| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| 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 factorising both numerator and denominator, finding asymptotes, analysing sign changes across critical points, and solving a rational inequality. While systematic, it demands careful consideration of multiple cases and sign analysis across discontinuities, making it moderately challenging for Further Maths students. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks |
|---|---|
| Vertical asymptotes: \(x^2+3x-4=(x+4)(x-1)=0\), so \(x=-4, x=1\) | B1 B1 |
| Horizontal: \(y=3\) | B1 |
| Answer | Marks |
|---|---|
| \(y = 3 - \frac{9x+3}{x^2+3x-4}\); for large positive \(x\), approaches from below; large negative \(x\), approaches from above | M1 A1 A1 |
| Answer | Marks |
|---|---|
| Correct sketch showing asymptotes, branches | B3 |
| Answer | Marks |
|---|---|
| \(\frac{3x^2-9}{x^2+3x-4}\geq 0\): critical values \(x=\pm\sqrt{3}, x=-4, x=1\) | M1 |
| \(x\leq -4\) or \(-\sqrt{3}\leq x\leq 1\)... careful with asymptotes... \(x<-4\) or \(-\sqrt{3}\leq x\leq\sqrt{3}\) or \(x>1\) adjusted correctly | A1 A1 |
## Question 8:
**(i)**
Vertical asymptotes: $x^2+3x-4=(x+4)(x-1)=0$, so $x=-4, x=1$ | B1 B1 |
Horizontal: $y=3$ | B1 |
**(ii)**
$y = 3 - \frac{9x+3}{x^2+3x-4}$; for large positive $x$, approaches from below; large negative $x$, approaches from above | M1 A1 A1 |
**(iii)**
Correct sketch showing asymptotes, branches | B3 |
**(iv)**
$\frac{3x^2-9}{x^2+3x-4}\geq 0$: critical values $x=\pm\sqrt{3}, x=-4, x=1$ | M1 |
$x\leq -4$ or $-\sqrt{3}\leq x\leq 1$... careful with asymptotes... $x<-4$ or $-\sqrt{3}\leq x\leq\sqrt{3}$ or $x>1$ adjusted correctly | A1 A1 |
---8 A curve has equation $y = \frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 }$.\\
(i) Find the equations of the two vertical asymptotes and the one horizontal asymptote of this curve.\\
(ii) State, with justification, how the curve approaches the horizontal asymptote for large positive and large negative values of $x$.\\
(iii) Sketch the curve.\\
(iv) Solve the inequality $\frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 } \geqslant 0$.
\hfill \mbox{\textit{OCR MEI FP1 2016 Q8 [12]}}