1.02o Sketch reciprocal curves: y=a/x and y=a/x^2

7

1. On the pair of axes in the Printed Answer Booklet, sketch the graphs of

4 The equation of a curve is \(\mathrm { y } = \frac { \mathrm { k } } { \mathrm { x } ^ { 2 } }\), where \(k\) is a constant.
The curve passes through the point \(( 2,1 )\).

1. Find the value of \(k\).
2. Sketch the curve.

8 For a particular value of \(a\), the curve \(\mathrm { y } = \frac { \mathrm { a } } { \mathrm { x } ^ { 2 } }\) passes through the point \(( 3,1 )\).
Find the coordinates of all the other points on the curve where both the \(x\)-coordinate and the \(y\)-coordinate are integers.

7 A curve has equation $$y = \frac { 3 x - 1 } { x + 2 }$$

1. Write down the equations of the two asymptotes to the curve.
2. Sketch the curve, indicating the coordinates of the points where the curve intersects the coordinate axes.
3. Hence, or otherwise, solve the inequality $$0 < \frac { 3 x - 1 } { x + 2 } < 3$$

7 A curve \(C\) has equation $$y = 7 + \frac { 1 } { x + 1 }$$

1. Define the translation which transforms the curve with equation \(y = \frac { 1 } { x }\) onto the curve \(C\).
2. 1. Write down the equations of the two asymptotes of \(C\).
   2. Find the coordinates of the points where the curve \(C\) intersects the coordinate axes.
3. Sketch the curve \(C\) and its two asymptotes.

8 A curve has equation $$y = \frac { x ^ { 2 } } { ( x - 1 ) ( x - 5 ) }$$

1. Write down the equations of the three asymptotes to the curve.
2. Show that the curve has no point of intersection with the line \(y = - 1\).
3. 1. Show that, if the curve intersects the line \(y = k\), then the \(x\)-coordinates of the points of intersection must satisfy the equation $$( k - 1 ) x ^ { 2 } - 6 k x + 5 k = 0$$
   2. Show that, if this equation has equal roots, then $$k ( 4 k + 5 ) = 0$$
4. Hence find the coordinates of the two stationary points on the curve.

7

1. 1. Write down the equations of the two asymptotes of the curve \(y = \frac { 1 } { x - 3 }\).
   2. Sketch the curve \(y = \frac { 1 } { x - 3 }\), showing the coordinates of any points of intersection with the coordinate axes.
   3. On the same axes, again showing the coordinates of any points of intersection with the coordinate axes, sketch the line \(y = 2 x - 5\).
2. 1. Solve the equation $$\frac { 1 } { x - 3 } = 2 x - 5$$
   2. Find the solution of the inequality $$\frac { 1 } { x - 3 } < 2 x - 5$$ □ \includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-08_367_197_2496_155}

8 A curve has equation \(y = \frac { 1 } { x ^ { 2 } - 4 }\).

1. 1. Write down the equations of the three asymptotes of the curve.
   2. Sketch the curve, showing the coordinates of any points of intersection with the coordinate axes.
2. Hence, or otherwise, solve the inequality $$\frac { 1 } { x ^ { 2 } - 4 } < - 2$$

1 A family of functions is defined as $$f ( x ) = a x + \frac { x ^ { 2 } } { 1 + x } , \quad x \neq - 1$$ where the parameter \(a\) is a real number. You may find it helpful to use a slider (for \(a\) ) to investigate the family of curves \(y = f ( x )\). \begin{enumerate}[label=(\alph*)] \item \begin{enumerate}[label=(\roman*)] \item On the axes in the Printed Answer Booklet, sketch the curve \(y = f ( x )\) in each of the following cases.

• \(a = - 2\)
• \(a = - 1\)
• \(a = 0\)
• State a feature which is common to the curve in all three cases, \(a = - 2\), \(a = - 1\) and \(a = 0\).
• State a feature of the curve for the cases \(a = - 2 , a = - 1\) that is not a feature of the curve in the case \(a = 0\).
• 1. Determine the equation of the oblique asymptote to the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) in terms of \(a\).
  2. For \(b \neq - 1,0,1\) let \(A\) be the point with coordinates ( \(- b , \mathrm { f } ( - b )\) ) and let \(B\) be the point with coordinates ( \(b , \mathrm { f } ( b )\) ).

Show that the \(y\)-coordinate of the point at which the chord to the curve \(y = f ( x )\) between \(A\) and \(B\) meets the \(y\)-axis is independent of \(a\).

With \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), determine the range of values of \(a\) for which

Find its coordinates and fully justify that it is a cusp.

