1.02q Use intersection points: of graphs to solve equations

7

1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } x = \frac { 1 } { 2 } x + 1$$ where \(x\) is in radians, has a root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify, by calculation, that this root lies between 0.5 and 1 .
3. Show that this root also satisfies the equation $$x = \sin ^ { - 1 } \left( \frac { 2 } { x + 2 } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 } { x _ { n } + 2 } \right)$$ with initial value \(x _ { 1 } = 0.75\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-2_355_601_1562_772} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \pi\). The area of triangle \(A O B\) is half the area of the sector.

1. Show that \(\alpha\) satisfies the equation $$x = 2 \sin x$$
2. Verify by calculation that \(\alpha\) lies between \(\frac { 1 } { 2 } \pi\) and \(\frac { 2 } { 3 } \pi\).
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( x _ { n } + 4 \sin x _ { n } \right)$$ converges, then it converges to a root of the equation in part (i).
4. Use this iterative formula, with initial value \(x _ { 1 } = 1.8\), to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 1 + x ^ { 2 }$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 0.5 and 0.8.
3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + x _ { n } ^ { 2 } } \right)$$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } \frac { 1 } { 2 } x = \frac { 1 } { 3 } x + 1$$ has one root in the interval \(0 < x \leqslant \pi\).
2. Show by calculation that this root lies between 1.4 and 1.6.
3. Show that, if a sequence of values in the interval \(0 < x \leqslant \pi\) given by the iterative formula $$x _ { n + 1 } = 2 \sin ^ { - 1 } \left( \frac { 3 } { x _ { n } + 3 } \right)$$ converges, then it converges to the root of the equation in part (i).
4. Use this iterative formula to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

9 \includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-3_481_483_1434_831} The diagram shows the curves \(y = x \cos x\) and \(y = \frac { k } { x }\), where \(k\) is a constant, for \(0 < x \leqslant \frac { 1 } { 2 } \pi\). The curves touch at the point where \(x = a\).

1. Show that \(a\) satisfies the equation \(\tan a = \frac { 2 } { a }\).
2. Use the iterative formula \(a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { a _ { n } } \right)\) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
3. Hence find the value of \(k\) correct to 2 decimal places.

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.

5

1. By sketching a suitable pair of graphs, show that the equation \(\ln ( x + 2 ) = 4 \mathrm { e } ^ { - x }\) has exactly one real root.
2. Show by calculation that this root lies between \(x = 1\) and \(x = 1.5\).
3. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 4 } { \ln \left( x _ { n } + 2 \right) } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7

1. By sketching a suitable pair of graphs, show that the equation $$\cos x = 2 - 2 x$$ where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 0.5 and 1 .
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = 1 - \frac { 1 } { 2 } \cos x _ { n }$$ converges, then it converges to the root of the equation in part (i).
4. Use this iterative formula, with initial value \(x _ { 1 } = 0.6\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5

1. By sketching a suitable pair of graphs, show that the equation $$\frac { 1 } { x } = \sin x$$ where \(x\) is in radians, has only one root for \(0 < x \leqslant \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between \(x = 1.1\) and \(x = 1.2\).
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { \sin x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 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.

7

1. By sketching a suitable pair of graphs, show that the equation \(4 - x ^ { 2 } = \sec \frac { 1 } { 2 } x\) has exactly one root in the interval \(0 \leqslant x < \pi\).
2. Verify by calculation that this root lies between 1 and 2 .
3. Use the iterative formula \(x _ { n + 1 } = \sqrt { 4 - \sec \frac { 1 } { 2 } x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

11. (a) On Diagram 1 sketch the graphs of

1. \(y = x ( 3 - x )\)
2. \(y = x ( x - 2 ) ( 5 - x )\) 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 $$y = x ( 3 - x ) \text { and } y = x ( x - 2 ) ( 5 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 13 \right) = 0\) The point \(P\) lies on both curves. Given that \(P\) lies in the first quadrant,
   (c) find, using algebra and showing your working, the exact coordinates of \(P\).
   
   \includegraphics[max width=\textwidth, alt={}]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-23_824_1211_296_370}
   \section*{Diagram 1}

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}

8. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 3 x - 2 ) ^ { 2 } ( x - 4 )$$

1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) > 0\)
2. Expand f(x) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found. The line \(l\), also shown in Figure 4, passes through the \(y\) intercept of \(C\) and is parallel to the \(x\)-axis. The line \(l\) cuts \(C\) again at points \(P\) and \(Q\), also shown in Figure 4 .
3. Using algebra and showing your working, find the length of line \(P Q\). Write your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be found.
   (Solutions relying entirely on calculator technology are not acceptable.)

4. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} Figure 1 shows a line \(l\) with equation \(x + y = 6\) and a curve \(C\) with equation \(y = 6 x - 2 x ^ { 2 } + 1\) The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\) as shown in Figure 1.

1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 1, is bounded by \(C , l\) and the \(x\)-axis.
2. Use inequalities to define the region \(R\).
   

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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)

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of

• the curve with equation \(y = \tan x\)
• the straight line l with equation \(y = \pi x\) in the interval \(- \pi < x < \pi\)

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )$$

1. Use the given information to state the values of \(x\) for which $$f ( x ) > 0$$
2. Expand \(( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )\), writing your answer as a polynomial in simplest form. The straight line \(l\) is the tangent to \(C\) at the point where \(C\) cuts the \(y\)-axis.
   Given that \(l\) cuts \(C\) at the point \(P\), as shown in Figure 4,
3. find, using algebra, the \(x\) coordinate of \(P\) (Solutions based on calculator technology are not acceptable.)

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 3 shows a plot of the curve with equation \(y = \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)

1. State the coordinates of the minimum point on the curve with equation $$y = 4 \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$$ A copy of Figure 3, called Diagram 1, is shown on the next page.
2. On Diagram 1, sketch and label the curves
   1. \(y = 1 + \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
   2. \(y = \tan \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
3. Hence find the number of solutions of the equation
   1. \(\tan \theta = 1 + \sin \theta\) that lie in the region \(0 \leqslant \theta \leqslant 2160 ^ { \circ }\)
   2. \(\tan \theta = 1 + \sin \theta\) that lie in the region \(0 \leqslant \theta \leqslant 1980 ^ { \circ }\)
      \includegraphics[max width=\textwidth, alt={}]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-25_746_808_577_575}
      \section*{Diagram 1}

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. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.} Figure 3 shows

• the line \(l\) with equation \(y - 5 x = 75\)
• the curve \(C\) with equation \(y = 2 x ^ { 2 } + x - 21\)

The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\), as shown in Figure 3 .

1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
2. Use inequalities to define the region \(R\).

1. The curve \(C _ { 1 }\) has equation

$$y = x \left( 4 - x ^ { 2 } \right)$$

1. Sketch the graph of \(C _ { 1 }\) showing the coordinates of any points of intersection with the coordinate axes. The curve \(C _ { 2 }\) has equation \(y = \frac { A } { x }\) where \(A\) is a constant.
2. Show that the \(x\) coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) satisfy the equation $$x ^ { 4 } - 4 x ^ { 2 } + A = 0$$
3. Hence find the range of possible values of \(A\) for which \(C _ { 1 }\) meets \(C _ { 2 }\) at 4 distinct points.