| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Binomial times linear coefficient |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring routine application of the binomial theorem formula for part (i), followed by simple algebraic multiplication in part (ii). The question involves standard techniques with no problem-solving insight needed, making it easier than average but not trivial since it requires careful coefficient tracking across two steps. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.04b Binomial probabilities: link to binomial expansion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain \(1 + 14x\) | B1 | Needs to be simplified, so \(1\) not \(1^7\) and \(14x\) not \(7 \times 2x\). B0 if other constant and/or \(x\) terms. Must be linked by \(+\) sign. |
| Attempt third term | M1 | Needs to be product of \(21\) and an attempt at squaring \(2x\) — allow even if brackets never seen, so \(42x^2\) gets M1. |
| Obtain \(84x^2\) | A1 (3) | Coefficient needs to be simplified. Ignore any further terms, right or wrong. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt at least one relevant product | M1 | Could be a single term or part of fuller expansion. Allow M1 even if second \(x^2\) term isn't from a relevant product. e.g. \(-70 + 84\) gets M1 A0. |
| Obtain two correct unsimplified terms (not necessarily summed) — either coefficients or still with powers of \(x\) | A1ft | Unsimplified terms e.g. \(-5x \times 14x\ldots\) If fuller expansion then A0 if other \(x^2\) terms present. |
| Obtain \(98\) | A1 (3) | Allow \(98x^2\). Allow if part of fuller expansion and not explicitly picked out. |
# Question 1:
## Part (i): $(1+2x)^7 = 1 + 14x + 84x^2$
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain $1 + 14x$ | B1 | Needs to be simplified, so $1$ not $1^7$ and $14x$ not $7 \times 2x$. B0 if other constant and/or $x$ terms. Must be linked by $+$ sign. |
| Attempt third term | M1 | Needs to be product of $21$ and an attempt at squaring $2x$ — allow even if brackets never seen, so $42x^2$ gets M1. |
| Obtain $84x^2$ | A1 (3) | Coefficient needs to be simplified. Ignore any further terms, right or wrong. |
## Part (ii): $(2-5x)(1+14x+84x^2)$, coeff of $x^2 = -70 + 168 = 98$
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt at least one relevant product | M1 | Could be a single term or part of fuller expansion. Allow M1 even if second $x^2$ term isn't from a relevant product. e.g. $-70 + 84$ gets M1 A0. |
| Obtain two correct unsimplified terms (not necessarily summed) — either coefficients or still with powers of $x$ | A1ft | Unsimplified terms e.g. $-5x \times 14x\ldots$ If fuller expansion then A0 if other $x^2$ terms present. |
| Obtain $98$ | A1 (3) | Allow $98x^2$. Allow if part of fuller expansion and not explicitly picked out. |
---1 (i) Find and simplify the first three terms, in ascending powers of $x$, in the binomial expansion of $( 1 + 2 x ) ^ { 7 }$.\\
(ii) Hence find the coefficient of $x ^ { 2 }$ in the expansion of $( 2 - 5 x ) ( 1 + 2 x ) ^ { 7 }$.
\hfill \mbox{\textit{OCR C2 2011 Q1 [6]}}