Rational curve intersections

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

1. Sketch the graph of \(C\). On your sketch
   • show the coordinates of any points of intersection with the coordinate axes
   • state clearly the equations of any asymptotes
   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 \(k\).

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 \(k\).

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}

4. (i) Sketch on the same diagram the curves \(y = x ^ { 2 } - 4 x\) and \(y = - \frac { 1 } { x }\).
(ii) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$

4. (a) Sketch on the same diagram the curves \(y = x ^ { 2 } - 4 x\) and \(y = - \frac { 1 } { x }\).
(b) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$

7. i. Sketch the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\) on the axes provided below.
\includegraphics[max width=\textwidth, alt={}, center]{e1b41613-a703-4eb3-9760-7b47b1dad099-06_849_921_1683_644}
ii. In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\).
iii. Hence, solve the inequality \(\frac { 3 } { x ^ { 2 } } \leq x ^ { 2 } - 2\), giving your answer in interval notation.

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 \(y = x ^ { 2 } - 2\).