Question 7 - A-Level Maths

OCR MEI FP1 2012 January — Question 7 14 marks

Exam Board	OCR MEI
Module	FP1 (Further Pure Mathematics 1)
Year	2012
Session	January
Marks	14
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Integration with Partial Fractions
Type	Rational function curve sketching
Difficulty	Standard +0.8 This is a comprehensive Further Maths rational function question requiring multiple techniques: finding intercepts, identifying asymptotes (including oblique behavior analysis), asymptotic approach direction, curve sketching, and solving a rational inequality. While each individual step uses standard FP1 techniques, the multi-part nature, requirement to synthesize information for an accurate sketch, and the inequality solving (requiring algebraic manipulation and sign analysis) elevate this above typical A-level questions.
Spec	1.02g Inequalities: linear and quadratic in single variable 1.02i Represent inequalities: graphically on coordinate plane 1.02k Simplify rational expressions: factorising, cancelling, algebraic division 1.02n Sketch curves: simple equations including polynomials

7 A curve has equation \(y = \frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 }\).

Find the coordinates of the points where the curve crosses the axes.
Write down the equations of the three asymptotes.
Determine whether the curve approaches the horizontal asymptote from above or from below for
(A) large positive values of \(x\),
(B) large negative values of \(x\).
Sketch the curve.
Solve the inequality \(\frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 } < 2\).

Show mark scheme Show mark scheme source

Question 7:

Part (i):

Answer	Marks	Guidance
Answer	Marks	Guidance
\((-1,\ 0)\), \(\left(\tfrac{1}{2},\ 0\right)\)	B1	Both \(x\)-intercepts
\(\left(0,\ \tfrac{1}{3}\right)\)	B1	\(y\)-intercept
[2]

Part (ii):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(x = -\sqrt{3},\ x = \sqrt{3},\ y = 2\)	B1, B1, B1*
[3]

Part (iii):

Answer	Marks	Guidance
Answer	Marks	Guidance
Evidence of method needed e.g. evaluation for large values or convincing algebraic argument	M1
(A) Large positive \(x\), \(y \to 2^+\) so from above	A1 dep*	Allow if \(y=2\) indicated but not explicit in (ii)
(B) Large negative \(x\), \(y \to 2^-\) so from below	A1 dep*	SC B1 dep* Correct (A) and (B) following M0
[3]

Part (iv):

Answer	Marks	Guidance
Answer	Marks	Guidance
[Sketch of graph]	B1	Correct asymptotes shown and labelled
B1	Correct central branch with intercepts labelled
B1	Correct shape. Allow asymptotes at \(x = \pm 3\) and \(y = k\), \(k > 0\); asymptotic behaviour shown with clear minimum in the LH branch
[3]

Part (v):

Answer	Marks	Guidance
Answer	Marks	Guidance
\((x+1)(2x-1) = 2(x^2-3)\)	M1	Finding where curve cuts \(y=2\) (or valid solution of an inequality)
\(x = -5\)	B1
\(x < -5\) or \(-\sqrt{3} < x < \sqrt{3}\)	B1
[3]

## Question 7:

### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(-1,\ 0)$, $\left(\tfrac{1}{2},\ 0\right)$ | B1 | Both $x$-intercepts |
| $\left(0,\ \tfrac{1}{3}\right)$ | B1 | $y$-intercept |
| **[2]** | | |

### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = -\sqrt{3},\ x = \sqrt{3},\ y = 2$ | B1, B1, B1* | |
| **[3]** | | |

### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Evidence of method needed e.g. evaluation for large values or convincing algebraic argument | M1 | |
| (A) Large positive $x$, $y \to 2^+$ so from above | A1 dep* | Allow if $y=2$ indicated but not explicit in (ii) |
| (B) Large negative $x$, $y \to 2^-$ so from below | A1 dep* | SC B1 dep* Correct (A) and (B) following M0 |
| **[3]** | | |

### Part (iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Sketch of graph] | B1 | Correct asymptotes shown and labelled |
| | B1 | Correct central branch with intercepts labelled |
| | B1 | Correct shape. Allow asymptotes at $x = \pm 3$ and $y = k$, $k > 0$; asymptotic behaviour shown with clear minimum in the LH branch |
| **[3]** | | |

### Part (v):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(x+1)(2x-1) = 2(x^2-3)$ | M1 | Finding where curve cuts $y=2$ (or valid solution of an inequality) |
| $x = -5$ | B1 | |
| $x < -5$ or $-\sqrt{3} < x < \sqrt{3}$ | B1 | |
| **[3]** | | |

Show LaTeX source

7 A curve has equation $y = \frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 }$.
\begin{enumerate}[label=(\roman*)]
\item Find the coordinates of the points where the curve crosses the axes.
\item Write down the equations of the three asymptotes.
\item Determine whether the curve approaches the horizontal asymptote from above or from below for\\
(A) large positive values of $x$,\\
(B) large negative values of $x$.
\item Sketch the curve.
\item Solve the inequality $\frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 } < 2$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI FP1 2012 Q7 [14]}}

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