Binary operations and their propertiesAQA A-Level Further Maths: Subtopic test

1. 1
   The binary operation $\ast$ is defined on the set of real numbers by $a\ast b=2a+3b$.

   (a)

   Find $4\ast(-1)$.

   [1 mark]

   • A$10$
   • B$5$
   • C$11$
   • D$-24$

   (b)

   Which pair of values shows that $\ast$ is not commutative?

   [1 mark]

   • A$a=2,\ b=2$
   • B$a=0,\ b=0$
   • C$a=-1,\ b=-1$
   • D$a=1,\ b=2$

   (c)

   Show that $\ast$ is not associative.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The binary operation $\otimes$ is defined on the set $\{0,1,2,3,4,5\}$ by $x\otimes y=$ the remainder when $xy$ is divided by 6, that is multiplication modulo 6.

   (a)

   Evaluate $4\otimes5$.

   [1 mark]

   • A$20$
   • B$3$
   • C$2$
   • D$9$

   (b)

   Which values of $x$ satisfy $x\otimes3=3$?

   [1 mark]

   • A$x=1,\ 3,\ 5$
   • B$x=1$ only
   • C$x=1,\ 5$
   • D$x=1,\ 3$

   (c)

   Prove that $\otimes$ is commutative.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The binary operation $\circ$ is defined on the set $S=\{1,2,3,4\}$ by $a\circ b=$ the remainder when $ab$ is divided by 5.

   (a)

   Construct the Cayley table for $\circ$ on $S$, with the first element of $a\circ b$ given by the row.

   [3 marks]

   (b)

   Use the Cayley table to show that $\circ$ is closed and commutative on $S$, and solve $x\circ x=1$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The binary operation $\ast$ is defined on the set of real numbers by $a\ast b=a+b-ab$.

   (a)

   (i) Show that $\ast$ is commutative.
   (ii) Prove that $\ast$ is associative.

   [6 marks]

   (b)

   (i) Show that $a\ast b=1-(1-a)(1-b)$.
   (ii) Hence show that the set of real numbers excluding 1 is closed under $\ast$.
   (iii) Solve $x\ast x=-3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions