| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| 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 This is a straightforward binomial expansion question requiring routine application of the binomial theorem for small n=5, followed by a simple algebraic manipulation to find when the x² coefficient equals zero. The question is slightly easier than average because it explicitly guides students through part (i) before asking them to use those results in part (ii), and the algebra involved is minimal. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((2 + 3x)^5 = 32 + 240x + 720x^2\) | \(3 \times\) B1 [3] | All co. |
| (ii) \((1 + ax)(2 + 3x)^5\) | ||
| \(\rightarrow (1 \times 720) + (a \times 240) = 0\) | M1 A1√ [2] | Must be considering exactly 2 terms. √ for (\(-\) coeff \(\div\) ÷ coeff \(x\)) |
| \(\rightarrow a = -3\) |
(i) $(2 + 3x)^5 = 32 + 240x + 720x^2$ | $3 \times$ B1 [3] | All co.
(ii) $(1 + ax)(2 + 3x)^5$ | |
$\rightarrow (1 \times 720) + (a \times 240) = 0$ | M1 A1√ [2] | Must be considering exactly 2 terms. √ for ($-$ coeff $\div$ ÷ coeff $x$)
$\rightarrow a = -3$ | |3 (i) Find the first 3 terms in the expansion of $( 2 + 3 x ) ^ { 5 }$ in ascending powers of $x$.\\
(ii) Hence find the value of the constant $a$ for which there is no term in $x ^ { 2 }$ in the expansion of $( 1 + a x ) ( 2 + 3 x ) ^ { 5 }$.
\hfill \mbox{\textit{CAIE P1 2009 Q3 [5]}}