Disproof by counter-exampleAQA A-Level Maths: Subtopic test

1. 1
   A student claims: 'For every positive integer $n$, the number $n^2+n+41$ is prime.'

   (a)

   Which value of $n$ is a counter-example to the claim?

   [1 mark]

   • A$n=1$
   • B$n=10$
   • C$n=39$
   • D$n=40$

   (b)

   What is the value of $n^2+n+41$ when $n=40$?

   [1 mark]

   • A$1763$
   • B$1681$
   • C$1601$
   • D$1640$

   (c)

   The student first tested $n=1,2,3,4,5$ and found a prime each time. Explain why this did not prove the claim, and what is needed to disprove it.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider two statements. Statement P: if $x$ and $y$ are irrational numbers then $x+y$ is irrational. Statement Q: if $x$ and $y$ are irrational numbers then $xy$ is irrational.

   (a)

   Which pair of numbers is a counter-example to statement P?

   [1 mark]

   • A$\sqrt2$ and $\sqrt3$
   • B$\sqrt2$ and $2$
   • C$\sqrt2$ and $-\sqrt2$
   • D$\pi$ and $1$

   (b)

   Which pair of numbers is a counter-example to statement Q?

   [1 mark]

   • A$\sqrt2$ and $\sqrt2$
   • B$\sqrt2$ and $\sqrt3$
   • C$\pi$ and $2$
   • D$\sqrt2$ and $3$

   (c)

   Give a counter-example to statement P in which $y\ne-x$, and show that it is a counter-example.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the statement: for every real number $x$, $x^2>x$.

   (a)

   Show that the statement is false by finding a counter-example and explaining clearly why it works.

   [3 marks]

   (b)

   Find the complete set of counter-examples, and write down a corrected statement that is true.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Four claims are made about positive integers. Claim 1: the sum of two prime numbers is even. Claim 2: $n^3>n^2$ for every $n$. Claim 3: if $n$ is prime then $2^n-1$ is prime. Claim 4: if $n$ is odd then $n^2-1$ is a multiple of 8.

   (a)

   Claims 1, 2 and 3 are false. For each, give a counter-example and show that it works.

   [6 marks]

   (b)

   A student tests $n=1,3,5,7,9$ for Claim 4 and says it is proved. (i) Explain why this is not a proof. (ii) Prove Claim 4. (iii) Explain why no counter-example to Claim 4 can exist.

   [6 marks]

   Total for question 4: 12 marks

End of questions