OCR MEI C4 — Question 2 5 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPartial Fractions
TypeBasic partial fractions decomposition
DifficultyEasy -2.5 This question appears to be incomplete or trivial as written - the expression is already a single fraction that cannot be decomposed into partial fractions (the denominator has only one linear factor). If the denominator should be quadratic or have multiple factors, this would be a standard C4 exercise worth around -1.0, but as stated it requires only recognizing that no decomposition is possible, making it exceptionally easy.
Spec1.02y Partial fractions: decompose rational functions
2 Express \(\frac { x + 1 } { ( 2 x - 1 ) }\) in partial fractions.
Question 2:
AnswerMarks Guidance
\(\frac{x+1}{x^2(2x-1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{2x-1} = \frac{Ax(2x-1)+B(2x-1)+Cx^2}{x^2(2x-1)}\)B1 correct partial fractions
\(x+1 = Ax(2x-1)+B(2x-1)+Cx^2\)M1 Using a correct method to find a coefficient (equating numerators and substituting, or cover-up). Condone omission of brackets only if implied by subsequent work. Must go as far as finding a coefficient. Not dependent on B1
\(x=0\): \(1=-B \Rightarrow B=-1\)A1 \(B=-1\) www
\(x=\frac{1}{2}\): \(\frac{3}{2}=\frac{C}{4} \Rightarrow C=6\)A1 \(C=6\) www
\(x^2\) coeffs: \(0=2A+C \Rightarrow A=-3\)A1 \(A=-3\) www; isw for incorrect assembly of partial fractions following correct \(A,B,C\)
\(\Rightarrow \frac{x+1}{x^2(2x-1)} = -\frac{3}{x} - \frac{1}{x^2} + \frac{6}{2x-1}\)[5] 
## Question 2:

$\frac{x+1}{x^2(2x-1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{2x-1} = \frac{Ax(2x-1)+B(2x-1)+Cx^2}{x^2(2x-1)}$ | B1 | correct partial fractions

$x+1 = Ax(2x-1)+B(2x-1)+Cx^2$ | M1 | Using a correct method to find a coefficient (equating numerators and substituting, or cover-up). Condone omission of brackets only if implied by subsequent work. Must go as far as finding a coefficient. Not dependent on B1

$x=0$: $1=-B \Rightarrow B=-1$ | A1 | $B=-1$ www

$x=\frac{1}{2}$: $\frac{3}{2}=\frac{C}{4} \Rightarrow C=6$ | A1 | $C=6$ www

$x^2$ coeffs: $0=2A+C \Rightarrow A=-3$ | A1 | $A=-3$ www; isw for incorrect assembly of partial fractions following correct $A,B,C$

$\Rightarrow \frac{x+1}{x^2(2x-1)} = -\frac{3}{x} - \frac{1}{x^2} + \frac{6}{2x-1}$ | [5] |

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2 Express $\frac { x + 1 } { ( 2 x - 1 ) }$ in partial fractions.

\hfill \mbox{\textit{OCR MEI C4  Q2 [5]}}
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