1.02n Sketch curves: simple equations including polynomials

5. The line \(l _ { 1 }\) has equation \(3 y - 2 x = 30\) The line \(l _ { 2 }\) passes through the point \(A ( 24,0 )\) and is perpendicular to \(l _ { 1 }\) Lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\)

1. Find, using algebra and showing your working, the coordinates of \(P\). Given that \(l _ { 1 }\) meets the \(x\)-axis at the point \(B\),
2. find the area of triangle \(B P A\).

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• \(\mathrm { f } ^ { \prime } ( \mathrm { x } ) = \frac { 12 } { \sqrt { \mathrm { x } } } + \frac { x } { 3 } - 4\)
• the point \(P ( 9,8 )\) lies on \(C\)
  1. find, in simplest form, \(\mathrm { f } ( x )\)

The line \(l\) is the normal to \(C\) at \(P\)

Find the coordinates of the point at which \(l\) crosses the \(y\)-axis.

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

Solutions relying on calculator technology are not acceptable.

1. Write \(\frac { 8 - \sqrt { 15 } } { 2 \sqrt { 3 } + \sqrt { 5 } }\) in the form \(a \sqrt { 3 } + b \sqrt { 5 }\) where \(a\) and \(b\) are integers to be found.
2. Hence, or otherwise, solve $$( x + 5 \sqrt { 3 } ) \sqrt { 5 } = 40 - 2 x \sqrt { 3 }$$ giving your answer in simplest form.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation \(y = \frac { 1 } { x + 2 }\)

1. State the equation of the asymptote of \(C\) that is parallel to the \(y\)-axis.
2. Factorise fully \(x ^ { 3 } + 4 x ^ { 2 } + 4 x\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.
3. On Diagram 1, add a sketch of the curve with equation $$y = x ^ { 3 } + 4 x ^ { 2 } + 4 x$$ On your sketch, state clearly the coordinates of each point where this curve cuts or meets the coordinate axes.
4. Hence state the number of real solutions of the equation $$( x + 2 ) \left( x ^ { 3 } + 4 x ^ { 2 } + 4 x \right) = 1$$ giving a reason for your answer.
   \includegraphics[max width=\textwidth, alt={}]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-09_800_1700_1053_185}
   Only use the copy of Diagram 1 if you need to redraw your answer to part (c).

11. \begin{figure}[h] \end{figure} Figure 5 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 x ^ { 2 } - 12 x + 14$$

1. Write \(2 x ^ { 2 } - 12 x + 14\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. Given that \(C\) has a minimum at the point \(P\)
2. state the coordinates of \(P\) The line \(l\) intersects \(C\) at \(( - 1,28 )\) and at \(P\) as shown in Figure 5.
3. Find the equation of \(l\) giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found. The finite region \(R\), shown shaded in Figure 5, is bounded by the \(x\)-axis, \(l\), the \(y\)-axis, and \(C\).
4. Use inequalities to define the region \(R\).

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

1. \(y = c ^ { 2 } - x ^ { 2 }\)
2. \(y = x ^ { 2 } ( x - 3 c )\) where \(c\) is a positive constant.
   Show clearly the coordinates of the points where each graph crosses or meets the \(x\)-axis and the \(y\)-axis.
   (b) Prove that the \(x\) coordinate of any point of intersection of $$y = c ^ { 2 } - x ^ { 2 } \text { and } y = x ^ { 2 } ( x - 3 c )$$ where \(c\) is a positive constant, is given by a solution of the equation $$x ^ { 3 } + ( 1 - 3 c ) x ^ { 2 } - c ^ { 2 } = 0$$ Given that the graphs meet when \(x = 2\) (c) find the exact value of \(c\), writing your answer as a fully simplified surd.

12. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 } C\) touches the \(x\)-axis at the origin and cuts the \(x\)-axis at the point \(A\).

1. Show that the coordinates of \(A\) are \(( 3,0 )\).
2. Show that the equation of the tangent to \(C\) at the point \(A\) is \(y = - 3 x + 9\) The tangent to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 5.
3. Use algebra to find the \(x\) coordinate of \(B\). The region \(R\), shown shaded in Figure 5, is bounded by the curve \(C\) and the tangent to \(C\) at \(A\).
4. Find, by using calculus, the area of region \(R\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

6. (a) Sketch the graph of \(y = \left( \frac { 1 } { 2 } \right) ^ { x } , x \in \mathbb { R }\), showing the coordinates of the point at which the graph crosses the \(y\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) The values of \(y\) are rounded to 3 decimal places. (b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for $$\int _ { - 0.9 } ^ { - 0.5 } \left( \frac { 1 } { 2 } \right) ^ { x } d x$$ II

10. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = x ^ { 2 } ( 9 - 2 x ) .$$ There is a minimum at the origin, a maximum at the point \(( 3,27 )\) and \(C\) cuts the \(x\)-axis at the point \(A\).

1. Write down the coordinates of the point \(A\).
2. On separate diagrams sketch the curve with equation
   1. \(y = \mathrm { f } ( x + 3 )\),
   2. \(y = \mathrm { f } ( 3 x )\). On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. The curve with equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant, has a maximum point at \(( 3,10 )\).
3. Write down the value of \(k\).

10. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 18 , \quad x \geqslant 0 ,$$

1. express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \geqslant 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).
2. In the space provided on page 19, sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). The line \(y = 41\) meets \(C\) at the point \(R\).
3. Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q \sqrt { } 2\), where \(p\) and \(q\) are integers.

10. $$x ^ { 2 } + 2 x + 3 \equiv ( x + a ) ^ { 2 } + b .$$

1. Find the values of the constants \(a\) and \(b\).
2. In the space provided below, sketch the graph of \(y = x ^ { 2 } + 2 x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes.
3. Find the value of the discriminant of \(x ^ { 2 } + 2 x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). The equation \(x ^ { 2 } + k x + 3 = 0\), where \(k\) is a constant, has no real roots.
4. Find the set of possible values of \(k\), giving your answer in surd form.

10. (a) On the same axes sketch the graphs of the curves with equations

1. \(y = x ^ { 2 } ( x - 2 )\),
2. \(y = x ( 6 - x )\),
   and indicate on your sketches the coordinates of all the points where the curves cross the \(x\)-axis.
   (b) Use algebra to find the coordinates of the points where the graphs intersect.

1. The curve \(C\) has equation

$$y = ( x + 3 ) ( x - 1 ) ^ { 2 }$$

1. Sketch \(C\) showing clearly the coordinates of the points where the curve meets the coordinate axes.
2. Show that the equation of \(C\) can be written in the form $$y = x ^ { 3 } + x ^ { 2 } - 5 x + k ,$$ where \(k\) is a positive integer, and state the value of \(k\). There are two points on \(C\) where the gradient of the tangent to \(C\) is equal to 3 .
3. Find the \(x\)-coordinates of these two points.

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}

1. (a) Factorise completely \(x ^ { 3 } - 4 x\) (b) Sketch the curve \(C\) with equation

$$y = x ^ { 3 } - 4 x ,$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. The point \(A\) with \(x\)-coordinate - 1 and the point \(B\) with \(x\)-coordinate 3 lie on the curve \(C\).
(c) Find an equation of the line which passes through \(A\) and \(B\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
(d) Show that the length of \(A B\) is \(k \sqrt { } 10\), where \(k\) is a constant to be found.

10. $$\mathrm { f } ( x ) = x ^ { 2 } + 4 k x + ( 3 + 11 k ) , \quad \text { where } k \text { is a constant. }$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found in terms of \(k\). Given that the equation \(\mathrm { f } ( x ) = 0\) has no real roots,
2. find the set of possible values of \(k\). Given that \(k = 1\),
3. sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any point at which the graph crosses a coordinate axis.

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}

5. The curve \(C\) has equation \(y = x ( 5 - x )\) and the line \(L\) has equation \(2 y = 5 x + 4\)

1. Use algebra to show that \(C\) and \(L\) do not intersect.
2. In the space on page 11, sketch \(C\) and \(L\) on the same diagram, showing the coordinates of the points at which \(C\) and \(L\) meet the axes.

8. The curve \(C _ { 1 }\) has equation $$y = x ^ { 2 } ( x + 2 )$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Sketch \(C _ { 1 }\), showing the coordinates of the points where \(C _ { 1 }\) meets the \(x\)-axis.
3. Find the gradient of \(C _ { 1 }\) at each point where \(C _ { 1 }\) meets the \(x\)-axis. The curve \(C _ { 2 }\) has equation $$y = ( x - k ) ^ { 2 } ( x - k + 2 )$$ where \(k\) is a constant and \(k > 2\)
4. Sketch \(C _ { 2 }\), showing the coordinates of the points where \(C _ { 2 }\) meets the \(x\) and \(y\) axes.

10. $$4 x ^ { 2 } + 8 x + 3 \equiv a ( x + b ) ^ { 2 } + c$$

1. Find the values of the constants \(a , b\) and \(c\).
2. On the axes on page 27, sketch the curve with equation \(y = 4 x ^ { 2 } + 8 x + 3\), showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{099016ad-e742-4679-9669-47dcd1d9cc5f-15_1283_1284_319_322}

10. The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3\). The point \(P\) has coordinates \(( 3,0 )\).

1. Show that \(P\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(Q\) also lies on \(C\). The tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).
3. Find the coordinates of \(Q\).

3. On separate diagrams, sketch the graphs of

1. \(y = ( x + 3 ) ^ { 2 }\),
2. \(y = ( x + 3 ) ^ { 2 } + k\), where \(k\) is a positive constant. Show on each sketch the coordinates of each point at which the graph meets the axes.

9. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } - 6 x \right) ( x - 2 ) + 3 x\),

1. express \(\mathrm { f } ( x )\) in the form \(x \left( a x ^ { 2 } + b x + c \right)\), where \(a\), \(b\) and \(c\) are constants.
2. Hence factorise \(\mathrm { f } ( x )\) completely.
3. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of each point at which the graph meets the axes.

9. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 5,65 )\). Given that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - 10 x - 12\),

1. use integration to find \(\mathrm { f } ( x )\).
2. Hence show that \(\mathrm { f } ( x ) = x ( 2 x + 3 ) ( x - 4 )\).
3. In the space provided on page 17, sketch \(C\), showing the coordinates of the points where \(C\) crosses the \(x\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{c0db3fe8-62ec-41e3-acaf-66b2c7b2754d-11_76_40_2646_1894}

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}