Higher January 2023 Paper 2 Q20
20 Solve the simultaneous equations
\(\begin{aligned} y &= 7 - 2x \\ x^2 + y^2 &= 34 \end{aligned}\)
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
\(x^2 + (7 - 2x)^2 = 34\) or \(\left(\dfrac{7 - y}{2}\right)^2 + y^2 = 34\) | M1 |
| \(5x^2 - 28x + 15\;[= 0]\) oe or \(5y^2 - 14y - 87\;[= 0]\) oe | M1 |
\((5x - 3)(x - 5)\;[= 0]\) or \(\dfrac{-(-28) \pm \sqrt{(-28)^2 - 4 \times 5 \times 15}}{2 \times 5}\) or \(5\left[\left(x - \dfrac{28}{10}\right)^2 - \dfrac{784}{100}\right] + 15 = 0\) oe or \(x = 0.6\) and \(x = 5\) (allow incorrect labels for \(x\)/\(y\)) or \((5y - 29)(y + 3)\;[= 0]\) or \(\dfrac{-(-14) \pm \sqrt{(-14)^2 - 4 \times 5 \times (-87)}}{2 \times 5}\) or \(5\left[\left(y - \dfrac{14}{10}\right)^2 - \dfrac{196}{100}\right] - 87 = 0\) oe or \(y = 5.8\) and \(y = -3\) (allow incorrect labels for \(x\)/\(y\)) | M1ft |
| eg \(y = 7 - 2 \times 5\) and \(y = 7 - 2 \times 0.6\) (correct labels for \(x\)/\(y\)) or eg \(5.8 = 7 - 2x\) and \(-3 = 7 - 2x\) (correct labels for \(x\)/\(y\)) | M1ft |
| Working must be shown Answer: \(x = 0.6\), \(y = 5.8\) \(x = 5\), \(y = -3\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: substitution of linear equation into quadratic
M1: dep on previous M1 for multiplying out and collecting terms, forming a three term quadratic in any form of \(ax^2 + bx + c\;(= 0)\) where at least 2 coefficients (\(a\) or \(b\) or \(c\)) are correct and all are non-zero
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 \(\dfrac{28 \pm \sqrt{784 - 300}}{10}\) or \(\dfrac{14 \pm \sqrt{196 + 1740}}{10}\))
(if completing the square allow as far as shown) or
correct values for \(x\) or correct values for \(y\) dep on correct quadratic
M1ft: dep on previous M1 for substituting their 2 found values of \(x\) or \(y\) in a suitable equation
or correct values for the other variable
A1: dep on M1 and the correct quadratic (allow coordinates) must be paired correctly