| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | State domain or range |
| Difficulty | Standard +0.8 This FP2 question requires finding an asymptote (straightforward) and proving range bounds using calculus or completing the square. The inequality proof demands algebraic manipulation and critical point analysis, going beyond routine exercises but using standard A-level techniques. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(y = 0\) (or seen if working given) | B1 | Must be \(=\); accept \(x\)-axis; ignore any others |
| (ii) Write as quad. in \(x^2\) | M1 | \((x^2 - x + (3y-1) = 0)\) |
| Use for real \(x\), \(b^2-4ac \geq 0\) | M1 | Allow \(>\) ; or \(<\) for no real \(x\) |
| Produce quad. inequality in \(y\) | M1 | \(1 \leq 12y^2 - 4y ; 12y^2 - 4y - 1 \leq 0\) |
| Attempt to solve inequality | M1 | Factorise/ quadratic formula |
| Justify A.G. | A1 | e.g. diagram / table of values of \(y\) |
| SC | Attempt diff. by product/quotient | M1 |
| Solve \(\frac{dy}{dx} = 0\) for two real \(x\) | M1 | |
| Get both \((-3, -\frac{\sqrt{6}}{8})\) and \((1, \frac{1}{2})\) | A1 | |
| Clearly prove min./max. | A1 | |
| Justify fully the inequality e.g. detailed graph | B1 |
**(i)** State $y = 0$ (or seen if working given) | B1 | Must be $=$; accept $x$-axis; ignore any others
**(ii)** Write as quad. in $x^2$ | M1 | $(x^2 - x + (3y-1) = 0)$
Use for real $x$, $b^2-4ac \geq 0$ | M1 | Allow $>$ ; or $<$ for no real $x$
Produce quad. inequality in $y$ | M1 | $1 \leq 12y^2 - 4y ; 12y^2 - 4y - 1 \leq 0$
Attempt to solve inequality | M1 | Factorise/ quadratic formula
Justify A.G. | A1 | e.g. diagram / table of values of $y$
| SC | Attempt diff. by product/quotient | M1
| | Solve $\frac{dy}{dx} = 0$ for two real $x$ | M1
| | Get both $(-3, -\frac{\sqrt{6}}{8})$ and $(1, \frac{1}{2})$ | A1
| | Clearly prove min./max. | A1
| | Justify fully the inequality e.g. detailed graph | B13 The equation of a curve is $y = \frac { x + 1 } { x ^ { 2 } + 3 }$.\\
(i) State the equation of the asymptote of the curve.\\
(ii) Show that $- \frac { 1 } { 6 } \leqslant y \leqslant \frac { 1 } { 2 }$.
\hfill \mbox{\textit{OCR FP2 2006 Q3 [6]}}