Mathematics · May / June 2023 · Paper 21 · Question 8

Eg

7

2

+

11

2

=

170

170 is a multiple of 10 so the statement is untrue *

Check first if they have multiplied the inequality e.g. by 4 to give

4

<

4

2

−

<

60

x

xy

x

y

xy

x −

2

or

4x

2

−

xy

4

2

2

3 12

2

4

2 8

4

2

14 56

4

4

12 48

xy

Concludes that

1

<

2

−

<

15

for all x and y that are positive even integers

x

4

less than 6 *

(i)

M1

Attempts to find

2

+

2

using two consecutive prime numbers a and b. It does not matter

a

b

whether this combination of prime numbers results in a multiple of 10 to score M1. Condone

slips in their evaluation of the sum of the squares of these two numbers. If a candidate has

several combinations, you only need to look for one calculation using two consecutive prime

numbers.

Note

1

2

+

2

2

=

...

as the only calculation is M0A0

A1*

Correctly:

o makes a comment that the answer is a multiple of 10

o makes a comment that 10 is a factor of their answer

o makes a comment that their answer is divisible by 10 (do not accept can be divided by

10)

o shows that the answer is a multiple of 10 (such as splitting the number into two factors,

one of which is 10, or showing the number is divisible by 10 e.g. 170

10 =

17

If a candidate has several combinations, you do not need to check all of them. Just look for one which

would lead to correct counter example.

Below are a list of common combinations which provide the required counter example but there will be

others

7

2

+

11

2

=

170

11

2

+

13

2

=

290

17

2

+

19

2

=

650

19

2

+

23

2

=

890

23

2

+

29

2

=

1370

(ii)

xy

M1

Attempts at least two valid combinations of x and y and evaluates

x −

2

condone slips in the

4

evaluation of the combinations.

Alternatively, attempts at least two valid combinations of x and y and evaluates

4x

2

−

xy

A1

All four valid combinations of x and y correctly evaluated for

(or in the alternative

x −

2

4

method

4x

2

−

xy

)

xy

A1*

A minimal conclusion that

2

(or condone

) for all x and y

1

<

x

−

<

15

4

<

4

x

2

−

xy

<

60

4

that are positive even integers less than 6.

e.g. “hence statement is true”, “they are all between 1 and 15” (or 4 and 60 in the alternative),

“proven”, “QED” or they may have a preamble which can be concluded with a tick or similar.