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

9 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-4_602_899_248_625} The diagram shows part of the curve \(y = x + \frac { 4 } { x }\) which has a minimum point at \(M\). The line \(y = 5\) intersects the curve at the points \(A\) and \(B\).

1. Find the coordinates of \(A , B\) and \(M\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

2 Find the set of values of \(a\) for which the curve \(y = - \frac { 2 } { x }\) and the straight line \(y = a x + 3 a\) meet at two distinct points.

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

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points on \(C\).
3. Sketch \(C\).

6 The curve \(C\) has equation \(\mathrm { y } = \frac { 10 + \mathrm { x } - 2 \mathrm { x } ^ { 2 } } { 2 \mathrm { x } - 3 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no turning points.
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right| < 4\).

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

1. Find the equations of the asymptotes of \(C\).
2. Find the 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 \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 } \right|\) and find the set of values of \(x\) for which \(2 \left| x ^ { 2 } + x + 9 \right| > 13 | x + 1 |\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The curve \(C\) has equation \(y = f ( x )\), where \(f ( x ) = \frac { x ^ { 2 } } { x + 1 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of any stationary points on \(C\).
3. Sketch \(C\).
4. Find the coordinates of any stationary points on the curve with equation \(\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }\).
5. Sketch the curve with equation \(y = \frac { 1 } { f ( x ) }\) and find, in exact form, the set of values for which $$\frac { 1 } { \mathrm { f } ( x ) } > \mathrm { f } ( x ) .$$ If you use the following page to complete the answer to any question, the question number must be clearly shown.

6. (a) Sketch the curve with equation $$y = - \frac { k } { x } \quad k > 0 \quad x \neq 0$$ (b) On a separate diagram, sketch the curve with equation $$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$ stating the coordinates of the point of intersection with the \(x\)-axis and, in terms of \(k\), the equation of the horizontal asymptote.
(c) Find the range of possible values of \(k\) for which the curve with equation $$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$ does not touch or intersect the line with equation \(y = 3 x + 4\) \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-21_72_47_2615_1886}

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

1. Sketch the graph of \(C\). On your sketch
   The curve \(D\) has equation \(y = k x ^ { 2 }\) where \(k\) is a constant. Given that \(C\) meets \(D\) at 4 distinct points,
2. find the range of possible values for \(k\).

1. (a) On Diagram 1, sketch a graph of the curve \(C\) with equation

$$y = \frac { 6 } { x } \quad x \neq 0$$ The curve \(C\) is transformed onto the curve with equation \(y = \frac { 6 } { x - 2 } \quad x \neq 2\) (b) Fully describe this transformation. The curve with equation $$y = \frac { 6 } { x - 2 } \quad x \neq 2$$ and the line with equation $$y = k x + 7 \quad \text { where } k \text { is a constant }$$ intersect at exactly two points, \(P\) and \(Q\).
Given that the \(x\) coordinate of point \(P\) is - 4
(c) find the value of \(k\),
(d) find, using algebra, the coordinates of point \(Q\).
(Solutions relying entirely on calculator technology are not acceptable.) \section*{Diagram 1} Only use this copy of Diagram 1 if you need to redraw your graph. \includegraphics[max width=\textwidth, alt={}, center]{bb21001f-fe68-4776-992d-ede1aae233d7-19_709_739_1802_664} Copy of Diagram 1
(Total for Question 7 is 10 marks)

1. (a) Sketch the graph of the curve \(C\) with equation

$$y = \frac { 4 } { x - k }$$ where \(k\) is a positive constant.
Show on your sketch

• the coordinates of any points where \(C\) cuts the coordinate axes
• the equation of the vertical asymptote to \(C\)

Given that the straight line with equation \(y = 9 - x\) does not cross or touch \(C\) (b) find the range of values of \(k\).

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation $$y = \tan x \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ The line \(l\), shown in Figure 4, is an asymptote to \(y = \tan x\)

1. State an equation for \(l\). A copy of Figure 4, labelled Diagram 1, is shown on the next page.
2. 1. On Diagram 1, sketch the curve with equation $$y = \frac { 1 } { x } + 1 \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ stating the equation of the horizontal asymptote of this curve.
   2. Hence, giving a reason, state the number of solutions of the equation
3. State the number of solutions of the equation \(\tan x = \frac { 1 } { x } + 1\) in the region
   1. \(0 \leqslant x \leqslant 40 \pi\)
   2. \(- 10 \pi \leqslant x \leqslant \frac { 5 } { 2 } \pi\) $$\begin{aligned} & \qquad \tan x = \frac { 1 } { x } + 1 \\ & \text { in the region } - 2 \pi \leqslant x \leqslant 2 \pi \end{aligned}$$" \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-31_725_1047_1078_447} \captionsetup{labelformat=empty} \caption{Diagram 1}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-32_2644_1837_118_114}

6. The curve \(C\) has equation \(y = \frac { 4 } { x } + k\), where \(k\) is a positive constant.

1. Sketch a graph of \(C\), stating the equation of the horizontal asymptote and the coordinates of the point of intersection with the \(x\)-axis. The line with equation \(y = 10 - 2 x\) is a tangent to \(C\).
2. Find the possible values for \(k\). \(\_\_\_\_\) -

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

1. Sketch the graph of \(C\). On your sketch you should show the coordinates of any points of intersection with the coordinate axes and state clearly the equations of any asymptotes. The line \(l\) has equation \(y = 4 x + k\), where \(k\) is a constant. Given that \(l\) meets \(C\) at two distinct points,
2. show that $$k ^ { 2 } + 16 k + 48 > 0$$
3. Hence find the range of possible values for \(k\).

