| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Stationary Points of Rational Functions |
| Difficulty | Standard +0.8 This FP1 question requires polynomial long division to identify the oblique asymptote, differentiation of a rational function using the quotient rule, and solving a quadratic inequality. While each technique is standard, the multi-step nature (4 parts building on each other), the need to connect asymptotic behavior with algebraic manipulation, and the final inequality requiring careful analysis of when a quadratic has no real roots makes this moderately challenging, above average difficulty for A-level but accessible to well-prepared FP1 students. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = 1\) | B1 | States vertical asymptote. Part total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = 2\) | B1 | States the value of \(a\) |
| \(y = ax + a + b + \frac{a+b+c}{(x-1)}\) | M1 | Divides |
| \(2 + b = -5 \Rightarrow b = -7\) (AG) | A1 | Compares coefficients to obtain \(b\). Part total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = 2x - 5 + \frac{a}{x-1}\) | (M1) | |
| \(= \frac{2x^2 - 7x + 5 + a}{x-1}\) | (B1A1) | |
| Equate coefficients: \(a=2,\; b=-7\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y' = 2 - \frac{(c-5)}{(x-1)^2} = 0\) | M1A1 | Differentiates and uses given value of \(x\) to obtain \(c\) |
| When \(x=2\) then \(c=7\) | A1 | Part total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Let \(y = \frac{2x^2 - 7x + 7}{(x-1)} = k\) | Forms quadratic in \(x\) | |
| \(\Rightarrow 2x^2 - (7+k)x + 7 + k = 0\) | B1 | |
| No real roots \(\Rightarrow (7+k)^2 - 8(7+k) < 0\) | M1 | Uses discriminant |
| \(\Rightarrow k^2 + 6k - 7 < 0\) | A1 | |
| \(\Rightarrow (k+7)(k-1) < 0\) | ||
| \(\Rightarrow -7 < k < 1\) | A1 | Obtains required result. Part total: 4 |
| Total: [11] |
## Question 9(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = 1$ | B1 | States vertical asymptote. **Part total: 1** |
## Question 9(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 2$ | B1 | States the value of $a$ |
| $y = ax + a + b + \frac{a+b+c}{(x-1)}$ | M1 | Divides |
| $2 + b = -5 \Rightarrow b = -7$ (AG) | A1 | Compares coefficients to obtain $b$. **Part total: 3** |
**Or alternatively:**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 2x - 5 + \frac{a}{x-1}$ | (M1) | |
| $= \frac{2x^2 - 7x + 5 + a}{x-1}$ | (B1A1) | |
| Equate coefficients: $a=2,\; b=-7$ | | |
## Question 9(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y' = 2 - \frac{(c-5)}{(x-1)^2} = 0$ | M1A1 | Differentiates and uses given value of $x$ to obtain $c$ |
| When $x=2$ then $c=7$ | A1 | **Part total: 3** |
## Question 9(iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Let $y = \frac{2x^2 - 7x + 7}{(x-1)} = k$ | | Forms quadratic in $x$ |
| $\Rightarrow 2x^2 - (7+k)x + 7 + k = 0$ | B1 | |
| No real roots $\Rightarrow (7+k)^2 - 8(7+k) < 0$ | M1 | Uses discriminant |
| $\Rightarrow k^2 + 6k - 7 < 0$ | A1 | |
| $\Rightarrow (k+7)(k-1) < 0$ | | |
| $\Rightarrow -7 < k < 1$ | A1 | Obtains required result. **Part total: 4** |
| | | **Total: [11]** |
---9 The curve $C$ with equation
$$y = \frac { a x ^ { 2 } + b x + c } { x - 1 }$$
where $a , b$ and $c$ are constants, has two asymptotes. It is given that $y = 2 x - 5$ is one of these asymptotes.\\
(i) State the equation of the other asymptote.\\
(ii) Find the value of $a$ and show that $b = - 7$.\\
(iii) Given also that $C$ has a turning point when $x = 2$, find the value of $c$.\\
(iv) Find the set of values of $k$ for which the line $y = k$ does not intersect $C$.
\hfill \mbox{\textit{CAIE FP1 2011 Q9 [11]}}