| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with reciprocal term binomial |
| Difficulty | Moderate -0.3 This is a straightforward binomial expansion question requiring identification of the correct term in part (i) using the binomial theorem formula, then a simple multiplication in part (ii). While it involves negative powers and careful algebraic manipulation, it's a standard textbook exercise with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Coeff of \(x^2\) is \(15 \times 16 \times (-\frac{1}{2})^2 = 60\) | B1 B1 M1 A1 | B1 for 2/3 parts. B1 unsimplified. Needs to consider the constant term |
| (ii) Constant term is \(20 \times 8x^3 \times (-1 + 8x^3) \times (1 + x^2)\) needs to consider 2 terms → \(60 - 20 = 40\) | [3] |
$\left(2x - \frac{1}{2x}\right)^6$
(i) Coeff of $x^2$ is $15 \times 16 \times (-\frac{1}{2})^2 = 60$ | B1 B1 M1 A1 | B1 for 2/3 parts. B1 unsimplified. Needs to consider the constant term
(ii) Constant term is $20 \times 8x^3 \times (-1 + 8x^3) \times (1 + x^2)$ needs to consider 2 terms → $60 - 20 = 40$ | [3] |
---2 Find the coefficient of $x ^ { 2 }$ in the expansion of\\
(i) $\left( 2 x - \frac { 1 } { 2 x } \right) ^ { 6 }$,\\
(ii) $\left( 1 + x ^ { 2 } \right) \left( 2 x - \frac { 1 } { 2 x } \right) ^ { 6 }$.
\hfill \mbox{\textit{CAIE P1 2013 Q2 [5]}}