| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Proving Rational Function Takes All Real Values |
| Difficulty | Standard +0.8 This FP2 question requires finding asymptotes (standard technique) and proving a rational function takes all real values by rearranging to form a quadratic in x and analyzing the discriminant. Part (ii) requires algebraic manipulation and discriminant analysis beyond routine A-level, but is a well-established FP2 technique rather than requiring novel insight. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt division/equate coeff. Get \(a = 2, b = -9\). Derive/quote \(x = 1\) | M1, A1, B1 | To lead to some \(ax+b\) (allow \(b=0\) here). Must be equations |
| (ii) Write as quadratic in \(x\). Use \(b^2 \geq 4ac\) (for real \(x\)). Get \(y^2 +14y+169 \geq 0\). Attempt to justify positive/negative. Get \((y+7)^2 +120 \geq 0\) – true for all \(y\) | M1, M1, A1, M1, A1 | \((2x^2-x(11+y)+(y-6)=0)\). Allow \(<, >\). Complete the square/sketch. SC: Attempt diff; quot./prod. rule M1. Attempt to solve \(dy/dx = 0\) M1. Show \(2x^2 - 4x + 17 = 0\) has no real roots e.g. \(b^2 - 4ac < 0\) A1. Attempt to use no t.p. M1. Justify all y e.g. consider asymptotes and approaches A1 |
**(i)** Attempt division/equate coeff. Get $a = 2, b = -9$. Derive/quote $x = 1$ | M1, A1, B1 | To lead to some $ax+b$ (allow $b=0$ here). Must be equations
**(ii)** Write as quadratic in $x$. Use $b^2 \geq 4ac$ (for real $x$). Get $y^2 +14y+169 \geq 0$. Attempt to justify positive/negative. Get $(y+7)^2 +120 \geq 0$ – true for all $y$ | M1, M1, A1, M1, A1 | $(2x^2-x(11+y)+(y-6)=0)$. Allow $<, >$. Complete the square/sketch. SC: Attempt diff; quot./prod. rule M1. Attempt to solve $dy/dx = 0$ M1. Show $2x^2 - 4x + 17 = 0$ has no real roots e.g. $b^2 - 4ac < 0$ A1. Attempt to use no t.p. M1. Justify all y e.g. consider asymptotes and approaches A16 The equation of a curve is $y = \frac { 2 x ^ { 2 } - 11 x - 6 } { x - 1 }$.\\
(i) Find the equations of the asymptotes of the curve.\\
(ii) Show that $y$ takes all real values.
\hfill \mbox{\textit{OCR FP2 2008 Q6 [8]}}