| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Challenging +1.2 This is a standard FP3 group theory question requiring proof of closure, identification of identity, and finding inverses by rationalizing denominators. While it involves abstract algebra (inherently harder than single maths), the techniques are routine for Further Maths students: multiply out brackets, rationalize using conjugates, and recognize that 5 being non-square ensures non-zero denominators. The conceptual demand is moderate but execution is straightforward. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group |
$H$ denotes the set of numbers of the form $a + b\sqrt{5}$, where $a$ and $b$ are rational. The numbers are combined under multiplication.
(i) Show that the product of any two members of $H$ is a member of $H$. [2]
It is now given that, for $a$ and $b$ not both zero, $H$ forms a group under multiplication.
(ii) State the identity element of the group. [1]
(iii) Find the inverse of $a + b\sqrt{5}$. [2]
(iv) With reference to your answer to part (iii), state a property of the number 5 which ensures that every number in the group has an inverse. [1]
\hfill \mbox{\textit{OCR FP3 2010 Q2 [6]}}