| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Factor polynomial then partial fractions |
| Difficulty | Moderate -0.3 This is a straightforward two-part question combining factor theorem and partial fractions. Part (i) uses substitution to find 'a' then factorisation (standard technique), while part (ii) is a routine partial fractions decomposition with linear factors. Both parts follow textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02y Partial fractions: decompose rational functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) Substitute \(x = \frac{1}{2}\) and equate to zero, or divide by \((2x-1)\), reach \(\frac{a}{2}x^2 + kx + \ldots\) and equate remainder to zero, or by inspection reach \(\frac{a}{2}x^2 + bx + c\) and an equation in b/c, or by inspection reach \(Ax^2 + Bx + a\) and an equation in A/B | M1 | |
| Obtain \(a = 2\) | A1 | |
| Attempt to find quadratic factor by division or inspection or equivalent | M1 | |
| Obtain \((2x-1)(x^2+2)\) | A1cwo | [4] |
| (ii) State or imply form \(\dfrac{A}{2x-1} + \dfrac{Bx+C}{x^2+2}\), following factors from part (i) | B1\(\sqrt{}\) | |
| Use relevant method to find a constant | M1 | |
| Obtain \(A = -4\), following factors from part (i) | A1\(\sqrt{}\) | |
| Obtain \(B = 2\) | A1 | |
| Obtain \(C = 5\) | A1 |
## Question 7:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** Substitute $x = \frac{1}{2}$ and equate to zero, or divide by $(2x-1)$, reach $\frac{a}{2}x^2 + kx + \ldots$ and equate remainder to zero, or by inspection reach $\frac{a}{2}x^2 + bx + c$ and an equation in b/c, or by inspection reach $Ax^2 + Bx + a$ and an equation in A/B | M1 | |
| Obtain $a = 2$ | A1 | |
| Attempt to find quadratic factor by division or inspection or equivalent | M1 | |
| Obtain $(2x-1)(x^2+2)$ | A1cwo | [4] |
| **(ii)** State or imply form $\dfrac{A}{2x-1} + \dfrac{Bx+C}{x^2+2}$, following factors from part **(i)** | B1$\sqrt{}$ | |
| Use relevant method to find a constant | M1 | |
| Obtain $A = -4$, following factors from part **(i)** | A1$\sqrt{}$ | |
| Obtain $B = 2$ | A1 | |
| Obtain $C = 5$ | A1 | |7 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = a x ^ { 3 } - x ^ { 2 } + 4 x - a$$
where $a$ is a constant. It is given that $( 2 x - 1 )$ is a factor of $\mathrm { p } ( x )$.\\
(i) Find the value of $a$ and hence factorise $\mathrm { p } ( x )$.\\
(ii) When $a$ has the value found in part (i), express $\frac { 8 x - 13 } { \mathrm { p } ( x ) }$ in partial fractions.
\hfill \mbox{\textit{CAIE P3 2011 Q7 [9]}}