| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2007 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Substitution into binomial expansion |
| Difficulty | Moderate -0.8 This is a straightforward two-part question requiring routine application of the binomial theorem for n=5, followed by a guided substitution and collection of terms. The substitution is explicitly given, and students only need to expand and identify the x² coefficient—no problem-solving insight required, just careful algebraic manipulation. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((2+u)^5 = 32 + 80u + 80u^2\) | B1 × 3 [3] | Co. Allow \(2^5\) for B1. |
| (ii) ... \(80(x + x^2) + 80(x + x^2)^2\) → coeff of \(x^2\) of \(80 + 80 = 160\) | M1, A1∨ [2] | Knows what to do – looks at more than 1 term. √ "coeff of \(x\) + coeff of \(x^2\)". |
**(i)** $(2+u)^5 = 32 + 80u + 80u^2$ | B1 × 3 [3] | Co. Allow $2^5$ for B1.
**(ii)** ... $80(x + x^2) + 80(x + x^2)^2$ → coeff of $x^2$ of $80 + 80 = 160$ | M1, A1∨ [2] | Knows what to do – looks at more than 1 term. √ "coeff of $x$ + coeff of $x^2$".3 (i) Find the first three terms in the expansion of $( 2 + u ) ^ { 5 }$ in ascending powers of $u$.\\
(ii) Use the substitution $u = x + x ^ { 2 }$ in your answer to part (i) to find the coefficient of $x ^ { 2 }$ in the expansion of $\left( 2 + x + x ^ { 2 } \right) ^ { 5 }$.
\hfill \mbox{\textit{CAIE P1 2007 Q3 [5]}}