1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta

5. \begin{figure}[h] \end{figure} Figure 3 shows the plan view of a viewing platform at a tourist site. The shape of the viewing platform consists of a sector \(A B C O A\) of a circle, centre \(O\), joined to a triangle \(A O D\). Given that

• \(O A = O C = 6 \mathrm {~m}\)
• \(A D = 14 \mathrm {~m}\)
• angle \(A D C = 0.43\) radians
• angle \(A O D\) is an obtuse angle
• \(O C D\) is a straight line
  find
  1. the size of angle \(A O D\), in radians, to 3 decimal places,
  2. the length of arc \(A B C\), in metres, to one decimal place,
  3. the total area of the viewing platform, in \(\mathrm { m } ^ { 2 }\), to one decimal place.

5. \begin{figure}[h] \end{figure} Figure 2 shows a plan view of a semicircular garden \(A B C D E O A\) The semicircle has

• centre \(O\)
• diameter \(A O E\)
• radius 3 m

The straight line \(B D\) is parallel to \(A E\) and angle \(B O A\) is 0.7 radians.

1. Show that, to 4 significant figures, angle \(B O D\) is 1.742 radians. The flowerbed \(R\), shown shaded in Figure 2, is bounded by \(B D\) and the arc \(B C D\).
2. Find the area of the flowerbed, giving your answer in square metres to one decimal place.
3. Find the perimeter of the flowerbed, giving your answer in metres to one decimal place.

6. \begin{figure}[h] \end{figure} Diagram NOT accurately drawn Figure 1 shows the plan view for the design of a stage.
The design consists of a sector \(O B C\) of a circle, with centre \(O\), joined to two congruent triangles \(O A B\) and \(O D C\). Given that

• angle \(B O C = 2.4\) radians
• area of sector \(B O C = 40 \mathrm {~m} ^ { 2 }\)
• \(A O D\) is a straight line of length 12.5 m
  1. find the radius of the sector, giving your answer, in m , to 2 decimal places,
  2. find the size of angle \(A O B\), in radians, to 2 decimal places.

Hence find

the total area of the stage, giving your answer, in \(\mathrm { m } ^ { 2 }\), to one decimal place,

the total perimeter of the stage, giving your answer, in m , to one decimal place.

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the plan view of a platform.
The plan view of the platform consists of a sector \(D O C\) of a circle centre \(O\) joined to a sector \(A O B E A\) of a different circle, also with centre \(O\). Given that

• angle \(A O B = 0.8\) radians
• arc length \(C D = 9 \mathrm {~m}\)
• \(D A : A O = 3 : 5\)
  1. show that \(A O = 7.03 \mathrm {~m}\) to 3 significant figures.
  2. Find the perimeter of the platform, in m , to 3 significant figures.
  3. Find the total area of the platform, giving your answer in \(\mathrm { m } ^ { 2 }\) to the nearest whole number.

7. \begin{figure}[h] \end{figure} The shape \(A B C D A\) consists of a sector \(A B C O A\) of a circle, centre \(O\), joined to a triangle \(A O D\), as shown in Figure 2. The point \(D\) lies on \(O C\).
The radius of the circle is 6 cm , length \(A D\) is 5 cm and angle \(A O D\) is 0.7 radians.

1. Find the area of the sector \(A B C O A\), giving your answer to one decimal place. Given angle \(A D O\) is obtuse,
2. find the size of angle \(A D O\), giving your answer to 3 decimal places.
3. Hence find the perimeter of shape \(A B C D A\), giving your answer to one decimal place.
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7. \begin{figure}[h] \end{figure} The line \(l _ { 1 }\) has equation \(4 y + 3 x = 48\) The line \(l _ { 1 }\) cuts the \(y\)-axis at the point \(C\), as shown in Figure 3.

