| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expand and state validity |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial expansion for negative/fractional powers with minimal algebraic manipulation. Students need only substitute into the standard formula (1+x)^n ≈ 1 + nx + ..., identify the validity condition |x²/2| < 1, and simplify coefficients—all routine procedures for P3 level with no problem-solving or insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State a correct unsimplified version of the \(x^2\) or \(x^4\) term: \((2-x^2)^{-2}\) or \(\left(1-\frac{1}{2}x^2\right)^{-2}\) | M1 | \(\frac{1}{4}\left(1+2\cdot\frac{x^2}{2}+\frac{-2\cdot-3}{2}\left(\frac{x^2}{2}\right)^2...\right)\) Symbolic binomial coefficients not sufficient for M1 |
| State correct first term \(\frac{1}{4}\) | B1 | Accept \(2^{-2}\) |
| Obtain next two terms \(\frac{1}{4}x^2+\frac{3}{16}x^4\) | A1 A1 | A1 for each one correct ISW. Full marks for \(\frac{1}{4}\left(1+x^2+\frac{3}{4}x^4\right)\) ISW. SC allow M1 A1 A1 for \(\frac{1}{4}\) and \(1+x^2+\frac{3}{4}x^4\) SOL. SC allow M1 A1 for \(1+x^2+\frac{3}{4}x^4\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State answer \( | x | <\sqrt{2}\) |
## Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State a correct unsimplified version of the $x^2$ or $x^4$ term: $(2-x^2)^{-2}$ or $\left(1-\frac{1}{2}x^2\right)^{-2}$ | M1 | $\frac{1}{4}\left(1+2\cdot\frac{x^2}{2}+\frac{-2\cdot-3}{2}\left(\frac{x^2}{2}\right)^2...\right)$ Symbolic binomial coefficients not sufficient for M1 |
| State correct first term $\frac{1}{4}$ | B1 | Accept $2^{-2}$ |
| Obtain next two terms $\frac{1}{4}x^2+\frac{3}{16}x^4$ | A1 A1 | A1 for each one correct ISW. Full marks for $\frac{1}{4}\left(1+x^2+\frac{3}{4}x^4\right)$ ISW. SC allow **M1 A1 A1** for $\frac{1}{4}$ and $1+x^2+\frac{3}{4}x^4$ SOL. SC allow **M1 A1** for $1+x^2+\frac{3}{4}x^4$ |
**Total: 4 marks**
## Question 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State answer $|x|<\sqrt{2}$ | B1 | Or $-\sqrt{2}<x<\sqrt{2}$ |
**Total: 1 mark**
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\begin{enumerate}[label=(\alph*)]
\item Expand $\left( 2 - x ^ { 2 } \right) ^ { - 2 }$ in ascending powers of $x$, up to and including the term in $x ^ { 4 }$, simplifying the coefficients.
\item State the set of values of $x$ for which the expansion is valid.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q2 [5]}}