Proof by deduction, exhaustion and counter-exampleEdexcel A-Level Maths: Subtopic test

1. 1
   A student claims that $n^2+n+41$ is a prime number for every positive integer $n$.

   (a)

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

   [1 mark]

   • A$n=2$
   • B$n=5$
   • C$n=40$
   • D$n=10$

   (b)

   The student checks $n=1,2,3,\ldots,10$ and finds that every value gives a prime. Which statement is correct?

   [1 mark]

   • AThis proves the claim, because ten cases is enough.
   • BThis is not a proof, because a finite number of cases does not cover every positive integer.
   • CThis proves the claim by exhaustion.
   • DThis disproves the claim.

   (c)

   Show that the claim is false for $n=41$, and state what type of proof this is.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f(n)=n^2-6n+10$ is defined for all real $n$.

   (a)

   Which is the completed-square form of $f(n)$?

   [1 mark]

   • A$(n-3)^2+1$
   • B$(n-6)^2+10$
   • C$(n-3)^2+19$
   • D$(n+3)^2+1$

   (b)

   What is the minimum value of $f(n)$ for real $n$?

   [1 mark]

   • A$0$
   • B$3$
   • C$10$
   • D$1$

   (c)

   Hence prove that $n^2-6n+10$ is positive for all real $n$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two consecutive integers are $n$ and $n+1$, where $n$ is an integer.

   (a)

   Prove that the sum of the squares of the two integers is always odd.

   [3 marks]

   (b)

   Prove that the sum of the squares of the two integers is always 1 more than a multiple of 4.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Consider the statement: for every prime number $p>3$, $p^2-1$ is a multiple of 24.

   (a)

   Prove by exhaustion that the statement is true for all primes $p$ with $3<p<25$.

   [6 marks]

   (b)

   Prove by deduction that the statement is true for every prime $p>3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions