Disproof by counter exampleEdexcel International A Level Maths: Subtopic test

1. 1
   A statement T is made: for all real numbers $x$, $x^2>x$.

   (a)

   Which value of $x$ is a counterexample to T?

   [1 mark]

   • A$x=2$
   • B$x=-3$
   • C$x=1$
   • D$x=5$

   (b)

   What is the smallest number of counterexamples needed to disprove T?

   [1 mark]

   • AOne
   • BTwo
   • CThree
   • DEvery real number

   (c)

   Find a value of $x$ with $0<x<1$ and use it to show that T is false.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A student states: 'If $p$ is a prime number, then $p+2$ is also a prime number.'

   (a)

   Which value of $p$ is a counterexample to the statement?

   [1 mark]

   • A$p=3$
   • B$p=5$
   • C$p=11$
   • D$p=7$

   (b)

   The student finds that $p=11$ gives $13$, which is prime. What does this show?

   [1 mark]

   • AThe statement is true for all primes
   • BThe statement is not disproved by $p=11$, but one agreeing case does not prove it
   • CThe statement is false
   • D$p=11$ is a counterexample

   (c)

   Find a counterexample to the statement with $p>10$. Show your working.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the statement U: $n^2-n+1$ is a prime number for all integers $n\ge2$.

   (a)

   Disprove U.

   [3 marks]

   (b)

   A student checks $n=2,3,4$ for U, finds $3,7,13$ (all prime) and concludes that U is true. (i) Explain why this conclusion is not valid. (ii) The student then claims that $n^2+n+1$ is prime for all integers $n\ge2$. Find a counterexample to this claim.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Statements about numbers can be disproved with a single counterexample. A calculator may be used.

   (a)

   Disprove each statement by giving a counterexample and showing that it works. (i) If $a$ and $b$ are real numbers with $a^2=b^2$, then $a=b$. (ii) $2^n+1$ is prime for all positive integers $n$. (iii) $\sqrt{x^2+9}=x+3$ for all real $x$.

   [6 marks]

   (b)

   Dana claims that $(x+1)^2>x^2+1$ for all real $x$. She tests $x=1,2,3$ and $10$ and the claim holds each time. (i) Explain why this is not a proof. (ii) Disprove the claim with a counterexample. (iii) Show that the claim is true only when $x>0$.

   [6 marks]

   Total for question 4: 12 marks

End of questions