| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion question requiring students to expand (a-x)^5 using the binomial theorem, then multiply by (1-ax) and equate the x^3 coefficient to -200. While it involves multiple steps and solving a quadratic, the techniques are standard and the problem structure is typical for this topic. Slightly easier than average due to clear instructions and routine application of formulas. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a^5 - 5a^4x + 10a^3x^2 - 10a^2x^3 + \ldots\) | B2,1,0 [2] | Ok full expansion (ignore extra terms). Descending: Ok if full expansion but max B1 for 4 terms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((1-ax)(\ldots 10a^3x^2 - 10a^2x^3\ldots) = (x^3)(-10a^4 - 10a^2)\) | M1 | Attempt to find coeff. of \(x^3\) from 2 terms |
| \(-10a^4 - 10a^2 = -200\) | A1\(\checkmark\) | Ft from *their* \(10a^3\), \(-10a^2\) from part (i) |
| \(a^2 = 4\) ignore \(a^2 = -5\) | M1 | Attempt soln. for \(a^2\) from 3-term quad. in \(a^2\) |
| \(a = \pm 2\) cao | A1 [4] | Ignore any imaginary solutions |
## Question 3(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a^5 - 5a^4x + 10a^3x^2 - 10a^2x^3 + \ldots$ | B2,1,0 [2] | Ok full expansion (ignore extra terms). Descending: Ok if full expansion but max B1 for 4 terms |
## Question 3(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(1-ax)(\ldots 10a^3x^2 - 10a^2x^3\ldots) = (x^3)(-10a^4 - 10a^2)$ | M1 | Attempt to find coeff. of $x^3$ from 2 terms |
| $-10a^4 - 10a^2 = -200$ | A1$\checkmark$ | Ft from *their* $10a^3$, $-10a^2$ from part (i) |
| $a^2 = 4$ ignore $a^2 = -5$ | M1 | Attempt soln. for $a^2$ from 3-term quad. in $a^2$ |
| $a = \pm 2$ cao | A1 [4] | Ignore any imaginary solutions |
---3 (i) Write down the first 4 terms, in ascending powers of $x$, of the expansion of $( a - x ) ^ { 5 }$.\\
(ii) The coefficient of $x ^ { 3 }$ in the expansion of $( 1 - a x ) ( a - x ) ^ { 5 }$ is - 200 . Find the possible values of the constant $a$.
\hfill \mbox{\textit{CAIE P1 2015 Q3 [6]}}