Past Exam Questions

Express \(\frac{2x^{2}-x+5}{x^{2}-1}\) in the form \(2+\frac{A}{x-1}+\frac{B}{x+1}\), where \(A\) and \(B\) are integers to be found.

The curve \(C\) has equation \(y=\frac{2x^{2}-x+5}{x^{2}-1}\). Show that there are two distinct values of \(x\) for which \(\frac{dy}{dx}=0\).

Sketch \(C\), stating the equations of the asymptotes and giving the coordinates of any points of intersection with the coordinate axes and with the asymptotes. You do not need to find the coordinates of the turning points.

The curve \(C\) has equation \(y=\frac{x+2}{x^{2}-9}\). Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}\lt 0\) at all points on \(C\).

State the equations of the asymptotes of \(C\).

Sketch \(C\), showing the coordinates of any points of intersection with the coordinate axes.

A curve \(C\) has equation \(y=\frac{x^{2}}{x-2}\). Find the equations of the asymptotes of \(C\).

Show that there are no points on \(C\) for which \(0\lt y\lt 8\).

Sketch \(C\), giving the coordinates of the turning points.

The curve \(C\) has equation
\(y=\frac{3 x-9}{(x-2)(x+1)}\)
(i) Find the equations of the asymptotes of \(C\).

(ii) Show that there is no point on \(C\) for which \(\frac{1}{3}\lt y\lt 3\).

(iii) Find the coordinates of the turning points of \(C\).

(iv) Sketch \(C\).

The curve C has equation \(y = \frac{x^2 + ax + 1}{x + 2}\), where \(a > \frac{5}{2}\).

1. Find the equations of the asymptotes of C.
2. Show that C has no stationary points.
3. Sketch C, stating the coordinates of the point of intersection with the y-axis and labelling the asymptotes.
4. 1. Sketch the curve with equation \(y = \left| \frac{x^2 + ax + 1}{x + 2} \right|\).
   2. On your sketch in part (i), draw the line \(y = a\).
   3. It is given that \(\left| \frac{x^2 + ax + 1}{x + 2} \right| < a\) for \(-5 - \sqrt{14} < x < -3\) and \(-5 + \sqrt{14} < x < 3\). Find the value of \(a\).

The curve \(C\) has equation \(y = \frac{x^2 + ax + 1}{x + 2}\), where \(a > \frac{5}{2}\).

1. (a) Find the equations of the asymptotes of \(C\).
2. (b) Show that \(C\) has no stationary points.
3. (c) Sketch \(C\), stating the coordinates of the point of intersection with the \(y\)-axis and labelling the asymptotes.
4. (d)
   1. Sketch the curve with equation \(y = \left| \frac{x^2 + ax + 1}{x + 2} \right|\).
   2. On your sketch in part (i), draw the line \(y = a\).
   3. It is given that \(\left| \frac{x^2 + ax + 1}{x + 2} \right| < a\) for \(-5 - \sqrt{14} < x < -3\) and \(-5 + \sqrt{14} < x < 3\). Find the value of \(a\).

The curve C has equation \(y = \frac{x^2 + x + 9}{x + 1}\).

(a) Find the equations of the asymptotes of C.

(b) Find the coordinates of the stationary points on C.

6 The curve \(C\) has equation \(y=\frac{x^{2}+x-1}{x-1}\).
(a) Find the equations of the asymptotes of \(C\).

(b) Show that there is no point on \(C\) for which \(1<y<5\).

(c) Find the coordinates of the intersections of \(C\) with the axes, and sketch \(C\).

(d) Sketch the curve with equation \(y=\left|\frac{x^{2}+x-1}{x-1}\right|\).

3 The curve \(C\) has equation \(y=\frac{x^{2}}{2 x+1}\).
(a) Find the equations of the asymptotes of \(C\).

(b) Find the coordinates of the stationary points on \(C\).

(c) Sketch \(C\).

6 The curve \(C\) has equation
\(y=\frac{x^{2}}{k x-1}\)
where \(k\) is a positive constant.
(i) Obtain the equations of the asymptotes of \(C\).

(ii) Find the coordinates of the stationary points of \(C\).

(iii) Sketch \(C\).

The line \(y=2 x+1\) is an asymptote of the curve \(C\) with equation
\(y=\frac{x^{2}+1}{a x+b}\)
(i) Find the values of the constants \(a\) and \(b\).

(ii) State the equation of the other asymptote of \(C\).

(iii) Sketch C. [Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.]

The curve \(C\) has equation
\(y=\frac{x^{2}+b}{x+b},\)
where \(b\) is a positive constant.
(i) Find the equations of the asymptotes of \(C\).

(ii) Show that \(C\) does not intersect the \(x\)-axis.

(iii) Justifying your answer, find the number of stationary points on \(C\).

(iv) Sketch \(C\). Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.

The curve \(C\) has equation
\(y=\frac{x^{2}+7 x+6}{x-2} .\)
(i) Find the coordinates of the points of intersection of \(C\) with the axes.

(ii) Find the equation of each of the asymptotes of \(C\).

(iii) Sketch \(C\).

The curve \(C\) has equation
\(y=\frac{x^{2}+a x-1}{x+1},\)
where \(a\) is constant and \(a\gt 1\).
(i) Find the equations of the asymptotes of \(C\).

(ii) Show that \(C\) intersects the \(x\)-axis twice.

(iii) Justifying your answer, find the number of stationary points on \(C\).

(iv) Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis.

The curve \(C\) has equation
\(y=\frac{5 x^{2}+5 x+1}{x^{2}+x+1} .\)
(i) Find the equation of the asymptote of \(C\).

(ii) Show that, for all real values of \(x,-\frac{1}{3} \leqslant y\lt 5\).

(iii) Find the coordinates of any stationary points of \(C\).

(iv) Sketch \(C\), stating the coordinates of any intersections with the \(y\)-axis.

The curve \(C\) has equation

\(y=\frac{ax^2+bx+c}{x+d},\)

where \(a\), \(b\), \(c\), and \(d\) are constants. The curve cuts the \(y\)-axis at \((0,-2)\) and has asymptotes \(x=2\) and \(y=x+1\).

(i) Write down the value of \(d\).

(ii) Determine the values of \(a\), \(b\), and \(c\).

(iii) Show that, at all points on \(C\), either \(y\leq3-2\sqrt6\) or \(y\geq3+2\sqrt6\).

A curve \(C\) has equation \(y=\frac{2 x^{2}+x-1}{x-1}\). Find the equations of the asymptotes of \(C\).

Show that there is no point on \(C\) for which \(1\lt y\lt 9\).