Moderate -0.5 This is a straightforward improper fraction requiring polynomial division followed by standard partial fraction decomposition. While it involves an extra step compared to proper fractions, the algebraic manipulation is routine and mechanical with no conceptual challenges beyond applying the standard algorithm.
State or imply the form \(\frac{D}{2x-1} + \frac{E}{x-3}\)
B1
Obtain one of \(D = -3\) and \(E = 2\)
A1
Obtain a second value
A1
Total: 5 marks
## Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $A + \frac{B}{2x-1} + \frac{C}{x-3}$ | **B1** | $\frac{Dx+E}{2x-1} + \frac{F}{x-3}$ and $\frac{P}{2x-1} + \frac{Qx+R}{x-3}$ are also valid |
| Use a correct method for finding a constant | **M1** | |
| Obtain one of $A = 2$, $B = -3$ and $C = 2$ | **A1** | Allow maximum **M1A1** for one or more 'correct' values after **B0** |
| Obtain a second value | **A1** | |
| Obtain the third value | **A1** | |
**Alternative method for Question 4:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Divide numerator by denominator | **M1** | |
| Obtain $2\left[+\frac{Px+Q}{(2x-1)(x-3)}\right]$ | **A1** | $\left[2 + \frac{x+7}{(2x-1)(x-3)}\right]$ |
| State or imply the form $\frac{D}{2x-1} + \frac{E}{x-3}$ | **B1** | |
| Obtain one of $D = -3$ and $E = 2$ | **A1** | |
| Obtain a second value | **A1** | |
**Total: 5 marks**
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