| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2006 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expansion with algebraic manipulation |
| Difficulty | Standard +0.8 This question requires algebraic manipulation of surds (part i), then applying the binomial expansion to two separate terms (1+x)^(-1/2) and (1-x)^(-1/2), followed by subtraction and division by 2x. While the individual techniques are standard A-level content, the multi-step coordination and algebraic complexity elevate this above average difficulty, though it remains a structured question with clear guidance. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks |
|---|---|
| (i) Simplify product and obtain \((1 + x) - (1 - x)\) | B1 |
| Complete the proof of the given result with no errors seen | B1 |
| (ii) Use correct method to obtain the first four terms of the expansion of \(\sqrt{1 + x}\) or \(\sqrt{1 - x}\) | M1 |
| EITHER: Obtain any correct unsimplified expansion of the numerator of the RHS of the identity up to the terms in \(x^3\) | A1 |
| Obtain final answer with constant term \(\frac{1}{2}\) | A1 |
| Obtain term \(\frac{1}{4}x^2\) and no term in \(x\) | A1 |
| OR: Obtain any correct unsimplified expansion of the denominator of the LHS of the identity up to the terms in \(x^2\) | A1 |
| Obtain final answer with constant term \(\frac{1}{2}\) | A1 |
| Obtain term \(\frac{1}{4}x^2\) and no term in \(x\) | A1 |
| [Symbolic binomial coefficients are not sufficient for the M1. Allow two correct separate expansions to earn the first A1 if the context is clear and appropriate.] | |
| [Allow the use of Maclaurin, giving M1A1 for \(f(0) = \frac{1}{2}\) and \(f'(0) = 0\), A1 for \(f''(0) = \frac{1}{4}\), and A1 for obtaining the correct final answer.] | |
| 4 |
(i) Simplify product and obtain $(1 + x) - (1 - x)$ | B1 |
Complete the proof of the given result with no errors seen | B1 |
(ii) Use correct method to obtain the first four terms of the expansion of $\sqrt{1 + x}$ or $\sqrt{1 - x}$ | M1 |
EITHER: Obtain any correct unsimplified expansion of the numerator of the RHS of the identity up to the terms in $x^3$ | A1 |
Obtain final answer with constant term $\frac{1}{2}$ | A1 |
Obtain term $\frac{1}{4}x^2$ and no term in $x$ | A1 |
OR: Obtain any correct unsimplified expansion of the denominator of the LHS of the identity up to the terms in $x^2$ | A1 |
Obtain final answer with constant term $\frac{1}{2}$ | A1 |
Obtain term $\frac{1}{4}x^2$ and no term in $x$ | A1 |
[Symbolic binomial coefficients are not sufficient for the M1. Allow two correct separate expansions to earn the first A1 if the context is clear and appropriate.] | | |
[Allow the use of Maclaurin, giving M1A1 for $f(0) = \frac{1}{2}$ and $f'(0) = 0$, A1 for $f''(0) = \frac{1}{4}$, and A1 for obtaining the correct final answer.] | | |
| | 4 |5 (i) Simplify $( \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) ) ( \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) )$, showing your working, and deduce that
$$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) } = \frac { \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) } { 2 x }$$
(ii) Using this result, or otherwise, obtain the expansion of
$$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) }$$
in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
\hfill \mbox{\textit{CAIE P3 2006 Q5 [6]}}