| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Proper subgroups identification |
| Difficulty | Standard +0.8 This is a Further Maths group theory question requiring understanding of polynomial groups with modular arithmetic. Parts (i)-(iii) are straightforward applications of definitions, but part (iv) requires generating a subgroup and recognizing it forms a vector space over Z_5, which demands structural insight beyond routine calculation. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group8.03e Order of elements: and order of groups8.03f Subgroups: definition and tests for proper subgroups |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(0\) | B1 | accept \(0 + 0x + 0x^2\) |
| [1] | ||
| (ii) \(2 + 3x + 4x^2\) | M1 | for 2 correct terms |
| A1 | ||
| [2] | ||
| (iii) \(125\) | B1 | or \(5^3\) |
| [1] | ||
| (iv) more than five elements are shown to be generated so \( | H | > 5\) |
| \( | H | \) is a factor of 125 |
| proper so \( | H | < 125\) |
| \( | H | = 25\) |
| [4] |
**(i)** $0$ | B1 | accept $0 + 0x + 0x^2$
| [1] |
**(ii)** $2 + 3x + 4x^2$ | M1 | for 2 correct terms
| A1 |
| [2] |
**(iii)** $125$ | B1 | or $5^3$
| [1] |
**(iv)** more than five elements are shown to be generated so $|H| > 5$ | B1 | e.g. elements generated by $1+x$ are $\{1+x, 2+2x, 3+3x, 4+4x, 0\}$ which does not include $2+x$
$|H|$ is a factor of 125 | B1 | or order subgroups 1, 5, 25 or 125
proper so $|H| < 125$ | B1 | or order is (1), 5, 25
$|H| = 25$ | B1 | penalise use of $H$ instead of $|H|$
| [4] |2 The elements of a group $G$ are polynomials of the form $a + b x + c x ^ { 2 }$, where $a , b , c \in \{ 0,1,2,3,4 \}$. The group operation is addition, where the coefficients are added modulo 5 .\\
(i) State the identity element.\\
(ii) State the inverse of $3 + 2 x + x ^ { 2 }$.\\
(iii) State the order of $G$.
The proper subgroup $H$ contains $2 + x$ and $1 + x$.\\
(iv) Find the order of $H$, justifying your answer.
\hfill \mbox{\textit{OCR FP3 2015 Q2 [8]}}