| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Single coefficient given directly |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring identification of the constant term using the general term formula, solving a simple equation for k, then finding another coefficient. It's routine application of a standard technique with no conceptual challenges, making it easier than average but not trivial since it requires correct setup of the general term. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Independent term \(= (2x)^3 \times \left(\frac{k}{x}\right)^3 \times {}_6C_3\) | B2,1,0 | Term must be isolated |
| \(= 540 \rightarrow k = 1\frac{1}{2}\) | B1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Term in \(x^2\) is \((2x)^4 \times \left(\frac{k}{x}\right)^2 \times {}_6C_2\) | B1 | All correct – even if \(k\) incorrect |
| \(15 \times 16 \times k^2 = 540\) (or \(540x^2\)) | B1 | FT for \(240k^2\) or \(240k^2x^2\) |
| Total: 2 |
## Question 1:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Independent term $= (2x)^3 \times \left(\frac{k}{x}\right)^3 \times {}_6C_3$ | **B2,1,0** | Term must be isolated |
| $= 540 \rightarrow k = 1\frac{1}{2}$ | **B1** | |
| **Total: 3** | | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Term in $x^2$ is $(2x)^4 \times \left(\frac{k}{x}\right)^2 \times {}_6C_2$ | **B1** | All correct – even if $k$ incorrect |
| $15 \times 16 \times k^2 = 540$ (or $540x^2$) | **B1** | **FT** for $240k^2$ or $240k^2x^2$ |
| **Total: 2** | | |1 The term independent of $x$ in the expansion of $\left( 2 x + \frac { k } { x } \right) ^ { 6 }$, where $k$ is a constant, is 540.\\
(i) Find the value of $k$.\\
(ii) For this value of $k$, find the coefficient of $x ^ { 2 }$ in the expansion.\\
\hfill \mbox{\textit{CAIE P1 2019 Q1 [5]}}