| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Ratio of coefficients condition |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring routine application of the binomial theorem formula and simple algebraic manipulation. Part (i) is pure recall of the formula, and part (ii) involves equating two coefficients and solving a linear equation for k. The question requires no problem-solving insight and is easier than a typical A-level question due to its mechanical nature. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(k^8 + 8k^7x + 28k^6x^2 + 56k^5x^3\) | B3, 2, 1 [3] | Loses 1 for each error. He can gain these marks if appropriate in (ii). |
| (ii) \(28k^6 = 56k^5 \rightarrow k = 2\) | M1 A1 [2] | Correct method of solving. co. nb \(k = 2x\) gets M1 A0. |
$(x + k)^8$
**(i)** $k^8 + 8k^7x + 28k^6x^2 + 56k^5x^3$ | B3, 2, 1 [3] | Loses 1 for each error. He can gain these marks if appropriate in (ii).
**(ii)** $28k^6 = 56k^5 \rightarrow k = 2$ | M1 A1 [2] | Correct method of solving. co. nb $k = 2x$ gets M1 A0.
**Total: [5]**
---2 (i) Find, in terms of the non-zero constant $k$, the first 4 terms in the expansion of $( k + x ) ^ { 8 }$ in ascending powers of $x$.\\
(ii) Given that the coefficients of $x ^ { 2 }$ and $x ^ { 3 }$ in this expansion are equal, find the value of $k$.
\hfill \mbox{\textit{CAIE P1 2009 Q2 [5]}}