| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions then binomial expansion |
| Difficulty | Standard +0.3 This is a standard two-part question combining partial fractions with binomial expansion. Part (i) requires routine partial fraction decomposition with a repeated linear factor, and part (ii) involves straightforward binomial expansions of three simple terms. While it requires multiple techniques and careful algebraic manipulation, both components are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Either State or imply form \(\frac{A}{1+x} + \frac{B}{(1+x)^2} + \frac{C}{2-3x}\) | B1 | |
| Use any relevant method to find at least one constant | M1 | |
| Obtain \(A = -1\) | A1 | |
| Obtain \(B = 3\) | A1 | |
| Obtain \(C = 4\) | A1 | |
| Or State or imply form \(\frac{A}{1+x} + \frac{Bx}{(1+x)^2} + \frac{C}{2-3x}\) | B1 | |
| Use any relevant method to find at least one constant | M1 | |
| Obtain \(A = 2\) | A1 | |
| Obtain \(B = -3\) | A1 | |
| Obtain \(C = 4\) | A1 | |
| Or State or imply form \(\frac{Dx+E}{(1+x)^2} + \frac{F}{2-3x}\) | B1 | |
| Use any relevant method to find at least one constant | M1 | |
| Obtain \(D = -1\) | A1 | |
| Obtain \(E = 2\) | A1 | |
| Obtain \(F = 4\) | A1 | [5] |
| (ii) Either Use correct method to find first two terms of expansion of \((1+x)^{-1}\) or \((1+x)^{-2}\) or \((2-3x)^{-1}\) or \(\left(1 - \frac{3}{2}x\right)^{-1}\) | M1 | |
| Obtain correct unsimplified expansion of first partial fraction up to \(x^2\) term | A1✱ | |
| Obtain correct unsimplified expansion of second partial fraction up to \(x^2\) term | A1✱ | |
| Obtain correct unsimplified expansion of third partial fraction up to \(x^2\) term | A1✱ | |
| Obtain final answer \(4 - 2x + \frac{25}{2}x^2\) | A1 | |
| Or 1 Use correct method to find first two terms of expansion of \((1+x)^{-2}\) or \((2-3x)^{-1}\) or \(\left(1 - \frac{3}{2}x\right)^{-1}\) | M1 | |
| Obtain correct unsimplified expansion of first partial fraction up to \(x^2\) term | A1✱ | |
| Obtain correct unsimplified expansion of second partial fraction up to \(x^2\) term | A1✱ | |
| Expand and obtain sufficient terms to obtain three terms | M1 | |
| Obtain final answer \(4 - 2x + \frac{25}{2}x^2\) | A1 | |
| Or 2 (expanding original expression) Use correct method to find first two terms of expansion of \((1+x)^{-2}\) or \((2-3x)^{-1}\) or \(\left(1 - \frac{3}{2}x\right)^{-1}\) | M1 | |
| Obtain correct expansion \(1 - 2x + 3x^2\) or unsimplified equivalent | A1 | |
| Obtain correct expansion \(\frac{1}{2}\left(1 + \frac{3}{2}x + \frac{9}{4}x^2\right)\) or unsimplified equivalent | A1 | |
| Expand and obtain sufficient terms to obtain three terms | M1 | |
| Obtain final answer \(4 - 2x + \frac{25}{2}x^2\) | A1 | |
| Or 3 (McLaurin expansion) Obtain first derivative \(f'(x) = (1+x)^{-2} - 6(1+x)^{-3} + 12(2-3x)^{-2}\) | M1 | |
| Obtain \(f'(0) = 1 - 6 + 3\) or equivalent | A1 | |
| Obtain \(f''(0) = -2 + 18 + 9\) or equivalent | A1 | |
| Use correct form for McLaurin expansion | M1 | |
| Obtain final answer \(4 - 2x + \frac{25}{2}x^2\) | A1 | [5] |
**(i)** **Either** State or imply form $\frac{A}{1+x} + \frac{B}{(1+x)^2} + \frac{C}{2-3x}$ | B1 |
Use any relevant method to find at least one constant | M1 |
Obtain $A = -1$ | A1 |
Obtain $B = 3$ | A1 |
Obtain $C = 4$ | A1 |
**Or** State or imply form $\frac{A}{1+x} + \frac{Bx}{(1+x)^2} + \frac{C}{2-3x}$ | B1 |
Use any relevant method to find at least one constant | M1 |
Obtain $A = 2$ | A1 |
Obtain $B = -3$ | A1 |
Obtain $C = 4$ | A1 |
**Or** State or imply form $\frac{Dx+E}{(1+x)^2} + \frac{F}{2-3x}$ | B1 |
Use any relevant method to find at least one constant | M1 |
Obtain $D = -1$ | A1 |
Obtain $E = 2$ | A1 |
Obtain $F = 4$ | A1 | [5]
**(ii)** **Either** Use correct method to find first two terms of expansion of $(1+x)^{-1}$ or $(1+x)^{-2}$ or $(2-3x)^{-1}$ or $\left(1 - \frac{3}{2}x\right)^{-1}$ | M1 |
Obtain correct unsimplified expansion of first partial fraction up to $x^2$ term | A1✱ |
Obtain correct unsimplified expansion of second partial fraction up to $x^2$ term | A1✱ |
Obtain correct unsimplified expansion of third partial fraction up to $x^2$ term | A1✱ |
Obtain final answer $4 - 2x + \frac{25}{2}x^2$ | A1 |
**Or 1** Use correct method to find first two terms of expansion of $(1+x)^{-2}$ or $(2-3x)^{-1}$ or $\left(1 - \frac{3}{2}x\right)^{-1}$ | M1 |
Obtain correct unsimplified expansion of first partial fraction up to $x^2$ term | A1✱ |
Obtain correct unsimplified expansion of second partial fraction up to $x^2$ term | A1✱ |
Expand and obtain sufficient terms to obtain three terms | M1 |
Obtain final answer $4 - 2x + \frac{25}{2}x^2$ | A1 |
**Or 2** (expanding original expression) Use correct method to find first two terms of expansion of $(1+x)^{-2}$ or $(2-3x)^{-1}$ or $\left(1 - \frac{3}{2}x\right)^{-1}$ | M1 |
Obtain correct expansion $1 - 2x + 3x^2$ or unsimplified equivalent | A1 |
Obtain correct expansion $\frac{1}{2}\left(1 + \frac{3}{2}x + \frac{9}{4}x^2\right)$ or unsimplified equivalent | A1 |
Expand and obtain sufficient terms to obtain three terms | M1 |
Obtain final answer $4 - 2x + \frac{25}{2}x^2$ | A1 |
**Or 3** (McLaurin expansion) Obtain first derivative $f'(x) = (1+x)^{-2} - 6(1+x)^{-3} + 12(2-3x)^{-2}$ | M1 |
Obtain $f'(0) = 1 - 6 + 3$ or equivalent | A1 |
Obtain $f''(0) = -2 + 18 + 9$ or equivalent | A1 |
Use correct form for McLaurin expansion | M1 |
Obtain final answer $4 - 2x + \frac{25}{2}x^2$ | A1 | [5]8 (i) Express $\frac { 7 x ^ { 2 } + 8 } { ( 1 + x ) ^ { 2 } ( 2 - 3 x ) }$ in partial fractions.\\
(ii) Hence expand $\frac { 7 x ^ { 2 } + 8 } { ( 1 + x ) ^ { 2 } ( 2 - 3 x ) }$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2013 Q8 [10]}}