1. State the \(y\) coordinate of \(C\). The point \(D ( 8,6 )\) lies on \(l _ { 1 }\) The line \(l _ { 2 }\) passes through \(D\) and is perpendicular to \(l _ { 1 }\) The line \(l _ { 2 }\) cuts the \(y\)-axis at the point \(E\) as shown in Figure 3.
2. Show that the \(y\) coordinate of \(E\) is \(- \frac { 14 } { 3 }\) A sector \(B C E\) of a circle with centre \(C\) is also shown in Figure 3. Given that angle \(B C E\) is 1.8 radians,
3. find the length of arc \(B E\). The region \(C B E D\), shown shaded in Figure 3, consists of the sector \(B C E\) joined to the triangle \(C D E\).
4. Calculate the exact area of the region \(C B E D\).

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the outline of the face of a ceiling fan viewed from below.
The fan consists of three identical sections congruent to \(O A B C D O\), shown in Figure 3, where

• \(O A B O\) is a sector of a circle with centre \(O\) and radius 9 cm
• \(O B C D O\) is a sector of a circle with centre \(O\) and radius 84 cm
• angle \(A O D = \frac { 2 \pi } { 3 }\) radians

Given that the length of the arc \(A B\) is 15 cm ,

1. show that the length of the arc \(C D\) is 35.9 cm to one decimal place. The face of the fan is modelled to be a flat surface.
   Find, according to the model,
2. the perimeter of the face of the fan, giving your answer to the nearest cm,
3. the surface area of the face of the fan. Give your answer to 3 significant figures and make your units clear.

5. \begin{figure}[h] \end{figure} Figure 1 shows the plan for a garden.
In the plan

• \(O A\) and \(C D\) are perpendicular to \(O D\)
• \(A B\) is an arc of the circle with centre \(O\) and radius 4 metres
• \(\quad B C\) is parallel to \(O D\)
• \(O D\) is 6 metres, \(O A\) is 4 metres and \(C D\) is 1.5 metres
  1. Show that angle \(A O B\) is 1.186 radians to 4 significant figures.
  2. Find the perimeter of the garden, giving your answer in metres to 3 significant figures.
  3. Find the area of the garden, giving your answer in square metres to 3 significant figures.

5. \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a garden.
The shape of the garden \(A B C D E A\) consists of a triangle \(A B E\) and a right-angled triangle \(B C D\) joined to a sector \(B D E\) of a circle with radius 6 m and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(A B = 10.8 \mathrm {~m}\) Angle \(B C D = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.3\) radians and \(A E = 12.2 \mathrm {~m}\)

1. Find the area of the sector \(B D E\), giving your answer in \(\mathrm { m } ^ { 2 }\)
2. Find the size of angle \(A B E\), giving your answer in radians to 2 decimal places.
3. Find the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.

1. \begin{figure}[h] \end{figure} Figure 1 shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(A O B\) is 1.25 radians. Given that the area of the sector \(A O B\) is \(15 \mathrm {~cm} ^ { 2 }\)

1. find the exact value of \(r\),
2. find the exact length of the perimeter of the sector. Write your answer in simplest form.
   

3. \begin{figure}[h] \end{figure} Figure 1 shows the design for a badge.
The design consists of two congruent triangles, \(A O C\) and \(B O C\), joined to a sector \(A O B\) of a circle centre \(O\).

• Angle \(A O B = \alpha\)
• \(A O = O B = 3 \mathrm {~cm}\)
• \(O C = 5 \mathrm {~cm}\)

Given that the area of sector \(A O B\) is \(7.2 \mathrm {~cm} ^ { 2 }\)

1. show that \(\alpha = 1.6\) radians.
2. Hence find
   1. the area of the badge, giving your answer in \(\mathrm { cm } ^ { 2 }\) to 2 significant figures,
   2. the perimeter of the badge, giving your answer in cm to one decimal place.

