OCR MEI C3 (Core Mathematics 3)

1 Either prove or disprove each of the following statements.

1. 'If \(m\) and \(n\) are consecutive odd numbers, then at least one of \(m\) and \(n\) is a prime number.'
2. 'If \(m\) and \(n\) are consecutive even numbers, then \(m n\) is divisible by 8 .'

2

1. Disprove the following statement: $$3 ^ { n } + 2 \text { is prime for all integers } n \geqslant 0 .$$
2. Prove that no number of the form \(3 ^ { n }\) (where \(n\) is a positive integer) has 5 as its final digit.

3

1. Factorise fully \(n ^ { 3 } - n\).
2. Hence prove that, if \(n\) is an integer, \(n ^ { 3 } - n\) is divisible by 6 .

4 Prove or disprove the following statement:
'No cube of an integer has 2 as its units digit.'

5 Use the triangle in Fig. 4 to prove that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta = 1\). For what values of \(\theta\) is this proof valid? \begin{figure}[h] \end{figure}

6

1. Multiply out \(\left( 3 ^ { n } + 1 \right) \left( 3 ^ { n } - 1 \right)\).
2. Hence prove that if \(n\) is a positive integer then \(3 ^ { 2 n } - 1\) is divisible by 8 .

7 State whether the following statements are true or false; if false, provide a counterexample.

1. If \(a\) is rational and \(b\) is rational, then \(a + b\) is rational.
2. If \(a\) is rational and \(b\) is irrational, then \(a + b\) is irrational.
3. If \(a\) is irrational and \(b\) is irrational, then \(a + b\) is irrational.

8

1. Disprove the following statement.
   'If \(p > q\), then \(\frac { 1 } { p } < \frac { 1 } { q }\).
2. State a condition on \(p\) and \(q\) so that the statement is true.
3. Show that

(A) \(( x - y ) \left( x ^ { 2 } + x y + y ^ { 2 } \right) = x ^ { 3 } - y ^ { 3 }\),
(B) \(\left( x + \frac { 1 } { 2 } y \right) ^ { 2 } + \frac { 3 } { 4 } y ^ { 2 } = x ^ { 2 } + x y + y ^ { 2 }\).
(ii) Hence prove that, for all real numbers \(x\) and \(y\), if \(x > y\) then \(x ^ { 3 } > y ^ { 3 }\).
(i) Verify the following statement: $$\text { ' } 2 ^ { p } - 1 \text { is a prime number for all prime numbers } p \text { less than } 11 \text { '. }$$ (ii) Calculate \(23 \times 89\), and hence disprove this statement:
' \(2 ^ { p } - 1\) is a prime number for all prime numbers \(p\) '.

11 Use the method of exhaustion to prove the following result.
No 1 - or 2 -digit perfect square ends in \(2,3,7\) or 8
State a generalisation of this result.

12 Prove that the following statement is false.
For all integers \(n\) greater than or equal to \(1 , n ^ { 2 } + 3 n + 1\) is a prime number.

13 Positive integers \(a , b\) and \(c\) are said to form a Pythagorean triple if \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).

1. Given that \(t\) is an integer greater than 1 , show that \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\) form a Pythagorean triple.
2. The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\).