TMUA · Paper 2: Mathematical Reasoning

Counterexamples

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

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A student claims: *'For every positive integer $n$ , $n^{2} - n + 11$ is prime.'* Which value of $n$ is a counterexample?
A student claims: *'If $p$ is a prime number, then $p + 4$ is also prime.'* Which value of $p$ is a counterexample?
A student claims: *'If $x > y$ , then $x^{2} > y^{2}$ .'* Which pair $(x, y)$ is a counterexample?
A student claims: 'If a whole number is divisible by both 4 and 6, then it is divisible by 24.' Which number is a counterexample?
A student claims: *'If a real number $x$ satisfies $x^{2} > x$ , then $x > 1$ .'* Which value of $x$ is a counterexample?
A student claims: *'If $a$ and $b$ are irrational numbers, then $a + b$ is irrational.'* Which pair $(a, b)$ is a counterexample?
A student claims: *'Every prime number greater than 3 can be written as $6 k + 1$ for some whole number $k$ .'* Which prime is a counterexample?
A student claims: 'Every cubic polynomial has at least one stationary point.' Which cubic is a counterexample?
A student claims: *'If $x > 0$ , then $x + \frac{1}{x} > 2$ .'* Which value of $x$ is a counterexample?
A student claims: *'If $n$ is a positive integer, then $2^{n} > n + 1$ .'* Which value of $n$ is a counterexample?
A student claims: *'If a sequence of positive terms satisfies $u_{n + 1} < u_{n}$ for all $n$ , then $u_{n} \to 0$ as $n \to \infty$ .'* Which sequence is a counterexample?
A student claims: *'If $a > 0$ , then the quadratic $a x^{2} + b x + c$ is positive for all real $x$ .'* Which quadratic is a counterexample?
A student claims: *'If $a^{2} = b^{2}$ , then $a = b$ .'* Which pair $(a, b)$ is a counterexample?
A student claims that $n^{2} + n + 41$ is prime for every positive integer $n$ . What is the smallest value of $n$ for which this is false?
For how many of the integers $n = 1, 2, \dots, 20$ is $n^{2} - n + 11$ not prime?
A student claims: *'If $n$ is a positive integer and $n^{2} + n$ is divisible by 6, then $n$ is divisible by 3.'* Which value of $n$ is a counterexample?

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Counterexamples

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