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6. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }$$

1. Sketch a graph of \(C\). Show on your graph the coordinates of the points where \(C\) cuts or meets the coordinate axes.
2. Write \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\), where \(a , b , c\) and \(d\) are constants to be found.
3. Hence, find the equation of the tangent to \(C\) at the point where \(x = \frac { 1 } { 3 }\)

9. Diagram NOT accurately drawn \begin{figure}[h] \end{figure} Figure 3 shows the plan view of the area being used for a ball-throwing competition.
Competitors must stand within the circle \(C\) and throw a ball as far as possible into the target area, \(P Q R S\), shown shaded in Figure 3. Given that

• circle \(C\) has centre \(O\)
• \(P\) and \(S\) are points on \(C\)
• \(O P Q R S O\) is a sector of a circle with centre \(O\)
• the length of arc \(P S\) is 0.72 m
• the size of angle \(P O S\) is 0.6 radians
  1. show that \(O P = 1.2 \mathrm {~m}\)

Given also that

$$5 x ^ { 2 } + 12 x - 1500 = 0$$

Hence calculate the total perimeter of the target area, \(P Q R S\), giving your answer to the nearest metre.

12. Diagram NOT drawn to scale Figure 1 shows the plan for a pond and platform. The platform is shown shaded in the figure and is labelled \(A B C D\). The pond and platform together form a circle of radius 22 m with centre \(O\). \(O A\) and \(O D\) are radii of the circle. Point \(B\) lies on \(O A\) such that the length of \(O B\) is 10 m and point \(C\) lies on \(O D\) such that the length of \(O C\) is 10 m . The length of \(B C\) is 15 m . The platform is bounded by the arc \(A D\) of the circle, and the straight lines \(A B , B C\) and \(C D\). Find

1. the size of the angle \(B O C\), giving your answer in radians to 3 decimal places,
2. the perimeter of the platform to 3 significant figures,
3. the area of the platform to 3 significant figures.

9. \begin{figure}[h] \end{figure} In Figure 3, the points \(A\) and \(B\) are the centres of the circles \(C _ { 1 }\) and \(C _ { 2 }\) respectively. The circle \(C _ { 1 }\) has radius 10 cm and the circle \(C _ { 2 }\) has radius 5 cm . The circles intersect at the points \(X\) and \(Y\), as shown in the figure. Given that the distance between the centres of the circles is 12 cm ,

1. calculate the size of the acute angle \(X A B\), giving your answer in radians to 3 significant figures,
2. find the area of the major sector of circle \(C _ { 1 }\), shown shaded in Figure 3,
3. find the area of the kite \(A Y B X\).

11. \begin{figure}[h] \end{figure} Figure 1 shows a triangle \(X Y Z\) with \(X Y = 10 \mathrm {~cm} , Y Z = 16 \mathrm {~cm}\) and \(Z X = 12 \mathrm {~cm}\).

1. Find the size of the angle \(Y X Z\), giving your answer in radians to 3 significant figures. The point \(A\) lies on the line \(X Y\) and the point \(B\) lies on the line \(X Z\) and \(A X = B X = 5 \mathrm {~cm} . A B\) is the arc of a circle with centre \(X\). The shaded region \(S\), shown in Figure 1, is bounded by the lines \(B Z , Z Y , Y A\) and the arc \(A B\). Find
2. the perimeter of the shaded region to 3 significant figures,
3. the area of the shaded region to 3 significant figures.

3. \begin{figure}[h] \end{figure} The shape \(P O Q A B C P\), as shown in Figure 1, consists of a triangle \(P O C\), a sector \(O Q A\) of a circle with radius 7 cm and centre \(O\), joined to a rectangle \(O A B C\). The points \(P , O\) and \(Q\) lie on a straight line. \(P O = 4 \mathrm {~cm} , C O = 5 \mathrm {~cm}\) and angle \(A O Q = 0.8\) radians.

