| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | March |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expand and state validity |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial expansion with a fractional power. Part (a) requires expanding (4-x)^(1/2) as 2(1-x/4)^(1/2), multiplying by (2x-5), and collecting the x² term—mechanical but requires careful algebra. Part (b) is direct recall that |x/4| < 1. Slightly easier than average due to being a standard textbook exercise with no conceptual surprises. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State unsimplified term in \(x\): \(4^{\frac{1}{2}} \times \frac{1}{2} \times \left(\frac{-x}{4}\right) = \frac{-x}{4}\) | B1 | |
| State unsimplified term in \(x^2\): \(4^{\frac{1}{2}} \times \frac{\frac{1}{2} \times \frac{-1}{2}}{2} \times \left(\frac{-x}{4}\right)^2 = \frac{-x^2}{64}\) | B1 | Allow \(\left(\frac{x}{4}\right)^2\) |
| Multiply by \((2x-5)\) and obtain 2 terms in \(x^2\), allow even if errors in \(4^{\frac{1}{2}}\), signs etc. | M1 | Allow unsimplified \(2x\): \(4^{\frac{1}{2}} \times \frac{1}{2} \times \left(\frac{-x}{4}\right) - 5\); and \(4^{\frac{1}{2}} \times \frac{\frac{1}{2} \times \frac{-1}{2}}{2} \times \left(\frac{-x}{4}\right)^2\); Allow \(\left(\frac{x}{4}\right)^2\); \(2x \times \left(\frac{-x}{4}\right)(-5) \times \frac{-x^2}{64}\) or \(2 \times \left(\frac{-1}{4}\right)(-5) \times \left(\frac{-1}{64}\right)\) |
| Obtain \(-\frac{27}{64}\) or \(-0.421875\) or \(-\frac{54}{128}\) | A1 | Allow in a full expansion up to \(x^2\), ignore extra terms even if they contain errors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \( | x | < 4\) |
## Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State unsimplified term in $x$: $4^{\frac{1}{2}} \times \frac{1}{2} \times \left(\frac{-x}{4}\right) = \frac{-x}{4}$ | B1 | |
| State unsimplified term in $x^2$: $4^{\frac{1}{2}} \times \frac{\frac{1}{2} \times \frac{-1}{2}}{2} \times \left(\frac{-x}{4}\right)^2 = \frac{-x^2}{64}$ | B1 | Allow $\left(\frac{x}{4}\right)^2$ |
| Multiply by $(2x-5)$ and obtain 2 terms in $x^2$, allow even if errors in $4^{\frac{1}{2}}$, signs etc. | M1 | Allow unsimplified $2x$: $4^{\frac{1}{2}} \times \frac{1}{2} \times \left(\frac{-x}{4}\right) - 5$; and $4^{\frac{1}{2}} \times \frac{\frac{1}{2} \times \frac{-1}{2}}{2} \times \left(\frac{-x}{4}\right)^2$; Allow $\left(\frac{x}{4}\right)^2$; $2x \times \left(\frac{-x}{4}\right)(-5) \times \frac{-x^2}{64}$ or $2 \times \left(\frac{-1}{4}\right)(-5) \times \left(\frac{-1}{64}\right)$ |
| Obtain $-\frac{27}{64}$ or $-0.421875$ or $-\frac{54}{128}$ | A1 | Allow in a full expansion up to $x^2$, ignore extra terms even if they contain errors |
## Question 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $|x| < 4$ | B1 | or $-4 < x < 4$ |
---2
\begin{enumerate}[label=(\alph*)]
\item Find the coefficient of $x ^ { 2 }$ in the expansion of $( 2 x - 5 ) \sqrt { 4 - x }$.
\item State the set of values of $x$ for which the expansion in part (a) is valid.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q2 [5]}}