June 2024 Paper 1 Q3

OCR ACurrent spec8 marksProof

(a) Find a counterexample to disprove the statement that the product of two prime numbers is always odd. [1]

(b) In each of the following cases write one of the symbols \(\Rightarrow\), \(\Leftrightarrow\), \(\Leftarrow\) in the box in the Printed Answer Booklet to make each statement correct.

(i) \(x^2 = 3x \quad \boxed{\phantom{\Leftrightarrow}} \quad x = 3\) [1]

(ii) \(x \gt 4 \quad \boxed{\phantom{\Leftrightarrow}} \quad x^3 \gt 64\) [1]

(iii) \(x^\circ = 45^\circ \quad \boxed{\phantom{\Leftrightarrow}} \quad \tan x^\circ = 1\) [1]

(c) Prove that the sum of the squares of any two odd numbers is always a multiple of 2 but never a multiple of 4. [4]

Mark scheme (a)

Scheme	Marks	AO
\(2 \times 3 = 6\) which is even, hence counterexample	B1	2.1
	[1]

Notes

B1: Any product involving 2 and a prime number, evaluated and contradiction identified.
eg \(2 \times 3 = 6\), which is not odd
Condone \(2 \times 2 = 4\), which is not odd

Mark scheme (b)

Scheme	Marks	AO
(i) \(x^2 = 3x \Leftarrow x = 3\)	B1	2.2a
	[1]
(ii) \(x \gt 4 \Leftrightarrow x^3 \gt 64\)	B1	2.2a
	[1]
(iii) \(x^\circ = 45^\circ \Rightarrow \tan x^\circ = 1\)	B1	2.2a
	[1]

Notes

(b)(i) B1: Correct symbol used. Condone \(\leftarrow\)

(b)(ii) B1: Correct symbol used. Condone \(\leftrightarrow\)

(b)(iii) B1: Correct symbol used. Condone \(\rightarrow\)

Mark scheme (c)

Scheme	Marks	AO
\((2m+1)^2 + (2n+1)^2\)	B1	2.1
\(= 4m^2 + 4m + 1 + 4n^2 + 4n + 1\) \(= 4m^2 + 4m + 4n^2 + 4n + 2\)	M1	2.1
\(2(2m^2 + 2n^2 + 2m + 2n + 1)\) hence multiple of 2	A1FT	2.4
\(4(m^2 + n^2 + m + n) + 2\) \(4(m^2 + n^2 + m + n)\) is multiple of 4, but 2 is not multiple of 4, so never multiple of 4	A1	2.4
	[4]

Notes

B1: Correct form seen for the sum of the squares of any two odd numbers.
ie two different variables

M1: Attempt to square, add and collect like terms for their two distinct odd numbers.
Their odd numbers must both be of the form \(2p \pm q\) (where \(q\) is odd)
May involve a single variable eg \((2n+1)^2 + (2n+3)^2\)
\(= 4n^2 + 4n + 1 + 4n^2 + 12n + 9\)
\(= 8n^2 + 16n + 10\)
Allow sign and/or coefficient errors only

A1FT: Show it is a multiple of 2, by taking out a common multiple or arguing that the coefficients in all terms are even.
FT on their two odd numbers
Factorising by 2 is sufficient for A1 ie no comment required
Condone dividing by 2 to show that the quotient would be an integer

A1: Not dep on previous A1, but must follow B1 M1.
Take out a common multiple from relevant terms, or argue using coefficients of terms, or take out common factor of 4 and argue that remaining factor is not an integer.
Must be from any two odd numbers (ie two different variables)
Condone dividing by 4 to show that the quotient is not an integer
Comment required – either refer to the remainder of 2 (including ‘2 more than a multiple of 4’), or that the entire quotient is not an integer, depending on method used

SC B1 for a complete worded argument about two distinct odd numbers eg odd\(^2\) = odd for both odd numbers; odd + odd = even, hence multiple of 2