DP Further Mathematics HL Questionbank

• Syllabus

4.6

Sub sections and their related questions

The identity element $e$.

• 10M.2.hl.TZ0.1a: (i)     Show that $*$ is associative. (ii)      Find the identity element. (iii)     Find...
• 16M.2.hl.TZ0.6b: State the identity in $J$.
• 16M.2.hl.TZ0.6c: Show that (i)     $1 - \sqrt 2$ has an inverse in $J$; (ii)     $2 + 4\sqrt 2$ has no...
• 16M.2.hl.TZ0.6d: Show that the subset, $G$, of elements of $J$ which have inverses, forms a group of infinite...
• 16M.2.hl.TZ0.6e: (i)     Find the inverse of $a + b\sqrt 2$. (ii)     Hence show that ${a^2} - 2{b^2}$...
• 18M.2.hl.TZ0.6a: Find the identity element of S with respect to $*$.
• 18M.2.hl.TZ0.6b: Show that every element of S has an inverse with respect to $*$.
• 18M.2.hl.TZ0.6c: State which elements of S are self-inverse with respect to $*$.

The inverse ${a^{ - 1}}$ of an element $a$.

• 16M.2.hl.TZ0.6b: State the identity in $J$.
• 16M.2.hl.TZ0.6c: Show that (i)     $1 - \sqrt 2$ has an inverse in $J$; (ii)     $2 + 4\sqrt 2$ has no...
• 16M.2.hl.TZ0.6d: Show that the subset, $G$, of elements of $J$ which have inverses, forms a group of infinite...
• 16M.2.hl.TZ0.6e: (i)     Find the inverse of $a + b\sqrt 2$. (ii)     Hence show that ${a^2} - 2{b^2}$...
• 18M.2.hl.TZ0.6a: Find the identity element of S with respect to $*$.
• 18M.2.hl.TZ0.6b: Show that every element of S has an inverse with respect to $*$.
• 18M.2.hl.TZ0.6c: State which elements of S are self-inverse with respect to $*$.

Proof that left-cancellation and right cancellation by an element $a$ hold, provided that $a$ has an inverse.

• 18M.2.hl.TZ0.6a: Find the identity element of S with respect to $*$.
• 18M.2.hl.TZ0.6b: Show that every element of S has an inverse with respect to $*$.
• 18M.2.hl.TZ0.6c: State which elements of S are self-inverse with respect to $*$.

Proofs of the uniqueness of the identity and inverse elements.

• 16M.2.hl.TZ0.6b: State the identity in $J$.
• 16M.2.hl.TZ0.6c: Show that (i)     $1 - \sqrt 2$ has an inverse in $J$; (ii)     $2 + 4\sqrt 2$ has no...
• 16M.2.hl.TZ0.6d: Show that the subset, $G$, of elements of $J$ which have inverses, forms a group of infinite...
• 16M.2.hl.TZ0.6e: (i)     Find the inverse of $a + b\sqrt 2$. (ii)     Hence show that ${a^2} - 2{b^2}$...
• 18M.2.hl.TZ0.6a: Find the identity element of S with respect to $*$.
• 18M.2.hl.TZ0.6b: Show that every element of S has an inverse with respect to $*$.
• 18M.2.hl.TZ0.6c: State which elements of S are self-inverse with respect to $*$.