| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Moderate -0.3 Part (i) is routine binomial expansion requiring direct application of the formula. Part (ii) requires multiplying two expansions and equating coefficients, which is a standard technique but involves slightly more algebraic manipulation than a basic binomial question. Overall, this is slightly easier than average due to being a well-practiced exam technique with straightforward computation. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((2+5x)^6 = 64 + 960x + 6000x^2\) | M1 | Attempt at least first 2 terms – products of binomial coeff and correct powers of 2 and \(5x\). Must be clear intention to use correct powers of 2 and \(5x\). Binomial coeff must be 6 soi; \(^6C_1\) is not yet enough |
| A1 | Obtain \(64 + 960x\). Allow \(2^6\) for 64 | |
| M1 | Attempt 3rd term – product of binomial coeff and correct powers of 2 and \(5x\). Binomial coeff must be 15 soi; \(^6C_2\) is not yet enough. \(1200x^2\) implies M1 | |
| A1 | Obtain \(6000x^2\). A0 if an otherwise correct expansion is subsequently spoiled by attempt to simplify eg \(4 + 60x + 375x^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((9 + 6cx\ldots)(64 + 960x + \ldots)\) | M1* | Expand first bracket and attempt at least one relevant product. Expansion of first bracket does not have to be correct, but must be attempted so M0 if using \((3+cx)(64+960x\ldots)\) |
| \((9 \times 960) + (6c \times 64) = 4416\) | M1d* | Equate sum of the two relevant terms to 4416 and attempt to solve for \(c\). Must now consider just the two relevant terms. M0 if additional terms, even if error has resulted in \(kx\) |
| \(384c = -4224\) | ||
| \(c = -11\) | A1 | Obtain \(c = -11\). A0 for \(c = -11x\) |
# Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(2+5x)^6 = 64 + 960x + 6000x^2$ | M1 | Attempt at least first 2 terms – products of binomial coeff and correct powers of 2 and $5x$. Must be clear intention to use correct powers of 2 and $5x$. Binomial coeff must be 6 soi; $^6C_1$ is not yet enough |
| | A1 | Obtain $64 + 960x$. Allow $2^6$ for 64 |
| | M1 | Attempt 3rd term – product of binomial coeff and correct powers of 2 and $5x$. Binomial coeff must be 15 soi; $^6C_2$ is not yet enough. $1200x^2$ implies M1 |
| | A1 | Obtain $6000x^2$. A0 if an otherwise correct expansion is subsequently spoiled by attempt to simplify eg $4 + 60x + 375x^2$ |
---
# Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(9 + 6cx\ldots)(64 + 960x + \ldots)$ | M1* | Expand first bracket and attempt at least one relevant product. Expansion of first bracket does not have to be correct, but must be attempted so M0 if using $(3+cx)(64+960x\ldots)$ |
| $(9 \times 960) + (6c \times 64) = 4416$ | M1d* | Equate sum of the two relevant terms to 4416 and attempt to solve for $c$. Must now consider just the two relevant terms. M0 if additional terms, even if error has resulted in $kx$ |
| $384c = -4224$ | | |
| $c = -11$ | A1 | Obtain $c = -11$. A0 for $c = -11x$ |
---3 (i) Find and simplify the first three terms in the expansion of $( 2 + 5 x ) ^ { 6 }$ in ascending powers of $x$.\\
(ii) In the expansion of $( 3 + c x ) ^ { 2 } ( 2 + 5 x ) ^ { 6 }$, the coefficient of $x$ is 4416. Find the value of $c$.
\hfill \mbox{\textit{OCR C2 2013 Q3 [7]}}