Higher June 2023 Paper 1R Q12
12 Moeen has a biased 6-sided dice.
The table gives information about the probability that, when the dice is thrown, it will land on each number.
| Number | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Probability | \(x\) | 0.15 | 0.5 | \(y\) | 0.13 | 0.03 |
Given that \(\;3x - y = 0.09\)
and \(\qquad\;\; x + y = 0.19\)
Show clear algebraic working. (3)
| Scheme | Marks |
|---|---|
| eg \(x + 0.15 + 0.5 + y + 0.13 + 0.03 = 1\) oe or \(x + y = 1 - 0.15 - 0.5 - 0.13 - 0.03\) oe or \(x + y + 0.81 = 1\) oe or \(x + y = 1 - 0.81\) oe or \(1 - 0.15 - 0.5 - 0.13 - 0.03 = 0.19\) oe or \(1 - 0.81 = 0.19\) oe | M1 |
| Working required Answer: Shown | A1 |
| (2) |
Notes
M1: for setting up an equation in \(x\) and \(y\) using the sum of probabilities equals 1
or
for showing that probabilities add up to 1
A1: correctly rearranges to \(x + y = 0.19\) (must be shown from a correct method)
or
a clear statement that \(x + y = 0.19\)
| Scheme | Marks |
|---|---|
\(x + y = 0.19\) \(3x - y = 0.09\) Adding (\(x + 3x = 0.19 + 0.09\) or \(4x = 0.28\)) or \(3x - (0.19 - x) = 0.09\) or \(x + 3x - 0.09 = 0.19\) or \(3x + 3y = 0.57\) \(3x - y = 0.09\) Subtracting (\(3y - -y = 0.57 - 0.09\) or \(4y = 0.48\)) or \(3(0.19 - y) - y = 0.09\) or \(\left(\dfrac{0.09 + y}{3}\right) + y = 0.19\) | M1 |
“0.07” + \(y\) = 0.19 or 3 × “0.07” − \(y\) = 0.09 or \(y\) = 0.19 − “0.07” or \(y\) = 3 × “0.07” − 0.09 or \(3x\) + 3 × “0.12” = 0.57 or \(3x\) − “0.12” = 0.09 or \(x\) = 0.19 − “0.12” or \(x = \left(\dfrac{0.09 + \text{``}0.12\text{''}}{3}\right)\) | M1 |
| Working required Answer: \(x = 0.07\) and \(y = 0.12\) | A1 |
| (3) | |
| (5 marks) |
Notes
M1: for a correct method to eliminate \(x\) or \(y\):
coefficients of \(x\) or \(y\) the same and correct operator to eliminate selected variable (condone any one arithmetic error in multiplication)
or
writing \(x\) or \(y\) in terms of the other variable and correctly substituting (condone missing brackets)
M1: dep on first M1 for a correct method to find other variable by substitution of found variable into one equation
or
for repeating the above method to find the second variable.
A1: oe dep on M1