| 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 | Coefficient zero after multiplying binomial |
| Difficulty | Moderate -0.3 Part (i) is straightforward binomial expansion requiring direct application of the formula for three terms. Part (ii) requires multiplying the expansion by (1-x) and setting the x² coefficient to zero, which involves collecting like terms and solving a simple equation. This is a standard textbook exercise with minimal problem-solving demand, making it slightly easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(1 - 6pyx + 15p^2x^2\) | B1B1 | Simplification of \(nCr\) can be scored in (ii) |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) \(15p^2 \times 1 - 6p \times -1\) or \(3p(5p + 2) = 0\) | M1 DM1 | Obtain & attempt to solve quadratic |
| \(p = -\frac{2}{5}\) oe | A1 | Allow \(p = 0\) in addition |
(i) $1 - 6pyx + 15p^2x^2$ | B1B1 | Simplification of $nCr$ can be scored in (ii)
[2]
(ii) $15p^2 \times 1 - 6p \times -1$ or $3p(5p + 2) = 0$ | M1 DM1 | Obtain & attempt to solve quadratic
$p = -\frac{2}{5}$ oe | A1 | Allow $p = 0$ in addition
[3]2 (i) In the expression $( 1 - p x ) ^ { 6 } , p$ is a non-zero constant. Find the first three terms when $( 1 - p x ) ^ { 6 }$ is expanded in ascending powers of $x$.\\
(ii) It is given that the coefficient of $x ^ { 2 }$ in the expansion of $( 1 - x ) ( 1 - p x ) ^ { 6 }$ is zero. Find the value of $p$.
\hfill \mbox{\textit{CAIE P1 2013 Q2 [5]}}