1
The binary operation ⨂ is given by
a ⨂ b = 3a(5 + b) (mod 8)
where a, b ∈ ℤ
Given that 2 ⨂ x = 6, which of the integers below is a possible value of x?
Circle your answer.
0
1
2
3
(Total 1 mark)
2
The function min (a, b) is defined by:
min(a, b) = a,a < b
= b,otherwise
For example, min (7, 2) = 2 and min (–4, 6) = –4.
Gary claims that the binary operation ∆, which is defined as
x ∆ y = min(x, y – 3)
where x and y are real numbers, is associative as finding the smallest number is not affected by
the order of operation.
Disprove Gary’s claim.
(Total 2 marks)
William+Perkin+CE+High+School
Page 1 of 3
3
The table shows some of the outcomes of performing a modular arithmetic operation.
2
2
3
3
1
1
Which pair are operations that could each be represented by the table?
Tick (✓
✓) one box.
Addition mod 6 and multiplication mod 5
Addition mod 6 and multiplication mod 6
Addition mod 4 and multiplication mod 5
Addition mod 4 and multiplication mod 6
(Total 1 mark)
William+Perkin+CE+High+School
Page 2 of 3
Mark schemes
1
Marking Instructions
AO
Marks
Circles correct answer
AO1.1b
B1
Typical Solution
0
Total 1 mark
2
Marking Instructions
AO
Marks
Searches for and finds correct
counter-example to the
associativity condition
AO1.1b
M1
Correctly argues that the binary
operation is not associative,
disproving Gary’s claim
AO2.3
Typical Solution
(3∆1)∆2 = –2
3∆(1∆2) = –4
R1
(3∆1)∆2≠3∆(1∆2)
Therefore ∆ is not an associative
binary operation, which
disproves Gary’s claim.
Total 2 marks
3
Marking Instructions
Ticks correct box
AO
Marks
AO1.1b
B1
Typical Solution
Addition mod 4 and Multiplication mod
5
Total 1 mark
William+Perkin+CE+High+School
Page 3 of 3