1. Find the length of arc \(A Q\).
2. Find the size of angle \(P O C\) in radians, giving your answer to 3 decimal places.
   (2)
3. Find the perimeter of the shape \(P O Q A B C P\), in cm , giving your answer to 2 decimal places.
   (4)

15. \begin{figure}[h] \end{figure} Figure 5 shows the design for a logo.
The logo is in the shape of an equilateral triangle \(A B C\) of side length \(2 r \mathrm {~cm}\), where \(r\) is a constant. The points \(L , M\) and \(N\) are the midpoints of sides \(A C , A B\) and \(B C\) respectively.
The shaded section \(R\), of the logo, is bounded by three curves \(M N , N L\) and \(L M\). The curve \(M N\) is the arc of a circle centre \(L\), radius \(r \mathrm {~cm}\).
The curve \(N L\) is the arc of a circle centre \(M\), radius \(r \mathrm {~cm}\).
The curve \(L M\) is the arc of a circle centre \(N\), radius \(r \mathrm {~cm}\). Find, in \(\mathrm { cm } ^ { 2 }\), the area of \(R\). Give your answer in the form \(k r ^ { 2 }\), where \(k\) is an exact constant to be determined.

10. \begin{figure}[h] \end{figure} Figure 1 shows the design for a shop sign \(A B C D A\). The sign consists of a triangle \(A O D\) joined to a sector of a circle \(D O B C D\) with radius 1.8 m and centre \(O\). The points \(A , B\) and \(O\) lie on a straight line.
Given that \(A D = 3.9 \mathrm {~m}\) and angle \(B O D\) is 0.84 radians,

1. calculate the size of angle \(D A O\), giving your answer in radians to 3 decimal places.
2. Show that, to one decimal place, the length of \(A O\) is 4.9 m .
3. Find, in \(\mathrm { m } ^ { 2 }\), the area of the shop sign, giving your answer to one decimal place.
4. Find, in m , the perimeter of the shop sign, giving your answer to one decimal place.

7. Figure 1 shows a circle with centre \(O\) and radius 9 cm . The points \(A\) and \(B\) lie on the circumference of this circle. The minor sector \(O A B\) has perimeter 30 cm and the angle between the radii \(O A\) and \(O B\) of this sector is \(\theta\) radians. Find

1. the length of the arc \(A B\),
2. the value of \(\theta\),
3. the area of the minor sector \(O A B\),
4. the area of triangle \(O A B\), giving your answer to 3 significant figures.

11. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 2 shows the design for a sail \(A P B C A\). The curved edge \(A P B\) of the sail is an arc of a circle centre \(O\) and radius \(r \mathrm {~m}\). The straight edge \(A C B\) is a chord of the circle. The height \(A B\) of the sail is 2.4 m . The maximum width \(C P\) of the sail is 0.4 m .

1. Show that \(r = 2\)
2. Show, to 4 decimal places, that angle \(A O B = 1.2870\) radians.
3. Hence calculate the area of the sail, giving your answer, in \(\mathrm { m } ^ { 2 }\), to 3 decimal places.

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of a design for a triangular garden \(A B C\). The garden has sides \(B A\) with length \(10 \mathrm {~m} , B C\) with length 6 m and \(C A\) with length 12 m . The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 10 m . A flowerbed \(B C D\) is shown shaded in Figure 2.

1. Find the size of angle \(B A C\), in radians, to 4 decimal places.
2. Find the perimeter of the flowerbed \(B C D\), in m , to 2 decimal places.
3. Find the area of the flowerbed \(B C D\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places.

10. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 1 shows a semicircle with centre \(O\) and radius \(3 \mathrm {~cm} . X Y\) is the diameter of this semicircle. The point Z is on the circumference such that angle \(X O Z = 1.3\) radians. The shaded region enclosed by the chord \(X Z\), the arc \(Z Y\) and the diameter \(X Y\) is a template for a badge. Find, giving each answer to 3 significant figures,

1. the length of the chord \(X Z\),
2. the perimeter of the template \(X Z Y X\),
3. the area of the template.

8. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 2 shows the design for a company logo. The design consists of a triangle \(A B E\) joined to a sector \(B C D E\) of a circle with radius 6 cm and centre \(E\). The line \(A E\) is perpendicular to the line \(D E\) and the length of \(A E\) is 9 cm . The size of angle \(D E B\) is 3.5 radians, as shown in Figure 2.

1. Find the length of the arc BCD. Find, to one decimal place,
2. the perimeter of the logo,
3. the area of the logo.