Identity and inversesAQA A-Level Further Maths: Subtopic test

1. 1
   The binary operation $*$ is defined on the set of real numbers $\mathbb{R}$ by $a*b=a+b-3$.

   (a)

   Find the identity element.

   [1 mark]

   • A$0$
   • B$3$
   • C$6$
   • D$-3$

   (b)

   Find the inverse of $7$.

   [1 mark]

   • A$-7$
   • B$\frac17$
   • C$-1$
   • D$4$

   (c)

   Find the element of $\mathbb{R}$ that is its own inverse.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The binary operation $\circ$ is defined on the set of real numbers $\mathbb{R}$ by $a\circ b=\frac{ab}{4}$.

   (a)

   Find the identity element.

   [1 mark]

   • A$1$
   • B$0$
   • C$\frac14$
   • D$4$

   (b)

   Find the inverse of $8$ in the set $\mathbb{R}\setminus\{0\}$.

   [1 mark]

   • A$2$
   • B$\frac12$
   • C$\frac18$
   • D$4$

   (c)

   Explain why the element $0$ has no inverse in $\mathbb{R}$ under $\circ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The set $S=\{1,2,3,4,5,6\}$ under the operation $\otimes$, multiplication modulo 7.

   (a)

   Show that $1$ is the identity element and find the inverse of each element of $S$.

   [3 marks]

   (b)

   Use inverses to solve $3\otimes x=4$ for $x\in S$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The binary operation $*$ is defined on the set $S=\mathbb{R}\setminus\{-1\}$ (all real numbers except $-1$) by $a*b=a+b+ab$.

   (a)

   (i) Show that $0$ is the identity element.
   (ii) Find, in terms of $a$, the inverse of $a\in S$.
   (iii) Show that this inverse is in $S$.

   [6 marks]

   (b)

   (i) Find the inverse of $3$.
   (ii) Solve $3*x=11$.
   (iii) Find all elements of $S$ that are their own inverse.

   [6 marks]

   Total for question 4: 12 marks

End of questions