| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Ratio of coefficients condition |
| Difficulty | Moderate -0.3 Part (i) is straightforward application of binomial expansion for n=3. Part (ii) requires setting up and solving a simple equation from coefficient comparison, leading to a quadratic. This is a standard C2 exercise with clear structure and routine algebraic manipulation, making it slightly easier than average but not trivial due to the two-part nature and algebraic solving required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3^3 + (3 \times 3^2 \times kx) + (3 \times 3 \times (kx)^2) + (kx)^3\) | M1 | Attempt expansion. Must attempt at least 3 of the 4 terms. Each term must be an attempt at the product of the relevant binomial coeff, the correct power of 3 and the correct power of \(kx\). Allow M1 if powers used incorrectly with \(kx\) ie only applied to \(x\) and not to \(k\) as well. Binomial coeff must be numerical, so \(^3C_2\) is M0 until evaluated. Allow M1 for expanding \(c(1 + \frac{kx}{3})^3\), any \(c\) |
| \(= 27 + 27kx + 9k^2x^2 + k^3x^3\) | A1 | Obtain at least two correct terms. Allow \(3^3\) for 27 and \(3^2\) for 9 |
| A1 | Obtain at least one further correct term | |
| A1 | Obtain fully correct simplified expansion. Must now be 27 and 9, not still index notation. Must be a correct expansion, with terms linked by '+' rather than just a list of 4 terms. No ISW if correct final answer is subsequently spoiled by attempt to 'simplify' eg dividing by 27 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(9k^2 = 27\) | M1 | Equate their coeff of \(x^2\) to their constant term and attempt to solve for \(k\). Must be equating coefficients not terms. Allow recovery if next line is \(k^2 = 3\), but M0 if \(x^2\) still present at this stage. Must attempt \(k\), but allow if only positive square root is considered |
| \(k^2 = 3\) | ||
| \(k = \pm\sqrt{3}\) | A1 | Must have \(\pm\), or two roots listed separately. Final answer must be given in exact form. A0 for \(\pm\sqrt{\frac{27}{9}}\). Must come from correct coefficients only. SR allow B1 if \(k = \pm\sqrt{3}\) is given as final answer, but inconsistent use of terms/coefficients within solution |
# Question 3:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3^3 + (3 \times 3^2 \times kx) + (3 \times 3 \times (kx)^2) + (kx)^3$ | M1 | Attempt expansion. Must attempt at least 3 of the 4 terms. Each term must be an attempt at the product of the relevant binomial coeff, the correct power of 3 and the correct power of $kx$. Allow M1 if powers used incorrectly with $kx$ ie only applied to $x$ and not to $k$ as well. Binomial coeff must be numerical, so $^3C_2$ is M0 until evaluated. Allow M1 for expanding $c(1 + \frac{kx}{3})^3$, any $c$ |
| $= 27 + 27kx + 9k^2x^2 + k^3x^3$ | A1 | Obtain at least two correct terms. Allow $3^3$ for 27 and $3^2$ for 9 |
| | A1 | Obtain at least one further correct term |
| | A1 | Obtain fully correct simplified expansion. Must now be 27 and 9, not still index notation. Must be a correct expansion, with terms linked by '+' rather than just a list of 4 terms. No ISW if correct final answer is subsequently spoiled by attempt to 'simplify' eg dividing by 27 |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $9k^2 = 27$ | M1 | Equate their coeff of $x^2$ to their constant term and attempt to solve for $k$. Must be equating coefficients not terms. Allow recovery if next line is $k^2 = 3$, but M0 if $x^2$ still present at this stage. Must attempt $k$, but allow if only positive square root is considered |
| $k^2 = 3$ | | |
| $k = \pm\sqrt{3}$ | A1 | Must have $\pm$, or two roots listed separately. Final answer must be given in exact form. A0 for $\pm\sqrt{\frac{27}{9}}$. Must come from correct coefficients only. **SR** allow **B1** if $k = \pm\sqrt{3}$ is given as final answer, but inconsistent use of terms/coefficients within solution |
---3 (i) Find the binomial expansion of $( 3 + k x ) ^ { 3 }$, simplifying the terms.\\
(ii) It is given that, in the expansion of $( 3 + k x ) ^ { 3 }$, the coefficient of $x ^ { 2 }$ is equal to the constant term. Find the possible values of $k$, giving your answers in an exact form.
\hfill \mbox{\textit{OCR C2 2016 Q3 [6]}}