OCR MEI FP1 2011 January — Question 7 12 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2011
SessionJanuary
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration with Partial Fractions
TypeRational function curve sketching
DifficultyStandard +0.3 This is a structured curve sketching question with clear sub-parts guiding students through standard techniques: finding intercepts, asymptotes, and behavior. While it's Further Maths content (FP1), the question is methodical with no novel insights required—students follow a routine procedure for rational function analysis. Slightly above average difficulty due to the Further Maths context and the inequality solving at the end, but well within standard textbook exercise territory.
Spec1.02g Inequalities: linear and quadratic in single variable1.02i Represent inequalities: graphically on coordinate plane1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials
7 Fig. 7 shows part of the curve with equation \(y = \frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d91a83d-971e-48ca-aa1a-09f2c1a8093a-3_894_890_447_625} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down the coordinates of the two points where the curve crosses the axes.
  2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
  3. Determine how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
  4. On the copy of Fig. 7, sketch the rest of the curve.
  5. Solve the inequality \(\frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) } < 0\).
Question 7:
Part (i):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\left(0, \frac{-1}{8}\right)\), \((-5, 0)\)B1, B1 [2] One mark for each point; SC1 for \(x=-5\), \(y=-\frac{1}{8}\)
Part (ii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(x=\frac{5}{2}\), \(x=\frac{-8}{3}\), \(y=0\)B1, B1, B1 [3] One mark for each equation
Part (iii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Large positive \(x\), \(y \rightarrow 0^+\) (e.g. consider \(x=100\))B1
Large negative \(x\), \(y \rightarrow 0^-\) (e.g. consider \(x=-100\))B1, M1 [3] Evidence of a valid method
Part (iv):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Graph sketchB1, B1 [2] RH branch correct; LH branch correct
Part (v):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(x < -5\)B1 cao
or \(\frac{-8}{3} < x < \frac{5}{2}\)B1 [2] cao 
# Question 7:

## Part (i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(0, \frac{-1}{8}\right)$, $(-5, 0)$ | B1, B1 [2] | One mark for each point; SC1 for $x=-5$, $y=-\frac{1}{8}$ |

## Part (ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x=\frac{5}{2}$, $x=\frac{-8}{3}$, $y=0$ | B1, B1, B1 [3] | One mark for each equation |

## Part (iii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Large positive $x$, $y \rightarrow 0^+$ (e.g. consider $x=100$) | B1 | |
| Large negative $x$, $y \rightarrow 0^-$ (e.g. consider $x=-100$) | B1, M1 [3] | Evidence of a valid method |

## Part (iv):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Graph sketch | B1, B1 [2] | RH branch correct; LH branch correct |

## Part (v):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x < -5$ | B1 | cao |
| or $\frac{-8}{3} < x < \frac{5}{2}$ | B1 [2] | cao |

---
7 Fig. 7 shows part of the curve with equation $y = \frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) }$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8d91a83d-971e-48ca-aa1a-09f2c1a8093a-3_894_890_447_625}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}

(i) Write down the coordinates of the two points where the curve crosses the axes.\\
(ii) Write down the equations of the two vertical asymptotes and the one horizontal asymptote.\\
(iii) Determine how the curve approaches the horizontal asymptote for large positive and large negative values of $x$.\\
(iv) On the copy of Fig. 7, sketch the rest of the curve.\\
(v) Solve the inequality $\frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) } < 0$.

\hfill \mbox{\textit{OCR MEI FP1 2011 Q7 [12]}}
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Q1 5 Q2 7 Q3 7 Q4 6 Q5 5 Q6 6 Q7 12 Q8 12 Q9 12