4MA0 Higher June 2015 Paper 4 Q23
23 Solve
\[\begin{aligned} x^2 + y^2 &= 20 \\ y &= 10 - 2x \end{aligned}\]Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
| \(x^2 + (10 - 2x)^2 = 20\) oe | M1 |
| \(100 - 20x - 20x + 4x^2\) or \(100 - 40x + 4x^2\) | M1 |
| \(x^2 + 100 - 20x - 20x + 4x^2 = 20\) or \(x^2 + 100 - 40x + 4x^2 = 20\) \(5x^2 - 40x + 80\;(= 0)\) or \(x^2 - 8x + 16\;(= 0)\) | A1 |
\(5(x - 4)(x - 4)\;(= 0)\) or \((x - 4)(x - 4)\;(= 0)\) or \((5x - 20)(x - 4)\;(= 0)\) or \(\dfrac{--8 \pm \sqrt{(-8)^2 - 4(1)(16)}}{2(1)}\) oe (may be partially evaluated; condone lack of brackets around negative numbers) | M1 |
| \(x = 4\) \(y = 2\) Answer: \(x = 4\) ; \(y = 2\) | A1 |
| (5) |
Notes
M1: or \(\left(\dfrac{10 - y}{2}\right)^2 + y^2 = 20\) oe
M1: (indep) for correct expansion of \((10 - 2x)^2\) or \(\left(\dfrac{10 - y}{2}\right)^2\) even if unsimplified
\(\left(\dfrac{100 - 10y - 10y + y^2}{4}\right) + y^2 = 20\) or \(\left(\dfrac{100 - 20y + y^2}{4}\right) + y^2 = 20\)
A1: \(5y^2 - 20y + 20\;(= 0)\)
or \(y^2 - 4y + 4\;(= 0)\)
M1: \(5(y - 2)(y - 2)\;(= 0)\) or \((y - 2)(y - 2)\;(= 0)\) or \((5y - 10)(y - 2)\;(= 0)\) or \(\dfrac{--4 \pm \sqrt{(-4)^2 - 4(1)(4)}}{2(1)}\) oe
(may be partially evaluated; condone lack of brackets around negative numbers)
A1: dep on all preceding method marks
No marks for \(x = 4\); \(y = 2\) with no working. Accept (4, 2)