1.02p Interpret algebraic solutions: graphically

3

1. By sketching a suitable pair of graphs, show that the equation \(x ^ { 3 } = 3 - x\) has exactly one real root.
2. Show that if a sequence of real values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 3 } { 3 x _ { n } ^ { 2 } + 1 }$$ converges, then it converges to the root of the equation in part (i).
3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

4

1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
2. Verify by calculation that \(4.5 < \alpha < 5.0\).
3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4

1. Find the quotient when \(3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 1\) is divided by ( \(x - 2\) ), and show that the remainder is 39 .
2. Hence show that the equation \(3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 40 = 0\) has exactly one real root.

16. \begin{figure}[h] \end{figure} The straight line \(l\) with equation \(y = \frac { 1 } { 2 } x + 1\) cuts the curve \(C\), with equation \(y = x ^ { 2 } - 4 x + 3\), at the points \(P\) and \(Q\), as shown in Figure 2

1. Use algebra to find the coordinates of the points \(P\) and \(Q\). The curve \(C\) crosses the \(x\)-axis at the points \(T\) and \(S\).
2. Write down the coordinates of the points \(T\) and \(S\). The finite region \(R\) is shown shaded in Figure 2. This region \(R\) is bounded by the line segment \(P Q\), the line segment \(T S\), and the \(\operatorname { arcs } P T\) and \(S Q\) of the curve.
3. Use integration to find the exact area of the shaded region \(R\).

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

1. \(y = x ( 4 - x )\)
2. \(y = x ^ { 2 } ( 7 - x )\) showing clearly the coordinates of the points where the curves cross the coordinate axes.
   (b) Show that the \(x\)-coordinates of the points of intersection of $$y = x ( 4 - x ) \text { and } y = x ^ { 2 } ( 7 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 4 \right) = 0\) The point \(A\) lies on both of the curves and the \(x\) and \(y\) coordinates of \(A\) are both positive.
   (c) Find the exact coordinates of \(A\), leaving your answer in the form ( \(p + q \sqrt { } 3 , r + s \sqrt { } 3\) ), where \(p , q , r\) and \(s\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{65d61b2c-2e47-402e-b08f-2d46bb00c188-14_1178_1203_1407_379}

1. (a) On the same diagram, sketch and clearly label the graphs with equations

$$y = \mathrm { e } ^ { x } \quad \text { and } \quad y = 10 - x$$ Show on your sketch the coordinates of each point at which the graphs cut the axes.
(b) Explain why the equation \(\mathrm { e } ^ { x } - 10 + x = 0\) has only one solution.
(c) Show that the solution of the equation $$\mathrm { e } ^ { x } - 10 + x = 0$$ lies between \(x = 2\) and \(x = 3\) (d) Use the iterative formula $$x _ { n + 1 } = \ln \left( 10 - x _ { n } \right) , \quad x _ { 1 } = 2$$ to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
Give your answers to 4 decimal places.

6. (a) On the same diagram, sketch the graphs of \(y = \left| x ^ { 2 } - 4 \right|\) and \(y = | 2 x - 1 |\), showing the coordinates of the points where the graphs meet the axes.
(b) Solve \(\left| x ^ { 2 } - 4 \right| = | 2 x - 1 |\), giving your answers in surd form where appropriate.
(c) Hence, or otherwise, find the set of values of \(x\) for which \(\left| x ^ { 2 } - 4 \right| > | 2 x - 1 |\).
(3)(Total 12 marks)

3. (a) Use algebra to find the exact solutions of the equation $$\left| 2 x ^ { 2 } + x - 6 \right| = 6 - 3 x$$ (b) On the same diagram, sketch the curve with equation \(y = \left| 2 x ^ { 2 } + x - 6 \right|\) and the line with equation \(y = 6 - 3 x\).
(c) Find the set of values of \(x\) for which $$\left| 2 x ^ { 2 } + x - 6 \right| > 6 - 3 x$$ (3)(Total 12 marks)

2. \includegraphics[max width=\textwidth, alt={}, center]{d6befd60-de40-41b6-8ae5-48656dbca40c-1_734_1228_888_479} The diagram above shows a sketch of the curve with equation $$y = \frac { x ^ { 2 } - 1 } { | x + 2 | } , \quad x \neq - 2$$ The curve crosses the \(x\)-axis at \(x = 1\) and \(x = - 1\) and the line \(x = - 2\) is an asymptote of the curve.

1. Use algebra to solve the equation \(\frac { x ^ { 2 } - 1 } { | x + 2 | } = 3 ( 1 - x )\).
2. Hence, or otherwise, find the set of values of \(x\) for which $$\frac { x ^ { 2 } - 1 } { | x + 2 | } < 3 ( 1 - x )$$ (Total 9 marks)

8

1. Describe completely the curve \(x ^ { 2 } + y ^ { 2 } = 25\).
2. Find the coordinates of the points of intersection of the curve \(x ^ { 2 } + y ^ { 2 } = 25\) and the line \(2 x + y - 5 = 0\).

10

1. Express \(x ^ { 2 } - 6 x + 2\) in the form \(( x - a ) ^ { 2 } - b\).
2. State the coordinates of the turning point on the graph of \(y = x ^ { 2 } - 6 x + 2\).
3. Sketch the graph of \(y = x ^ { 2 } - 6 x + 2\). You need not state the coordinates of the points where the graph intersects the \(x\)-axis.
4. Solve the simultaneous equations \(y = x ^ { 2 } - 6 x + 2\) and \(y = 2 x - 14\). Hence show that the line \(y = 2 x - 14\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 2\).

6. \includegraphics[max width=\textwidth, alt={}, center]{00364339-8108-4031-8e67-6100810e8297-2_549_885_251_370} The diagram shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).

