| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Solving Inequalities with Rational Functions |
| Difficulty | Standard +0.8 This FP1 question requires understanding of rational functions including asymptotes, curve sketching, and solving rational inequalities by sign analysis. While the individual components (finding intercepts, asymptotes) are routine, part (iv) requires systematic analysis of sign changes across multiple discontinuities, which is more demanding than standard A-level work and typical of Further Maths. |
| Spec | 1.02g 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 polynomials1.02y Partial fractions: decompose rational functions |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(0, -\dfrac{5}{6}\right)\) | B1 | Allow for both \(x=0\) and \(y=-\dfrac{5}{6}\) seen |
| \((\sqrt{5},\, 0),\;(-\sqrt{5},\, 0)\) | B1 [2] | Allow \((\pm\sqrt{5}, 0)\) or for both \(y=0\) and \(x=\pm\sqrt{5}\) seen |
| Answer | Marks | Guidance |
|---|---|---|
| \(a = 2\) | B1 | |
| \(y = 0\) | B1 | |
| \(x = -3,\; x = 2\) | B1 [3] | Must be two equations |
| Answer | Marks | Guidance |
|---|---|---|
| Graph with two outer branches correctly placed | B1 | |
| Inner branches correctly placed | B1 | For good drawing, dep all 3 marks above |
| Correct asymptotes and intercepts labelled | B1 | Look for clear maximum point on right-hand branch |
| Condone turning points in \(-\sqrt{5} < x < \dfrac{1}{2},\; y < 0\) | B1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| \(-3 < x < -\sqrt{5},\quad \dfrac{1}{2} < x < 2,\quad x > \sqrt{5}\) | B3 [3] | One mark for each. Strict inequalities. Allow \(2.24\) for \(\sqrt{5}\). (If B3 then \(-1\) if more than 3 inequalities) |
## Question 7(i):
$\left(0, -\dfrac{5}{6}\right)$ | B1 | Allow for both $x=0$ and $y=-\dfrac{5}{6}$ seen
$(\sqrt{5},\, 0),\;(-\sqrt{5},\, 0)$ | B1 **[2]** | Allow $(\pm\sqrt{5}, 0)$ or for both $y=0$ and $x=\pm\sqrt{5}$ seen
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## Question 7(ii):
$a = 2$ | B1 |
$y = 0$ | B1 |
$x = -3,\; x = 2$ | B1 **[3]** | Must be two equations
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## Question 7(iii):
Graph with two outer branches correctly placed | B1 |
Inner branches correctly placed | B1 | For good drawing, dep all 3 marks above
Correct asymptotes and intercepts labelled | B1 | Look for clear maximum point on right-hand branch
Condone turning points in $-\sqrt{5} < x < \dfrac{1}{2},\; y < 0$ | B1 **[4]** |
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## Question 7(iv):
$-3 < x < -\sqrt{5},\quad \dfrac{1}{2} < x < 2,\quad x > \sqrt{5}$ | B3 **[3]** | One mark for each. Strict inequalities. Allow $2.24$ for $\sqrt{5}$. (If B3 then $-1$ if more than 3 inequalities)7 A curve has equation $y = \frac { x ^ { 2 } - 5 } { ( x + 3 ) ( x - 2 ) ( a x - 1 ) }$, where $a$ is a constant.\\
(i) Find the coordinates of the points where the curve crosses the $x$-axis and the $y$-axis.\\
(ii) You are given that the curve has a vertical asymptote at $x = \frac { 1 } { 2 }$. Write down the value of $a$ and the equations of the other asymptotes.\\
(iii) Sketch the curve.\\
(iv) Find the set of values of $x$ for which $y > 0$.
\hfill \mbox{\textit{OCR MEI FP1 2014 Q7 [12]}}