June 2022 Paper 2 Q7
7 It is given that any integer can be expressed in the form \(3m + r\), where \(m\) is an integer and \(r\) is 0, 1 or 2.
Use this fact to answer the following.
By considering the different values of \(r\), determine the probability that the sum of these three integers is divisible by 3. [4]
| Scheme | Marks | AO |
|---|---|---|
| \((3m + 0)^2 = 9m^2\) | B1 | 3.1a |
| \((3m + 1)^2 = 9m^2 + 6m + 1\) \((3m + 2)^2 = 9m^2 + 12m + 4\) | M1 | 1.1 |
| \(= 3(3m^2 + 2m) + 1\) \(= 3(3m^2 + 4m + 1) + 1\) or \(3(3m^2 + 4m) + 4\) | A1 | 2.1 |
| None of these is of the form \(3n + 2\) Allow “\(\ne 3n + 2\)” | A1 | 3.2a |
| [4] |
Notes
NB Other correct methods may be seen
B1: \(9m^2\) alone, not as part of longer expression
M1: At least one of these expansions attempted using \(r = 1\) or 2.
Must include three (or four) terms, Allow one error
A1: At least one of these seen explicitly
A1: Must see the statement oe.
Can be seen once at end or with each separate case
Dep complete method, with all three cases seen
Alternative method 1
| Scheme | Marks |
|---|---|
| \((3m + r)^2\) \(\quad(= 9m^2 + 6mr + r^2)\) | M1 |
| \(= 3(3m^2 + 2mr) + r^2\) \(= 3n + r^2\) | A1 |
| But \(r^2 = 0\), 1 or 4 | B1 |
| Hence not in the form \(3n + 2\) for any \(r\) | A1 |
M1: Attempted. Must include 3 (or 4) terms, Allow one error
A1: Explicit
A1: Must see the statement oe Dep complete method
Alternative method 2
| Scheme | Marks |
|---|---|
| Let \((3m + r)^2 = 3n + 2\) | M1 |
| \(3(3m^2 + 2mr - n) = 2 - r^2\) | A1 |
| Hence \(2 - r^2\) is divisible by 3 | B1 |
| But \(2 - 0^2 = 2\), \(2 - 1^2 = 1\), \(2 - 2^2 = -2\) None of these is divisible by 3 | A1 |
Alternative method 3
| Scheme | Marks |
|---|---|
| \((3m)^2 = (9m^2 - 2) + 2\) | B1 |
| \((3m + 1)^2 = (9m^2 + 6m - 1) + 2\) | M1 |
| \((3m + 2)^2 = (9m^2 + 12m + 2) + 2\) | A1 |
| \((9m^2 - 2) = 3(3m^2) - 2\) or \(3\left(3m^2 - \frac{2}{3}\right) + 2\) \((9m^2 + 6m - 1) = 3(3m^2 + 2m) - 1\) or \(3\left(3m^2 + 2m - \frac{1}{3}\right) + 2\) \((9m^2 + 12m + 2) = 3(3m^2 + 4m) + 2\) or \(3\left(3m^2 + 4m + \frac{2}{3}\right) + 2\) Hence none is divisible by 3 | A1 |
M1: Allow one arithmetical error
A1: Both correct
A1: None of the brackets is an integer
| Scheme | Marks | AO |
|---|---|---|
| Either imply three digits all of the same type or imply three digits all of different types | M1* | 3.1a |
| P(0, 0, 0) or P(1, 1, 1) or P(2, 2, 2): \(\left(\frac{1}{3}\right)^3\) | M1dep | 1.1 |
| P(0, 1, 2): \(\left(\frac{1}{3}\right)^3 \times 6\) or \(1 \times \frac{2}{3} \times \frac{1}{3}\) oe | M1dep | 2.1 |
| \(\frac{9}{27}\) or \(\frac{1}{3}\) or 0.333 (3 sf) | A1 | 1.1 |
| [4] |
Notes
M1*: Could be numerical or algebraic or in words
If listed, must be clear which ones are selected
M1dep: M1 for \(\left(\frac{1}{3}\right)^3\) associated with at least one of these
M1dep: M1 for \(\left(\frac{1}{3}\right)^3 \times k\) where \(k = 4\), 5 or 6, associated with (0, 1, 2)
A1: Correct answer with no working: M0M0M0A0
Alternative method for 2nd & 3rd M1M1
| Scheme | Marks |
|---|---|
| No. of cases \(= 3^3 = 27\) | M1 |
| No. divisible by \(3 = (3 + 6 =)\ 9\) | M1 |
M1: Allow 7 or 8