OCR MEI C3 (Core Mathematics 3)

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]
2. 'If \(m\) and \(n\) are consecutive even numbers, then \(mn\) is divisible by 8.' [2]

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

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

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? [3] \includegraphics{figure_4}

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

State whether the following statements are true or false; if false, provide a counter-example.

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. [3]

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

1. Show that
   1. \((x - y)(x^2 + xy + y^2) = x^3 - y^3\),
   2. \((x + \frac{1}{2}y)^2 + \frac{3}{4}y^2 = x^2 + xy + y^2\). [4]
2. Hence prove that, for all real numbers \(x\) and \(y\), if \(x > y\) then \(x^3 > y^3\). [3]

1. Verify the following statement: $$\text{'} 2^p - 1 \text{ is a prime number for all prime numbers } p \text{ less than 11'.} [2]
2. Calculate \(23 \times 89\), and hence disprove this statement: $$\text{'} 2^p - 1 \text{ is a prime number for all prime numbers } p \text{'.} [2]

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. [3]

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

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 \(2t\), \(t^2 - 1\) and \(t^2 + 1\) form a Pythagorean triple. [3]
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 \(2t\), \(t^2 - 1\) and \(t^2 + 1\). [3]