Higher June 2023 Paper 2 Q21
21 Solve the simultaneous equations
\[\begin{aligned} 2x^2 + 3y^2 &= 11 \\ x &= 3y - 1 \end{aligned}\]Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
\(2(3y - 1)^2 + 3y^2 = 11\) or \(2x^2 + 3\left(\dfrac{x + 1}{3}\right)^2 = 11\) | M1 |
| \(21y^2 - 12y - 9 = 0\) oe or \(7x^2 + 2x - 32 = 0\) oe | M1 |
eg \((7y + 3)(y - 1) = 0\) \(\dfrac{-(-12) \pm \sqrt{(-12)^2 - 4 \times 21 \times (-9)}}{2 \times 21}\) \(21\left[\left(y - \dfrac{2}{7}\right)^2 - \dfrac{4}{49}\right] - 9 = 0\) oe (gives \(y = 1\), \(y = -\dfrac{3}{7}\)) or eg \((7x + 16)(x - 2) = 0\) \(\dfrac{-(2) \pm \sqrt{(2)^2 - 4 \times 7 \times -32}}{2 \times 7}\) \(7\left[\left(x + \dfrac{1}{7}\right)^2 - \dfrac{1}{49}\right] - 32 = 0\) (corrected from the printed mark scheme: \(\left(x - \dfrac{1}{7}\right)^2\)) (gives \(x = 2\), \(x = -\dfrac{16}{7}\)) | M1 |
eg \(3 \times 1 - 1\) and \(3 \times -\dfrac{3}{7} - 1\) or eg \(\dfrac{2 + 1}{3}\) and \(\dfrac{-\frac{16}{7} + 1}{3}\) | M1ft |
Working required Answer: \(x = 2\), \(y = 1\) and \(x = -\dfrac{16}{7}\), \(y = -\dfrac{3}{7}\) | 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) with at least 2 coefficients (\(a\) or \(b\) or \(c\)) correct
M1: 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 in subst terms and some simplification – allow as far as eg \(\dfrac{12 \pm \sqrt{144 + 756}}{42}\) or \(\dfrac{-2 \pm \sqrt{4 + 896}}{14}\) )(if completing the square allow as far as shown (allow error in final constant) or correct values for \(x\) or correct values for \(y\)
M1ft: dep on previous M1 for substituting (must be shown) their 2 found values of \(x\) or \(y\) in a suitable equation (use 2dp or better for substitution) or fully correct values for the other variable (correct labels for \(x\) / \(y\))
A1: dep on M2 (allow coordinates)
must be paired correctly
allow \(x = -2.28(57…)\) and \(y = -0.42(85…)\) (even if obtained from premature rounding of the other variable.)