Higher June 2021 Paper 2 Q23
23 Pippa has a box containing \(N\) pens.
There are only black pens and red pens in the box.
The number of black pens in the box is 3 more than the number of red pens.
Pippa is going to take at random 2 pens from the box.
The probability that she will take a black pen followed by a red pen is \(\dfrac{9}{35}\)
Find the possible values of \(N\).
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
eg \(\dfrac{\left(\frac{N + 3}{2}\right)}{N}\ \left(= \dfrac{N + 3}{2N}\right)\) or eg where \(b\) = number of black pens \(\dfrac{b}{2b - 3}\) or \(\dfrac{b}{N}\) and \(N = 2b - 3\) (or \(b = \dfrac{N + 3}{2}\)) or eg where \(r\) = number of red pens \(\dfrac{r + 3}{2r + 3}\) or \(\dfrac{r + 3}{N}\) and \(N = 2r + 3\) (or \(r = \dfrac{N - 3}{2}\)) | M1 |
eg \(\dfrac{N + 3}{2N} \times \dfrac{N - 3}{2(N - 1)} = \dfrac{9}{35}\) or eg \(\dfrac{b}{2b - 3} \times \dfrac{b - 3}{2b - 4} = \dfrac{9}{35}\) or \(\dfrac{b}{N} \times \dfrac{b - 3}{N - 1} = \dfrac{9}{35}\) or eg \(\dfrac{r + 3}{2r + 3} \times \dfrac{r}{2r + 2} = \dfrac{9}{35}\) or \(\dfrac{r + 3}{N} \times \dfrac{r}{N - 1} = \dfrac{9}{35}\) and \(N = 2r + 3\) | M1 |
| eg \(35(N + 3)(N - 3) = 9(2N(2N - 2))\) or \(35(N^2 - 9) = 9(4N^2 - 4N)\) or eg \(35(b^2 - 3b) = 9(4b^2 - 14b + 12)\) or eg \(35(r^2 + 3r) = 9(4r^2 + 10r + 6)\) | M1 |
| eg \(N^2 - 36N + 315\ (= 0)\) or eg \(b^2 - 21b + 108\ (= 0)\) or eg \(r^2 - 15r + 54\ (= 0)\) | M1 |
| Working must be seen Answer: 21, 15 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for making a correct start by finding the probability of the first pen being black for their method. If in 2 variables, one must also be defined in terms of the other. (any letter may be used for the variable)
M1: oe dep on previous M1 for a correct equation for black, red – must be in one variable or if 2 variables, one must be defined in terms of other.
M1: dep on previous marks
for a correct equation in one variable with no algebraic fractions – brackets may or may not be expanded
M1: For correctly rearranging their equation to a 3 term quadratic
A1: cao dep on M4