Past Exam Questions

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]
2. Show that \(1 < y \leq 3\) for all real values of \(x\). [4]
3. Find the coordinates of any stationary points on C. [2]
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 \(\frac{x^2 + 1}{x^2 + 3} < \frac{1}{2}\). [4]

The curve C has equation \(y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}\).

1. Find the equations of the asymptotes of C.
2. Find the coordinates of any stationary points on C.
3. Sketch C, stating the coordinates of any intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac{4x^2 + x + 1}{2x^2 - 7x + 3} \right| = k\) and state the set of values of \(k\) for which it has 4 distinct real solutions.

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

1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote. [2]
2. Find the coordinates of any stationary points on \(C\). [4]
3. Sketch \(C\), stating the coordinates of the intersections with the axes. [3]
4. Sketch \(y^2 = \frac{x+1}{x^2+3}\), stating the coordinates of the stationary points and the intersections with the axes. [4]

The curve C has equation \(y = f(x)\), where \(f(x) = \frac{x^2 + 2}{x^2 - x - 2}\).

1. Find the equations of the asymptotes of C.
2. Find the coordinates of any stationary points on C, giving your answers correct to 1 decimal place.
3. Sketch C, stating the coordinates of any intersections with the axes.
4. Sketch the curve with equation \(y = \frac{1}{f(x)}\).
5. Find the set of values for which \(\frac{1}{f(x)} < f(x)\).

The curve C has equation \(y = f(x)\), where \(f(x) = \frac{x^2 + 2}{x^2 - x - 2}\).

1. (a) Find the equations of the asymptotes of C.
2. (b) Find the coordinates of any stationary points on C, giving your answers correct to 1 decimal place.
3. (c) Sketch C, stating the coordinates of any intersections with the axes.
4. (d) Sketch the curve with equation \(y = \frac{1}{f(x)}\).
5. (e) Find the set of values for which \(\frac{1}{f(x)} < f(x)\).

The curve C has equation \(y = \frac{x^2 + 2x - 15}{x - 2}\).

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 intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac{x^2 - 2x - 15}{x - 2} \right|\).
5. Find the set of values of \(x\) for which \(\left| \frac{2x^2 + 4x - 30}{x - 2} \right| < 15\).

The curve \(C\) has equation \(y = \frac{x^2 + 2x - 15}{x - 2}\).

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 intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac{x^2 - 2x - 15}{x - 2} \right|\).
5. Find the set of values of \(x\) for which \(\left| \frac{2x^2 + 4x - 30}{x - 2} \right| < 15\).

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

1. Find the equations of the asymptotes of C.
2. Find the coordinates of the turning points on C.
3. Sketch C.
4. Sketch the curves with equations \(y = \left| \frac{x^2 + 2x + 1}{x - 3} \right|\) and \(y^2 = \frac{x^2 + 2x + 1}{x - 3}\) on a single diagram, clearly identifying each curve.

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

1. Show that C has no vertical asymptotes and state the equation of the horizontal asymptote of C.
2. Find the coordinates of the stationary points on C.
3. Sketch C, stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac{2x^2 - x - 1}{x^2 + x + 1} \right|\) and state the set of values of \(k\) for which \(\left| \frac{2x^2 - x - 1}{x^2 + x + 1} \right| = k\) has 4 distinct real solutions.

(a) Sketch the curve with equation \(y = \frac{x+1}{x-1}\).

(b) Sketch the curve with equation \(y = \frac{|x|+1}{|x|-1}\) and find the set of values of \(x\) for which \(\frac{|x|+1}{|x|-1} < -2\).

A curve \(C\) has equation \(y = \frac{ax^2 + x - 1}{x - 1}\), 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 \(1 < y < 1 + 4a\).
3. Sketch \(C\). You do not need to find the coordinates of the intersections with the axes.

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

1. Find the equations of the asymptotes of C.
2. Find the exact coordinates of the stationary points on C.
3. Sketch C, stating the coordinates of any intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac{x^2 - x}{x + 1} \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac{x^2 - x}{x + 1} \right| < 6\).

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

1. (a) Find the equations of the asymptotes of \(C\).
2. (b) Find the coordinates of any stationary points on \(C\).
3. (c) Sketch \(C\), stating the coordinates of the intersections with the axes.
4. (d) Sketch the curve with equation \(y = \left| \frac{x^2 - x - 3}{1 + x - x^2} \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac{x^2 - x - 3}{1 + x - x^2} \right| < 3\).

The curve \(C\) has equation \(y = \frac{4x+5}{4-4x^2}\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of any stationary points on \(C\).
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac{4x+5}{4-4x^2} \right|\) and find in exact form the set of values of \(x\) for which \(4|4x+5| > 5|4-4x^2|\).

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

1. (a) Find the equations of the asymptotes of C.
2. (b) Show that there is no point on C for which \(0 < y < 12\).
3. (c) Sketch C.
4. (d)
   1. Sketch the graphs of \(y = \left| \frac{x^2}{x-3} \right|\) and \(y = |x| - 3\) on a single diagram, stating the coordinates of the intersections with the axes.
   2. Use your sketch to find the set of values of \(c\) for which \(\left| \frac{x^2}{x-3} \right| \leq |x| + c\) has no solution.

10 The curves \(C_{1}\) and \(C_{2}\) have equations
\(y=\frac{a x}{x+5} \quad \text { and } \quad y=\frac{x^{2}+(a+10) x+5 a+26}{x+5}\)
respectively, where \(a\) is a constant and \(a>2\).
(i) Find the equations of the asymptotes of \(C_{1}\).

(ii) Find the equation of the oblique asymptote of \(C_{2}\).

(iii) Show that \(C_{1}\) and \(C_{2}\) do not intersect.

(iv) Find the coordinates of the stationary points of \(C_{2}\).

(v) Sketch \(C_{1}\) and \(C_{2}\) on a single diagram. [You do not need to calculate the coordinates of any points where \(C_{2}\) crosses the axes.]

Solve the simultaneous equations

\(\frac{x}{2y}-\frac{4y}{x}=-1,\qquad x=1-6y.\)

(a) Solve the equation \(x^{\frac13}-x^{\frac16}=2\).

(b) Solve the simultaneous equations \(\lg(x+2y)=0\) and \(x^2+4xy+y=1\).

(a) Given that \(256^{x+y} \times 16^{-2x}=8^{-x+3y}\), show that \(y=3x\).

(b) Hence find the exact solutions of the following simultaneous equations.

\(256^{x+y} \times 16^{-2x}=8^{-x+3y}\)

\(x^2+3y^2=56\)

Solve the following simultaneous equations. \(\begin{aligned} \frac{y}{x} & =\frac{3}{2} \\ \frac{y^{4}}{x^{5}} & =\frac{27}{16} \end{aligned}\)