| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with reciprocal term binomial |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring identification of the correct term using the binomial theorem formula. Part (i) is routine application with r=2 giving the x term. Part (ii) adds one simple multiplication step but requires no novel insight—just careful bookkeeping of which terms from part (i) contribute to the final x coefficient. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Coefficient of \(x = 80(x)\) | B2 | Correct value must be selected for both marks. SR +80 seen in an expansion gets B1 or \(-80\) gets B1 if selected. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Coefficient of \(\frac{1}{x} = -40\left(\frac{1}{x}\right)\) | B2 | Correct value soi in (ii), if powers unsimplified only allow if selected. SR +40 soi in (ii) gets B1. |
| Coefficient of \(x = (1 \times \text{their } 80) + (3 \times \text{their } -40) = -40(x)\) | M1 A1 | Links the appropriate 2 terms only for M1. |
## Question 1:
**Part (i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Coefficient of $x = 80(x)$ | **B2** | Correct value must be selected for both marks. SR +80 seen in an expansion gets **B1** or $-80$ gets **B1** if selected. |
**Part (ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Coefficient of $\frac{1}{x} = -40\left(\frac{1}{x}\right)$ | **B2** | Correct value soi in (ii), if powers unsimplified only allow if selected. SR +40 soi in (ii) gets **B1**. |
| Coefficient of $x = (1 \times \text{their } 80) + (3 \times \text{their } -40) = -40(x)$ | **M1 A1** | Links the appropriate 2 terms only for **M1**. |
---1 (i) Find the coefficient of $x$ in the expansion of $\left( 2 x - \frac { 1 } { x } \right) ^ { 5 }$.\\
(ii) Hence find the coefficient of $x$ in the expansion of $\left( 1 + 3 x ^ { 2 } \right) \left( 2 x - \frac { 1 } { x } \right) ^ { 5 }$.\\
\hfill \mbox{\textit{CAIE P1 2017 Q1 [6]}}