AQA FP1 2013 June — Question 9 14 marks

Exam BoardAQA
ModuleFP1 (Further Pure Mathematics 1)
Year2013
SessionJune
Marks14
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Mark schemeDownload PDF ↗
TopicCurve Sketching
TypeRange restriction with excluded interval (linear/mixed denominator)
DifficultyChallenging +1.2 This is a structured FP1 rational function question with clear scaffolding through parts (a)-(c). Finding asymptotes is routine; part (b) guides students through algebraic manipulation and discriminant analysis to find the stationary point without calculus. While it requires multiple techniques (factoring, discriminant, completing the square), the heavy scaffolding and standard methods make it moderately above average difficulty but not requiring significant novel insight.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials
9 A curve has equation $$y = \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 2 } - 2 x - 3 }$$
  1. Find the equations of the three asymptotes of the curve.
    1. Show that if the line \(y = k\) intersects the curve then $$( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$$
    2. Given that the equation \(( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0\) has real roots, show that $$k ^ { 2 } - k \geqslant 0$$
    3. Hence show that the curve has only one stationary point and find its coordinates.
      (No credit will be given for solutions based on differentiation.)
  3. Sketch the curve and its asymptotes.
A curve has equation
\[y = \frac{x^2 - 2x + 1}{x^2 - 2x - 3}\]
(a) Find the equations of the three asymptotes of the curve. (3 marks)
(b)(i) Show that if the line \(y = k\) intersects the curve then
\[(k - 1)x^2 - 2(k - 1)x - (3k + 1) = 0\]
(1 mark)
(ii) Given that the equation \((k - 1)x^2 - 2(k - 1)x - (3k + 1) = 0\) has real roots, show that
\[k^2 - k \leq 0\]
(3 marks)
(iii) Hence show that the curve has only one stationary point and find its coordinates.
(No credit will be given for solutions based on differentiation.) (4 marks)
(c) Sketch the curve and its asymptotes. (3 marks)
A curve has equation
$$y = \frac{x^2 - 2x + 1}{x^2 - 2x - 3}$$

**(a) Find the equations of the three asymptotes of the curve. (3 marks)**

**(b)(i) Show that if the line $y = k$ intersects the curve then**
$$(k - 1)x^2 - 2(k - 1)x - (3k + 1) = 0$$
**(1 mark)**

(ii) Given that the equation $(k - 1)x^2 - 2(k - 1)x - (3k + 1) = 0$ has real roots, show that
$$k^2 - k \leq 0$$
**(3 marks)**

(iii) Hence show that the curve has only one stationary point and find its coordinates.
(No credit will be given for solutions based on differentiation.) **(4 marks)**

**(c) Sketch the curve and its asymptotes. (3 marks)**
9 A curve has equation

$$y = \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 2 } - 2 x - 3 }$$
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the three asymptotes of the curve.
\item \begin{enumerate}[label=(\roman*)]
\item Show that if the line $y = k$ intersects the curve then

$$( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$$
\item Given that the equation $( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$ has real roots, show that

$$k ^ { 2 } - k \geqslant 0$$
\item Hence show that the curve has only one stationary point and find its coordinates.\\
(No credit will be given for solutions based on differentiation.)
\end{enumerate}\item Sketch the curve and its asymptotes.
\end{enumerate}

\hfill \mbox{\textit{AQA FP1 2013 Q9 [14]}}
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Q1 3 Q2 7 Q3 8 Q4 7 Q5 8 Q6 11 Q7 11 Q8 6 Q9 14