1 A family of curves is given by the equation $$y = \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 }$$ where the parameter \(a\) is a real number.

1. 1. On the axes in the Printed Answer Booklet, sketch the curve in each of these cases.
      • \(a = - 0.5\)
      • \(a = - 0.1\)
      • \(a = 0.5\)
      • State one feature of the curve for the cases \(a = - 0.5\) and \(a = - 0.1\) that is not a feature of the curve in the case \(a = 0.5\).
      • By using a slider for \(a\), or otherwise, write down the non-zero value of \(a\) for which the points on the curve (\textit{) all lie on a straight line.
      • Write down the equation of the vertical asymptote of the curve (}).
      The equation of the curve (*) can be written in the form \(y = x + A + \frac { a ^ { 2 } - a } { x - 1 }\), where \(A\) is a constant.
   2. Show that \(A = 0\).
   3. Hence, or otherwise, find the value of $$\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 } - x \right) .$$
   4. Explain the significance of the result in part (a)(vi) in terms of a feature of the curve (*).
   5. In this part of the question the value of the parameter \(a\) satisfies \(0 < a < 1\). For values of \(a\) in this range the curve intersects the \(x\)-axis at points X and Y . The point Z has coordinates \(( 0 , - 1 )\). These three points form a triangle XYZ.
      1. Determine, in terms of \(a\), the area of the triangle XYZ.
      2. Find the maximum area of the triangle XYZ.

4

1. Sketch the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\) on the axes provided in the Printed Answer Booklet.
2. In this question you must show detailed reasoning. Find the exact coordinates of the points of intersection of the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\).

9 A curve \(C\) has equation $$y = \frac { 2 } { x ( x - 4 ) }$$

1. Write down the equations of the three asymptotes of \(C\).
2. The curve \(C\) has one stationary point. By considering an appropriate quadratic equation, find the coordinates of this stationary point.
   (No credit will be given for solutions based on differentiation.)
3. Sketch the curve \(C\).

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

1. 1. Write down the equations of the asymptotes of the curve \(C\).
   2. Sketch the curve \(C\).
2. The line \(y = x - 3\) intersects the curve \(C\) at a point which has \(x\)-coordinate \(\alpha\).
   1. Show that \(\alpha\) lies within the interval \(3 < x < 4\).
   2. Starting from the interval \(3 < x < 4\), use interval bisection twice to obtain an interval of width 0.25 within which \(\alpha\) must lie.

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.}

1. Sketch the curve \(C\) with equation $$y = \frac { 1 } { 2 - x } \quad x \neq 2$$ State on your sketch
   • the equation of the vertical asymptote
   • the coordinates of the intersection of \(C\) with the \(y\)-axis
   The straight line \(l\) has equation \(y = k x - 4\), where \(k\) is a constant.
   Given that \(l\) cuts \(C\) at least once,
2. 1. show that $$k ^ { 2 } - 5 k + 4 \geqslant 0$$
   2. find the range of possible values for \(k\).

Given that \(f(x) = \frac{1}{x}\), \(x \neq 0\),

1. sketch the graph of \(y = f(x) + 3\) and state the equations of the asymptotes. [4]
2. Find the coordinates of the point where \(y = f(x) + 3\) crosses a coordinate axis. [2]

Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).

1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]

\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{4}{x^2}\).

1. On the copy of Fig. 12, draw accurately the line \(y = 2x + 5\) and hence find graphically the three roots of the equation \(\frac{4}{x^2} = 2x + 5\). [3]
2. Show that the equation you have solved in part (i) may be written as \(2x^3 + 5x^2 - 4 = 0\). Verify that \(x = -2\) is a root of this equation and hence find, in exact form, the other two roots. [6]
3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x^3 + 2x^2 - 4 = 0\). [3]

\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{-1}{x - 3}\).

1. Draw accurately, on the copy of Fig. 12, the graph of \(y = x^2 - 4x + 1\) for \(-1 < x < 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac{-1}{x - 3}\) and \(y = x^2 - 4x + 1\). [5]
2. Show algebraically that, where the curves intersect, \(x^3 - 7x^2 + 13x - 4 = 0\). [3]
3. Use the fact that \(x = 4\) is a root of \(x^3 - 7x^2 + 13x - 4 = 0\) to find a quadratic factor of \(x^3 - 7x^2 + 13x - 4\). Hence find the exact values of the other two roots of this equation. [5]

\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{1}{x - 2}\).

1. Draw accurately the graph of \(y = 2x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\). [3]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = -x + k\). Hence find the exact values of \(k\) for which \(y = -x + k\) is a tangent to \(y = \frac{1}{x - 2}\). [4]