| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Polynomial Expansion and Simplification |
| Difficulty | Easy -1.2 This is a straightforward C1 question testing basic algebraic manipulation. Part (i) requires expanding two binomial squares and simplifying—a routine skill. Part (ii) asks for a single coefficient in a polynomial product, which can be found by identifying relevant terms without full expansion. Both parts are mechanical exercises with no problem-solving or conceptual depth required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((3x+1)^2 - 2(2x-3)^2 = (9x^2 + 6x + 1) - 2(4x^2 - 12x + 9) = x^2 + 30x - 17\) | M1, A1, A1, 3 | Square to get at least one 3 or 4 term quadratic: \(9x^2 + 6x + 1\) or \(4x^2 - 12x + 9\) soi; \(x^2 + 30x - 17\) |
| (ii) \(2x^3 + 6x^3 + 4x^3 = 12x^3\); \(12\) | B1, B1, 2 | 2 of \(2x^3\), \(6x^3\), \(4x^3\) soi; N.B. www for these terms, must be positive; \(12\) or \(12x^3\) |
**(i)** $(3x+1)^2 - 2(2x-3)^2 = (9x^2 + 6x + 1) - 2(4x^2 - 12x + 9) = x^2 + 30x - 17$ | M1, A1, A1, 3 | Square to get at least one 3 or 4 term quadratic: $9x^2 + 6x + 1$ or $4x^2 - 12x + 9$ soi; $x^2 + 30x - 17$
**(ii)** $2x^3 + 6x^3 + 4x^3 = 12x^3$; $12$ | B1, B1, 2 | 2 of $2x^3$, $6x^3$, $4x^3$ soi; **N.B. www for these terms, must be positive**; $12$ or $12x^3$2 (i) Simplify $( 3 x + 1 ) ^ { 2 } - 2 ( 2 x - 3 ) ^ { 2 }$.\\
(ii) Find the coefficient of $x ^ { 3 }$ in the expansion of
$$\left( 2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 3 \right) \left( x ^ { 2 } - 2 x + 1 \right)$$
\hfill \mbox{\textit{OCR C1 2006 Q2 [5]}}