Standard +0.3 This is a straightforward proof requiring students to express odd integers as 2k+1, expand and simplify to show the sum equals 2(2k²+2k+2l²+2l+1), demonstrating divisibility by 2 but not 4. While it requires algebraic manipulation and understanding of proof structure, it's a standard AS-level proof technique with clear steps and no novel insight needed.
Question 9:
9 | States algebraic expressions for
two distinct non-consecutive odd
numbers. | 3.1a | M1 | Let n = 2p + 1 m =2q + 1
Where p and q are integers
m2 + n2 = (2p + 1)2 + (2q + 1)2
= 4p2 + 4p + 1 + 4q2 + 4q + 1
= 2(2p2 + 2q2 + 2p + 2q + 1)
Factor 2 shows it is a multiple of 2
Factor (2p2 + 2q2 + 2p + 2q + 1) is
1 more than a multiple of 2 so
m2 + n2 is not a multiple of 4
Expands their two-termed
expression for m and n in
m2 + n2 | 1.1a | M1
Obtains their correct expanded
expression.
Do not allow if substitutions
define the same odd number. | 1.1b | A1F
Concludes correctly that the
expression is a multiple of 2
Do not allow if substitutions
define consecutive odd numbers
or substitutions which generate
the same odd number. | 2.4 | E1
Completes a reasoned
argument to conclude correctly
the expression is not a multiple
of 4. CAO
OE
Must not have used
substitutions which involve m or
n or define consecutive odd
numbers or which generate the
same odd number. CAO | 2.1 | R1
Question 9 Total | 5
Q | Marking instructions | AO | Marks | Typical solution
Integers $m$ and $n$ are both odd.
Prove that $m^2 + n^2$ is a multiple of 2 but not a multiple of 4
[5 marks]
\hfill \mbox{\textit{AQA AS Paper 1 2022 Q9 [5]}}