TMUA · Paper 2: Mathematical Reasoning

Errors in proofs

11 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

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A student argues: *'Let $a = b$ . Then $a^{2} = ab$ , so $a^{2} - b^{2} = ab - b^{2}$ . Factorising, $(a + b) (a - b) = b (a - b)$ . Dividing by $(a - b)$ gives $a + b = b$ , so $2 b = b$ , hence $2 = 1$ .'* Which step is in…
A student solves $x + 2 = x$ as follows: *'Squaring both sides gives $x + 2 = x^{2}$ , so $(x - 2) (x + 1) = 0$ and $x = 2$ or $x = - 1$ . Both are solutions.'* What is wrong with this argument?
A student argues: *' $n^{2} + n$ equals $2$ , $6$ , $12$ , $20$ when $n = 1, 2, 3, 4$ , which are all even, so $n^{2} + n$ is even for every positive integer $n$ .'* What is wrong with this argument?
A student argues: *'Since $x^{2} > 4$ , taking square roots gives $x > 2$ .'* What is wrong with this argument?
A student argues: *'For $x \neq = 0$ , $\frac{1}{x} < 1$ . Multiplying both sides by $x$ gives $1 < x$ . So $x > 1$ .'* What is wrong with this argument?
A student argues: *'Since $lo g x^{2} = 2 lo g x$ , the equation $lo g x^{2} = 4$ has the same solutions as $2 lo g x = 4$ , namely $x = 100$ (using base 10).'* What is wrong with this argument?
A student argues: 'If it is raining then the ground is wet. The ground is wet. Therefore it is raining.' What is wrong with this argument?
A student argues: *'Since $(- 2)^{2} > (- 1)^{2}$ , it follows that $- 2 > - 1$ .'* What is wrong with this argument?
A student argues: *'The function $f (x) = x^{3}$ has $f^{'} (0) = 0$ , so $f$ has a maximum at $x = 0$ .'* What is wrong with this argument?
A student solves $x^{2} = 2 x$ as follows: *'Dividing both sides by $x$ gives $x = 2$ . So the only solution is $x = 2$ .'* What is wrong with this argument?
A student argues: *' $1 = 1 = (- 1) (- 1) = - 1 \times - 1 = - 1$ .'* Which step is invalid?

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Errors in proofs

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