Proof by deduction and exhaustionAQA A-Level Maths: Subtopic test

1. 1
   Let $n$ be a positive integer.

   (a)

   Which expression is always even?

   [1 mark]

   • A$2n+1$
   • B$n^2+1$
   • C$n^2+n$
   • D$3n$

   (b)

   Which expression represents the product of two consecutive odd positive integers?

   [1 mark]

   • A$(2n-1)(2n+1)$
   • B$2n(2n+1)$
   • C$(2n+1)(2n+2)$
   • D$(2n+1)^2$

   (c)

   Show that the sum of the three consecutive integers $n$, $n+1$ and $n+2$ is a multiple of 3.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Every integer has the form $3k$, $3k+1$ or $3k+2$, where $k$ is an integer.

   (a)

   Expanding $(3k+1)^2$, which form does the result take, for some integer $m$?

   [1 mark]

   • A$3m$
   • B$3m+2$
   • C$6m+1$
   • D$3m+1$

   (b)

   What is the remainder when $(3k+2)^2$ is divided by 3?

   [1 mark]

   • A0
   • B1
   • C2
   • D4

   (c)

   Hence prove that the square of any integer is either a multiple of 3 or one more than a multiple of 3.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student wants to prove that $n^3-n$ is divisible by 6 for every positive integer $n$.

   (a)

   Factorise $n^3-n$ fully and use your factorisation to explain why it is always even.

   [3 marks]

   (b)

   Hence complete the proof that $n^3-n$ is divisible by 6 for every positive integer $n$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Work with integers throughout. A square number is the square of an integer.

   (a)

   (i) Prove that the sum of the squares of two consecutive integers is odd. (ii) Show that the sum of the squares of three consecutive integers is never a multiple of 3.

   [6 marks]

   (b)

   Prove by exhaustion that no square number ends in the digit 2, 3, 7 or 8.

   [6 marks]

   Total for question 4: 12 marks

End of questions