3. (i) Solve the simultaneous equations $$\begin{aligned} & y = x ^ { 2 } - 6 x + 7 \\ & y = 2 x - 9 \end{aligned}$$ (ii) Hence, describe the geometrical relationship between the curve \(y = x ^ { 2 } - 6 x + 7\) and the straight line \(y = 2 x - 9\).

3 The curve with equation \(y = \frac { 1 } { 5 } x ( 10 - x )\) is used to model the arch of a bridge over a road, where \(x\) and \(y\) are distances in metres, with the origin as shown in Fig. 12.1. The \(x\)-axis represents the road surface. \begin{figure}[h] \end{figure}

1. State the value of \(x\) at A , where the arch meets the road.
2. Using symmetry, or otherwise, state the value of \(x\) at the maximum point B of the graph. Hence find the height of the arch.
3. Fig. 12.2 shows a lorry which is 4 m high and 3 m wide, with its cross-section modelled as a rectangle. Find the value of \(d\) when the lorry is in the centre of the road. Hence show that the lorry can pass through this arch. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fed65420-9ef9-41d6-a58f-3b0f801d6225-3_528_870_1558_717} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
   \end{figure}
4. Another lorry, also modelled as having a rectangular cross-section, has height 4.5 m and just touches the arch when it is in the centre of the road. Find the width of this lorry, giving your answer in surd form.

5 A curve has equation \(y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }\), where \(k\) is a positive constant and \(k \neq 2\).

1. Find the equations of the three asymptotes.
2. Use your graphical calculator to obtain rough sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\).
3. In the case \(k < 2\), your sketch may not show clearly the shape of the curve near \(x = 0\). Use calculus to show that the curve has a minimum point when \(x = 0\).
4. In the case \(k > 2\), your sketch may not show clearly how the curve approaches its asymptote as \(x \rightarrow + \infty\). Show algebraically that the curve crosses this asymptote.
5. Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\). These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates. RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{MEI STRUCTURED MATHEMATICS} Further Methods for Advanced Mathematics (FP2)
   Tuesday

5 A curve has equation \(y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }\), where \(k\) is a positive constant and \(k \neq 2\).

1. Find the equations of the three asymptotes.
2. Use your graphical calculator to obtain rough sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\).
3. In the case \(k < 2\), your sketch may not show clearly the shape of the curve near \(x = 0\). Use calculus to show that the curve has a minimum point when \(x = 0\).
4. In the case \(k > 2\), your sketch may not show clearly how the curve approaches its asymptote as \(x \rightarrow + \infty\). Show algebraically that the curve crosses this asymptote.
5. Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\). These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates.

10

1. Find the coordinates of the stationary points of the curve \(y = 2 x ^ { 3 } + 5 x ^ { 2 } - 4 x\).
2. State the set of values for \(x\) for which \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 x\) is a decreasing function.
3. Show that the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 2 }\) is \(10 x - 4 y - 7 = 0\).
4. Hence, with the aid of a sketch, show that the equation \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 x = \frac { 5 } { 2 } x - \frac { 7 } { 4 }\) has two distinct real roots.

6. \begin{figure}[h] \end{figure} The figure 1 shows sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\). $$f ( x ) = a x ( x - b ) ^ { 2 } , x \in R$$ where \(a\) and \(b\) are constants.
The curve passes through the origin and touches the \(x\)-axis at the point \(( 3,0 )\).
There is a minimum point at \(( 1 , - 4 )\) and a maximum point at \(( 3,0 )\).
a. Find the equation of \(C\).
b. Deduce the values of \(x\) for which $$\mathrm { f } ^ { \prime } ( x ) > 0$$ Given that the line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at exactly one point,
c. State the possible values for \(k\).

2 Two curves have equations \(y = \ln x\) and \(y = \frac { k } { x }\), where \(k\) is a positive constant.

1. Sketch the curves on a single diagram.
2. Explain how your diagram shows that the equation \(x \ln x - k = 0\) has exactly one real root.

3

1. Determine, in terms of \(k\), the coordinates of the point where the lines with the following equations intersect. $$\begin{array} { r } x + y = k \\ 2 x - y = 1 \end{array}$$
2. Determine, in terms of \(k\), the coordinates of the points where the line \(\mathrm { x } + \mathrm { y } = \mathrm { k }\) crosses the curve \(y = x ^ { 2 } + k\).

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}

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \left| x ^ { 2 } - 8 \right|\) and a sketch of the straight line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. The equation $$\left| x ^ { 2 } - 8 \right| = m x + c$$ has exactly 3 roots, as shown in Figure 1.

1. Show that $$m ^ { 2 } - 4 c + 32 = 0$$ Given that \(c = 3 m\)
2. determine the value of \(m\) and the value of \(c\)
3. Hence solve $$\left| x ^ { 2 } - 8 \right| \geqslant m x + c$$

3. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ^ { 2 } - 10 x \quad x \in \mathbb { R }$$

1. Solve the equation $$\mathrm { f } ( | x | ) = 48$$
2. Find the set of values of \(x\) for which $$| f ( x ) | \geqslant \frac { 5 } { 2 } x$$

7 A curve has equation \(y = \frac { 4 x + 11 } { ( x + 3 ) ^ { 2 } }\).

1. Show that the curve meets the line \(y = k\) if and only if \(k \leq 4\), and deduce the coordinates of the turning point on the curve.
2. Sketch the curve, stating the coordinates of the points where it cuts the axes, and showing clearly its asymptotes and the turning point.

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

1. Find the equations of the asymptotes of \(C\).
2. Sketch \(C\), indicating clearly the asymptotes and any points where \(C\) meets the coordinate axes.