| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2005 |
| 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 recall of the formula. Part (ii) requires identifying coefficients and solving a linear equation to make the x² term vanish—a standard technique but involves an extra conceptual step beyond pure recall, making it slightly easier than average overall. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((2 - x)^6 = 64 - 192x + 240x^2\) | 3 × B1 [3] | One for each term. Allow \(2^6\). |
| (ii) \((1 + kx)(2 - x)^6\) coeff of \(x^2 = 240 - 192k = 0 \Rightarrow k = 5/4\) or \(1.25\) | M1 A1 \(\checkmark\) [2] | Must be considering sum of 2 terms. ft for his expansion. (allow M1 if looking for coeff of \(x\)). |
(i) $(2 - x)^6 = 64 - 192x + 240x^2$ | 3 × B1 [3] | One for each term. Allow $2^6$.
(ii) $(1 + kx)(2 - x)^6$ coeff of $x^2 = 240 - 192k = 0 \Rightarrow k = 5/4$ or $1.25$ | M1 A1 $\checkmark$ [2] | Must be considering sum of 2 terms. ft for his expansion. (allow M1 if looking for coeff of $x$).4 (i) Find the first 3 terms in the expansion of $( 2 - x ) ^ { 6 }$ in ascending powers of $x$.\\
(ii) Find the value of $k$ for which there is no term in $x ^ { 2 }$ in the expansion of $( 1 + k x ) ( 2 - x ) ^ { 6 }$.
\hfill \mbox{\textit{CAIE P1 2005 Q4 [5]}}