Range restriction with discriminant (quadratic denominator)

6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } + 1 }\).

1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote.
2. Show that \(1 < y \leqslant 3\) for all real values of \(x\).
3. Find the coordinates of any stationary points on \(C\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-12_2718_42_107_2007} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-13_2720_40_106_18}
4. Sketch \(C\), stating the coordinates of any intersections with the axes and labelling the asymptote.
5. Sketch the curve with equation \(y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 }\) and find the set of values of \(x\) for which \(\frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 } < \frac { 1 } { 2 }\).

10 A curve \(C\) has equation $$y = \frac { 5 \left( x ^ { 2 } - x - 2 \right) } { x ^ { 2 } + 5 x + 10 }$$ Find the coordinates of the points of intersection of \(C\) with the axes. Show that, for all real values of \(x , - 1 \leqslant y \leqslant 15\). Sketch \(C\), stating the coordinates of any turning points and the equation of the horizontal asymptote.
[0pt] [Question 11 is printed on the next page.]

8 A curve \(C\) has equation $$y = \frac { x ( x - 3 ) } { x ^ { 2 } + 3 }$$

1. State the equation of the asymptote of \(C\).
2. The line \(y = k\) intersects the curve \(C\). Show that \(4 k ^ { 2 } - 4 k - 3 \leqslant 0\).
3. Hence find the coordinates of the stationary points of the curve \(C\). (No credit will be given for solutions based on differentiation.) \includegraphics[max width=\textwidth, alt={}, center]{e45b07a3-e303-4caf-8f3a-5341bad7560a-24_2488_1728_219_141}

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

The curve \(C\) has equation $$y = \frac{5x^2 + 5x + 1}{x^2 + x + 1}.$$

1. Find the equation of the asymptote of \(C\). [2]
2. Show that, for all real values of \(x\), \(-\frac{1}{5} \leqslant y < 5\). [4]
3. Find the coordinates of any stationary points of \(C\). [2]
4. Sketch \(C\), stating the coordinates of any intersections with the \(y\)-axis. [2]