| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Standard product of two binomials |
| Difficulty | Moderate -0.3 This is a straightforward binomial expansion question requiring systematic application of the binomial theorem to find specific coefficients. Parts (i) and (ii) are routine calculations, while part (iii) adds a minor twist by requiring students to recognize they can use their previous answer and find one additional term. The algebraic manipulation is standard and the question structure guides students through the solution methodically. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(81(x^8)\) | B1 | |
| [1] | ||
| (ii) \(10 \times 3^1(x^3)\) soi leading to their answer | B1B1 | B1 for 10, 5C2 or 5C3. B1 for \(3^3\). But must be multiplied. |
| \(270(x^8)\) | B1 | |
| [3] | ||
| (iii) \(k \times\) (i) | M1 | \(k \neq 1,0\) |
| 405 soi | A1 | |
| \(+\) (ii) | DM1 | |
| \(675(x^8)\) | A1 | |
| [4] |
**(i)** $81(x^8)$ | B1 |
| | [1]
**(ii)** $10 \times 3^1(x^3)$ soi leading to their answer | B1B1 | B1 for 10, 5C2 or 5C3. B1 for $3^3$. But must be multiplied.
$270(x^8)$ | B1 |
| | [3]
**(iii)** $k \times$ (i) | M1 | $k \neq 1,0$
405 soi | A1 |
$+$ (ii) | DM1 |
$675(x^8)$ | A1 |
| | [4]8 (i) Find the coefficient of $x ^ { 8 }$ in the expansion of $\left( x + 3 x ^ { 2 } \right) ^ { 4 }$.\\
(ii) Find the coefficient of $x ^ { 8 }$ in the expansion of $\left( x + 3 x ^ { 2 } \right) ^ { 5 }$.\\
(iii) Hence find the coefficient of $x ^ { 8 }$ in the expansion of $\left[ 1 + \left( x + 3 x ^ { 2 } \right) \right] ^ { 5 }$.
\hfill \mbox{\textit{CAIE P1 2013 Q8 [8]}}