3. Given that \(\quad \mathrm { f } ( x ) = \frac { 1 } { x } , \quad x \neq 0\),

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

8. The point \(P ( 1 , a )\) lies on the curve with equation \(y = ( x + 1 ) ^ { 2 } ( 2 - x )\).

1. Find the value of \(a\).
2. On the axes below sketch the curves with the following equations:
   1. \(y = ( x + 1 ) ^ { 2 } ( 2 - x )\),
   2. \(y = \frac { 2 } { x }\). On your diagram show clearly the coordinates of any points at which the curves meet the axes.
3. With reference to your diagram in part (b) state the number of real solutions to the equation $$( x + 1 ) ^ { 2 } ( 2 - x ) = \frac { 2 } { x } .$$
   \includegraphics[max width=\textwidth, alt={}]{871f5957-180d-4379-88ce-186432f57bad-10_1347_1344_1245_297}

10. (a) On the axes below, sketch the graphs of

1. \(y = x ( x + 2 ) ( 3 - x )\)
2. \(y = - \frac { 2 } { x }\) showing clearly the coordinates of all the points where the curves cross the coordinate axes.
   (b) Using your sketch state, giving a reason, the number of real solutions to the equation $$x ( x + 2 ) ( 3 - x ) + \frac { 2 } { x } = 0$$ \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-13_994_997_1270_479}

6. Figure 1 shows a sketch of the curve with equation \(y = \frac { 2 } { x } , x \neq 0\) The curve \(C\) has equation \(y = \frac { 2 } { x } - 5 , x \neq 0\), and the line \(l\) has equation \(y = 4 x + 2\)

1. Sketch and clearly label the graphs of \(C\) and \(l\) on a single diagram. On your diagram, show clearly the coordinates of the points where \(C\) and \(l\) cross the coordinate axes.
2. Write down the equations of the asymptotes of the curve \(C\).
3. Find the coordinates of the points of intersection of \(y = \frac { 2 } { x } - 5\) and \(y = 4 x + 2\)

6. The curve \(C\) has equation \(y = \frac { 3 } { x }\) and the line \(l\) has equation \(y = 2 x + 5\).

1. On the axes below, sketch the graphs of \(C\) and \(l\), indicating clearly the coordinates of any intersections with the axes.
2. Find the coordinates of the points of intersection of \(C\) and \(l\). \includegraphics[max width=\textwidth, alt={}, center]{9451ec48-d955-44a8-9988-68f7c0fb9821-07_1137_1141_1046_397}

11. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(H\) with equation \(y = \frac { 3 } { x } + 4 , x \neq 0\).

1. Give the coordinates of the point where \(H\) crosses the \(x\)-axis.
2. Give the equations of the asymptotes to \(H\).
3. Find an equation for the normal to \(H\) at the point \(P ( - 3,3 )\). This normal crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
4. Find the length of the line segment \(A B\). Give your answer as a surd.

9. (a) On separate axes sketch the graphs of

1. \(y = - 3 x + c\), where \(c\) is a positive constant,
2. \(y = \frac { 1 } { x } + 5\) On each sketch show the coordinates of any point at which the graph crosses the \(y\)-axis and the equation of any horizontal asymptote. Given that \(y = - 3 x + c\), where \(c\) is a positive constant, meets the curve \(y = \frac { 1 } { x } + 5\) at two distinct points,
   (b) show that \(( 5 - c ) ^ { 2 } > 12\) (c) Hence find the range of possible values for \(c\).

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { 3 } { x } , x \neq 0\).

1. On a separate diagram, sketch the curve with equation \(y = \frac { 3 } { x + 2 } , x \neq - 2\), showing the coordinates of any point at which the curve crosses a coordinate axis.
2. Write down the equations of the asymptotes of the curve in part (a).

5 On separate diagrams,

1. sketch the curve \(y = \frac { 1 } { x }\),
2. sketch the curve \(y = x \left( x ^ { 2 } - 1 \right)\), stating the coordinates of the points where it crosses the \(x\)-axis,
3. sketch the curve \(y = - \sqrt { } x\).

4

1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
2. Hence sketch the curve \(y = \frac { 1 } { ( x - 3 ) ^ { 2 } }\).
3. Describe fully a transformation that transforms the curve \(y = \frac { 1 } { x ^ { 2 } }\) to the curve \(y = \frac { 2 } { x ^ { 2 } }\).

2

1. On separate diagrams, sketch the graphs of
   1. \(\mathrm { y } = \frac { 1 } { \mathrm { x } }\),
   2. \(y = x ^ { 4 }\).
2. Describe a transformation that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = 8 x ^ { 3 }\).

6

1. Sketch the graph of \(y = \frac { 1 } { x }\), where \(x \neq 0\), showing the parts of the graph corresponding to both positive and negative values of \(x\).
2. Describe fully the geometrical transformation that transforms the curve \(y = \frac { 1 } { x }\) to the curve \(y = \frac { 1 } { x + 2 }\). Hence sketch the curve \(y = \frac { 1 } { x + 2 }\).
3. Differentiate \(\frac { 1 } { x }\) with respect to \(x\).
4. Use parts (ii) and (iii) to find the gradient of the curve \(y = \frac { 1 } { x + 2 }\) at the point where it crosses the \(y\)-axis.