Rational functions with parameters

3 A curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { a } \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 }\), where \(a\) is a positive constant.

1. Find the equations of the asymptotes of \(C\).
2. Show that there is no point on \(C\) for which \(1 < \mathrm { y } < 1 + 4 \mathrm { a }\).
3. Sketch C. You do not need to find the coordinates of the intersections with the axes.

5．The function \(f\) is given by $$f ( x ) = \frac { 1 } { \lambda } \left( x ^ { 2 } - 4 \right) \left( x ^ { 2 } - 25 \right)$$ where \(x\) is real and \(\lambda\) is a positive integer．
（a）Sketch the graph of \(y = \mathrm { f } ( x )\) showing clearly where the graph crosses the coordinate axes．
（b）Find，in terms of \(\lambda\) ，the range of f ．
（c）Find the sets of positive integers \(k\) and \(\lambda\) such that the equation $$k = | \mathrm { f } ( x ) |$$ has exactly \(k\) distinct real roots．

10 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant. Obtain the equation of each of the asymptotes of \(C\). In separate diagrams, sketch \(C\) for the cases \(\lambda > 0\) and \(\lambda < 0\). In both cases the coordinates of the turning points must be indicated.

7 The curve \(C\) has equation $$y = \lambda x + \frac { x } { x - 2 }$$ where \(\lambda\) is a non-zero constant. Find the equations of the asymptotes of \(C\). Show that \(C\) has no turning points if \(\lambda < 0\). Sketch \(C\) in the case \(\lambda = - 1\), stating the coordinates of the intersections with the axes.

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.
2. 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.

16 Curve \(C\) has equation \(y = \frac { a x } { x + b }\) where \(a\) and \(b\) are constants.
The equations of the asymptotes to \(C\) are \(x = - 2\) and \(y = 3\)
\includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-20_796_819_459_609} 16

1. Write down the value of \(a\) and the value of \(b\) 16
2. The gradient of \(C\) at the origin is \(\frac { 3 } { 2 }\)
   With reference to the graph, explain why there is exactly one root of the equation $$\frac { a x } { x + b } = \frac { 3 x } { 2 }$$ 16
3. Using the values found in part (a), solve the inequality $$\frac { a x } { x + b } \leq 1 - x$$ [4 marks]

14 The function f is defined by $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 3 } { x ^ { 2 } + p x + 7 } \quad x \in \mathbb { R }$$ where \(p\) is a constant.
The graph of \(y = \mathrm { f } ( x )\) has only one asymptote.
14

1. Write down the equation of the asymptote.
   14
2. Find the set of possible values of \(p\)
   □
   14
3. Find the coordinates of the points at which the graph of \(y = \mathrm { f } ( x )\) intersects the axes. \section*{Question 14 continues on the next page} 14
4. \(\quad A\) curve \(C\) has equation $$y = \frac { x ^ { 2 } - 3 } { x ^ { 2 } - 3 x + 7 }$$ The curve \(C\) has a local minimum at the point \(M\) as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-24_371_835_587_605} The line \(y = k\) intersects curve \(C\)
   14
5. 1. Show that $$19 k ^ { 2 } - 16 k - 12 \leq 0$$ 14
6. (ii) Hence, find the \(y\)-coordinate of point \(M\)

12 A curve, \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 12 x + 12 } { x ^ { 2 } + 4 x - 4 }\)
The line \(y = k\) intersects the curve, \(C _ { 1 }\) 12

1. 1. Show that \(( k + 3 ) ( k - 1 ) \geq 0\)
      [0pt] [5 marks]
      12
2. (ii) Hence find the coordinates of the stationary point of \(C _ { 1 }\) that is a maximum point.
   [0pt] [4 marks] 12
3. Show that the curve \(C _ { 2 }\) whose equation is \(y = \frac { 1 } { \mathrm { f } ( x ) }\), has no vertical asymptotes.
   [0pt] [2 marks]
   12
4. State the equation of the line that is a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\).
   [0pt] [1 mark]