P2: ProofEdexcel International A Level Maths: Topic test

1. 1
   Statement $P$: for every integer $n$ with $1\le n\le5$, the number $n^2+2$ is not a multiple of $4$.

   (a)

   How many different values of $n$ must be checked to prove $P$ by exhaustion?

   [1 mark]

   • A$4$
   • B$5$
   • C$6$
   • D$25$

   (b)

   Which list gives the values of $n^2+2$ for $n=1,2,3,4,5$ in order?

   [1 mark]

   • A$3,\ 6,\ 11,\ 18,\ 27$
   • B$3,\ 8,\ 13,\ 18,\ 23$
   • C$9,\ 16,\ 25,\ 36,\ 49$
   • D$2,\ 8,\ 18,\ 32,\ 50$

   (c)

   Prove statement $P$ by exhaustion.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Statement $Q$: $n^2+n+11$ is a prime number for every positive integer $n$.

   (a)

   Which value of $n$ shows that $Q$ is false?

   [1 mark]

   • A$5$
   • B$7$
   • C$8$
   • D$10$

   (b)

   Which of the following would be sufficient to disprove $Q$?

   [1 mark]

   • AShowing that $n^2+n+11$ is prime for $n=1,2,\dots,9$
   • BShowing that $n^2+n+11$ is odd for every positive integer $n$
   • CFinding one positive integer $n$ for which $n^2+n+11$ is not prime
   • DShowing that $n^2+n+11$ is prime for a very large value of $n$

   (c)

   Show that $n=11$ is also a counterexample to statement $Q$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Every integer $n$ can be written in exactly one of the forms $3k$, $3k+1$ or $3k+2$, where $k$ is an integer.

   (a)

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

   [3 marks]

   (b)

   A student makes two claims about the sum of the squares of any three consecutive integers. Claim $X$: the sum is always a multiple of $3$. Claim $Y$: the sum is always even. Disprove each claim by giving a counterexample.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $x$ and $y$ are positive integers with $x+y=10$.

   (a)

   Prove by exhaustion that $xy<30$ for every such pair.

   [6 marks]

   (b)

   Two statements are made about these integers. Statement $1$: $xy$ is a multiple of $3$. Statement $2$: $x^2+y^2>40$. Determine whether each statement is true or false, justifying your answers, and explain why different methods are needed to settle them.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The number of diagonals of a polygon with $n$ sides is $\frac12n(n-3)$. Statement $R$: for every integer $n$ with $3\le n\le6$, a polygon with $n$ sides has fewer than $2n$ diagonals.

   (a)

   How many diagonals does a polygon with $5$ sides have?

   [1 mark]

   • A$10$
   • B$2$
   • C$5$
   • D$8$

   (b)

   For which value of $n$ in the range $3\le n\le6$ is the number of diagonals closest to $2n$?

   [1 mark]

   • A$3$
   • B$4$
   • C$5$
   • D$6$

   (c)

   Prove statement $R$ by exhaustion.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A student states that $\sqrt{x}<x$ for every real number $x>0$.

   (a)

   Which value of $x$ is a counterexample to the student's statement?

   [1 mark]

   • A$0.25$
   • B$4$
   • C$16$
   • D$2.25$

   (b)

   For which of the following statements, each made for all $x>0$, is $x=0.25$ a counterexample?

   [1 mark]

   • A$x<\sqrt{x}$
   • B$x^2>x$
   • C$x^3<x^2$
   • D$\dfrac1x>x$

   (c)

   The student tests $x=4$, $x=9$ and $x=16$, finds that $\sqrt{x}<x$ each time, and concludes that the statement is true. Explain why this is not a valid proof, and what would be enough to show the statement is false.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Statement $T$: for every prime number $p$ with $5\le p<20$, $p^2-1$ is a multiple of $24$.

   (a)

   Prove statement $T$ by exhaustion.

   [3 marks]

   (b)

   A student claims that (i) $p^2-1$ is a multiple of $24$ for every integer $p>3$, and (ii) $p^2-1$ is a multiple of $48$ for every prime $p\ge5$. Disprove each claim with a counterexample.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Statement $M$: $2^n-1$ is a prime number for every prime number $n$.

   (a)

   Show that statement $M$ is true for $n=2,3,5$ and $7$, justifying that each value of $2^n-1$ is prime.

   [6 marks]

   (b)

   (i) Disprove statement $M$ by considering $n=11$. [3 marks] (ii) A student claims that $2^n-1$ is prime for every integer $n\ge2$. Find the smallest value of $n$ that shows this claim is false, justifying that it is the smallest. [3 marks]

   [6 marks]

   Total for question 8: 12 marks

End of questions