P4: ProofEdexcel International A Level Maths: Topic test

1. 1
   A proof that $\sqrt3$ is irrational begins: Assume that $\sqrt3$ is rational, so $\sqrt3=\frac{p}{q}$, where $p$ and $q$ are positive integers with highest common factor $1$.

   (a)

   Which equation follows from this assumption?

   [1 mark]

   • A$p^{2}=3q$
   • B$p^{2}=3q^{2}$
   • C$p=3q$
   • D$p^{2}=9q^{2}$

   (b)

   Which conclusion about $p$ follows from $p^{2}=3q^{2}$?

   [1 mark]

   • A$p$ is even
   • B$p$ is a multiple of $9$
   • C$p=3q$
   • D$p$ is a multiple of $3$

   (c)

   Complete the proof that $\sqrt3$ is irrational.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the statement: there are no integers $x$ and $y$ such that $9x+15y=1$.

   (a)

   Which of the following is the correct first line of a proof of this statement by contradiction?

   [1 mark]

   • AAssume that there are integers $x$ and $y$ such that $9x+15y=1$.
   • BAssume that $9x+15y=1$ for all integers $x$ and $y$.
   • CAssume that $9x+15y\ne1$ for all integers $x$ and $y$.
   • DAssume that $x$ and $y$ are not integers.

   (b)

   Which of the following completes the proof?

   [1 mark]

   • A$9x+15y$ is a multiple of $5$, but $1$ is not a multiple of $5$
   • B$9x+15y$ is even, but $1$ is odd
   • C$9x+15y=3(3x+5y)$ is a multiple of $3$, but $1$ is not a multiple of $3$
   • D$9x+15y$ is a multiple of $9$, but $1$ is not a multiple of $9$

   (c)

   Use a similar method to prove that there are no integers $x$ and $y$ such that $14x+21y=5$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   You may use the fact that $\sqrt5$ is irrational. Let $q$ be a rational number.

   (a)

   Prove by contradiction that $q+\sqrt5$ is irrational.

   [3 marks]

   (b)

   Prove by contradiction that $\dfrac{3+\sqrt5}{2}$ is irrational.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question $x=\log_{2}3$.

   (a)

   Prove by contradiction that $x$ is irrational.

   [6 marks]

   (b)

   (i) Show that $\log_{2}12=2+x$.
   (ii) Prove by contradiction that $\log_{2}12$ is irrational.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Consider the statement: there are no integers $m$ and $n$ such that $m^{2}-n^{2}=10$.

   (a)

   Assume that there are such integers $m$ and $n$. Which of the following must be true of $m-n$ and $m+n$?

   [1 mark]

   • AOne is even and the other is odd
   • BThey are both always even
   • CThey are either both even or both odd
   • DThey are both always odd

   (b)

   Which of the following completes the proof?

   [1 mark]

   • A$(m-n)(m+n)=10$; if both factors are even the product is a multiple of $4$, and if both are odd the product is odd; neither can equal $10$
   • B$(m-n)(m+n)=10$; in both cases the product is odd, so it cannot equal $10$
   • C$(m-n)(m+n)=10$; in both cases the product is a multiple of $4$, so it cannot equal $10$
   • D$(m-n)(m+n)=10$; both factors are even, so the product is a multiple of $8$, which $10$ is not

   (c)

   Prove by contradiction that there are no integers $m$ and $n$ such that $m^{2}-n^{2}=4k+2$ for any integer $k$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Consider the statement: there is no smallest positive rational number.

   (a)

   Which of the following is the correct first line of a proof of this statement by contradiction?

   [1 mark]

   • AAssume that there is a largest positive rational number.
   • BAssume that there is no smallest positive rational number.
   • CAssume that $1$ is the smallest positive rational number.
   • DAssume that there is a smallest positive rational number, $r$.

   (b)

   Which positive rational number is smaller than $r$ and so gives the contradiction?

   [1 mark]

   • A$2r$
   • B$\dfrac{r}{2}$
   • C$r^{2}$
   • D$r-1$

   (c)

   Use the same method to prove that there is no largest even integer.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $n$ be an integer. Statement P is: if $n^{2}$ is even, then $n$ is even.

   (a)

   Prove statement P by contradiction.

   [3 marks]

   (b)

   Use statement P to prove by contradiction that $\sqrt6$ is irrational.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   In this question you may use the fact that the cube of an integer is odd if and only if the integer is odd. In each part, $p$ and $q$ are integers with $q>0$ and highest common factor $1$.

   (a)

   Prove by contradiction that the equation $x^{3}+x+1=0$ has no rational roots.

   [6 marks]

   (b)

   Prove by contradiction that $\sqrt[3]{2}$ is irrational.

   [6 marks]

   Total for question 8: 12 marks

End of questions