1.01a Proof: structure of mathematical proof and logical steps

Prove that \(8^n - 7n + 6\) is divisible by 7 for all integers \(n \geq 0\) [5 marks]

A student claims: "Given any two non-zero square matrices, A and B, then \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\)"

1. Explain why the student's claim is incorrect giving a counter example. [2 marks]
2. Refine the student's claim to make it fully correct. [1 mark]
3. Prove that your answer to part (b) is correct. [3 marks]

Given that the vectors a and b are perpendicular, prove that \(|(\mathbf{a} + 5\mathbf{b}) \times (\mathbf{a} - 4\mathbf{b})| = k|\mathbf{a}||\mathbf{b}|\), where \(k\) is an integer to be found. Explicitly state any properties of the vector product that you use within your proof. [9 marks]

Given that \(n\) is an integer such that \(1 \leq n \leq 4\), prove that \(2n^2 + 5\) is a prime number. [3]

Prove that \(f(x) = x^3 - 6x^2 + 13x - 7\) is an increasing function. [4]

In each of the two statements below, \(x\) and \(y\) are real numbers. One of the statements is true while the other is false. A: \(x^2 + y^2 \geqslant 2xy\), for all real values of \(x\) and \(y\). B: \(x + y \geqslant 2\sqrt{xy}\), for all real values of \(x\) and \(y\).

1. Identify the statement which is false. Find a counter example to show that this statement is in fact false. [3]
2. Identify the statement which is true. Give a proof to show that this statement is in fact true. [2]

Given that \(n\) is an integer such that \(1 \leqslant n \leqslant 6\), use proof by exhaustion to show that \(n^2 - 2\) is not divisible by 3. [3]

Prove that \(x - 10 < x^2 - 5x\) for all real values of \(x\). [4]

In each of the two statements below, \(c\) and \(d\) are real numbers. One of the statements is true while the other is false. A: Given that \((2c + 1)^2 = (2d + 1)^2\), then \(c = d\). B: Given that \((2c + 1)^3 = (2d + 1)^3\), then \(c = d\).

1. Identify the statement which is false. Find a counter example to show that this statement is in fact false.
2. Identify the statement which is true. Give a proof to show that this statement is in fact true. [5]

The quadratic equation \(4x^2 - 12x + m = 0\), where \(m\) is a positive constant, has two distinct real roots. Show that the quadratic equation \(3x^2 + mx + 7 = 0\) has no real roots. [7]

Prove that $$\log_a a \times \log_a 19 = \log_a 19$$ whatever the value of the positive constant \(a\). [3]

Prove that \(\sqrt{2}\cos(2\theta + 45°) \equiv \cos^2\theta - 2\sin\theta\cos\theta - \sin^2\theta\), where \(\theta\) is measured in degrees. [3]

Shona makes the following claim. "\(n\) is an even positive integer greater than \(2 \Rightarrow 2^n - 1\) is not prime" Prove that Shona's claim is true. [4]

In this question you must show detailed reasoning.

1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(e^x > 1 + x\). [2]
2. Hence, by using a suitable substitution, deduce that \(e^t > et\) for \(t > 1\). [1]
3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(e^{\pi}\) or \(\pi^e\). [3]