Dep on M1 simplified to a 3 term quadratic with 2 or 3 of 3 terms correct
M1ft: dep on M1 for solving their 3 term quadratic equation using any correct method (if factorising, allow brackets which expanded give 2 out of 3 terms correct ) (if using formula allow one sign error and some simplification – allow as far as \(\dfrac{3 \pm \sqrt{9 - 8}}{4}\) or \(\dfrac{-5 \pm \sqrt{25 - 24}}{4}\) ) or if completing the square then as far as shown on LHS OR the correct values for \(x\) OR the correct values for \(y\)
M1: dep on previous M1 for correct method to find both other values or
a correct pair of values
A1: oe dep on M2
for all 4 values