Structure of proof and proof by exhaustionEdexcel International A Level Maths: Subtopic test

1. 1
   A statement S is made: for every integer $n$ with $1\le n\le5$, the value of $n^2+n$ is even.

   (a)

   Which of the following is needed for a proof by exhaustion of S?

   [1 mark]

   • AShowing that S is true for $n=3$ only
   • BShowing that S is true for each of $n=1,2,3,4,5$
   • CFinding one value of $n$ for which S is true
   • DShowing that S is true for $n=1$ and $n=5$, and assuming the values in between must also work

   (b)

   Find the value of $n^2+n$ when $n=4$.

   [1 mark]

   • A$16$
   • B$12$
   • C$25$
   • D$20$

   (c)

   Prove that S is true by exhaustion.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $x$ and $y$ are odd positive integers, each less than 7.

   (a)

   How many different ordered pairs $(x,y)$ are possible?

   [1 mark]

   • A$3$
   • B$6$
   • C$9$
   • D$27$

   (b)

   Which of the following is a possible value of $x+y$?

   [1 mark]

   • A$10$
   • B$7$
   • C$12$
   • D$9$

   (c)

   Prove that $x+y$ is divisible by $2$ for every possible pair.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(n)=n^3-n$, where $n$ is an integer.

   (a)

   Use proof by exhaustion to show that $f(n)$ is divisible by $6$ for every integer $n$ with $2\le n\le6$.

   [3 marks]

   (b)

   A student claims that $f(n)$ is divisible by $12$ for every integer $n$ with $2\le n\le6$. Test this claim, and explain what your result shows about proof by exhaustion.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Maya tests two statements about integers by checking every case. A calculator may be used.

   (a)

   Statement 1: for every odd integer $n$ with $3\le n\le9$, $n^2-1$ is divisible by $8$. (i) Use proof by exhaustion to prove Statement 1. (ii) Explain why this does not prove that $n^2-1$ is divisible by $8$ for every odd integer $n$.

   [6 marks]

   (b)

   Statement 2: $n^2+n+11$ is prime for every integer $n$ with $0\le n\le10$. Use proof by exhaustion to determine whether Statement 2 is true or false. Justify your answer fully.

   [6 marks]

   Total for question 4: 12 marks

End of questions