4MA0 Higher June 2013 Paper 3R Q21
21 A bag contains \(x\) counters.
7 of the counters are blue.
Sam takes at random a counter from the bag and does not replace it.
Jill then takes a counter from the bag.
The probability they both take a blue counter is 0.2
(a) Form an equation involving \(x\).
Show that your equation can be expressed as \(x^2 - x - 210 = 0\) (2)
Show that your equation can be expressed as \(x^2 - x - 210 = 0\) (2)
(b) Solve \(x^2 - x - 210 = 0\)
Show clear algebraic working. (3)
Show clear algebraic working. (3)
| Scheme | Marks |
|---|---|
\(\dfrac{7}{x} \times \dfrac{6}{x - 1} = 0.2\) \(42 = 0.2x(x - 1)\) \(210 = x^2 - x\) | M1 |
| \(x^2 - x - 210\ (=0)\) | A1* |
| (2) |
Notes
M1: for \(\dfrac{7}{x} \times \dfrac{6}{x - 1} = 0.2\) or \(\dfrac{7}{x} \times \dfrac{6}{x - 1} = \dfrac{1}{5}\)
(corrected from the printed mark scheme: the working column prints \(\dfrac{6}{x+1}\))
A1*: * answer given; sufficient steps must be seen to get to correct quadratic
| Scheme | Marks |
|---|---|
| \((x - 15)(x + 14)\ (=0)\) | M2 |
| −14 , 15 | A1 |
| (3) |
Notes
M2: M1 for \((x \pm 15)(x \pm 14)\)
A1: (dep on M2) for −14, 15 or 15
M1 \(\dfrac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 1 \times (-210)}}{2}\)
(may be partially evaluated, condone no brackets around negative numbers, accept \(1^2\))
M1 (indep) for \(\sqrt{841}\) or 29
A1 (dep on M1) for −14, 15 or 15