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

10. \begin{figure}[h] \end{figure} Circle \(C _ { 1 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 64\) with centre \(O _ { 1 }\).
Circle \(C _ { 2 }\) has equation \(( x - 6 ) ^ { 2 } + y ^ { 2 } = 100\) with centre \(O _ { 2 }\).
The circles meet at points \(A\) and \(B\) as shown in Figure 3.
a. Show that angle \(A O _ { 2 } B = 1.85\) radians to 3 significant figures.
(3)
b. Find the area of the shaded region, giving your answer correct to 1 decimal place.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sector \(P O Q\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
The angle \(P O Q\) is 0.5 radians.
The area of the sector is \(9 \mathrm {~cm} ^ { 2 }\).
Show that the perimeter of the sector is \(k\) times the length of the arc, where \(k\) is an integer.

3. \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 \(\theta\) radians.
The area of the sector \(A O B\) is \(11 \mathrm {~cm} ^ { 2 }\) Given that the perimeter of the sector is 4 times the length of the arc \(A B\), find the exact value of \(r\).

15. \begin{figure}[h] \end{figure} A company makes toys for children.
Figure 5 shows the design for a solid toy that looks like a piece of cheese.
The toy is modelled so that

• face \(A B C\) is a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(A\)
• angle \(B A C = 0.8\) radians
• faces \(A B C\) and \(D E F\) are congruent
• edges \(A D , C F\) and \(B E\) are perpendicular to faces \(A B C\) and \(D E F\)
• edges \(A D , C F\) and \(B E\) have length \(h \mathrm {~cm}\)

Given that the volume of the toy is \(240 \mathrm {~cm} ^ { 3 }\)

1. show that the surface area of the toy, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 0.8 r ^ { 2 } + \frac { 1680 } { r }$$ making your method clear. Using algebraic differentiation,
2. find the value of \(r\) for which \(S\) has a stationary point.
3. Prove, by further differentiation, that this value of \(r\) gives the minimum surface area of the toy.

8. \begin{figure}[h] \end{figure} Figure 1 shows the plan view of a stage.
The plan view shows two congruent triangles \(A B O\) and \(G F O\) joined to a sector \(O C D E O\) of a circle, centre \(O\), where

• angle \(C O E = 2.3\) radians
• arc length \(C D E = 27.6 \mathrm {~m}\)
• \(A O G\) is a straight line of length 15 m
  1. Show that \(O C = 12 \mathrm {~m}\).
  2. Show that the size of angle \(A O B\) is 0.421 radians correct to 3 decimal places.

Given that the total length of the front of the stage, \(B C D E F\), is 35 m ,

find the total area of the stage, giving your answer to the nearest square metre.

11. \begin{figure}[h] \end{figure} Figure 4 shows the design of a badge.
The shape \(A B C O A\) is a semicircle with centre \(O\) and diameter 10 cm . \(O B\) is the arc of a circle with centre \(A\) and radius 5 cm .
The region \(R\), shown shaded in Figure 4, is bounded by the arc \(O B\), the arc \(B C\) and the line \(O C\). Find the exact area of \(R\).
Give your answer in the form \(( a \sqrt { 3 } + b \pi ) \mathrm { cm } ^ { 2 }\), where \(a\) and \(b\) are rational numbers.

11. \begin{figure}[h] \end{figure} Circle \(C _ { 1 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 100\) Circle \(C _ { 2 }\) has equation \(( x - 15 ) ^ { 2 } + y ^ { 2 } = 40\) The circles meet at points \(A\) and \(B\) as shown in Figure 3.

1. Show that angle \(A O B = 0.635\) radians to 3 significant figures, where \(O\) is the origin. The region shown shaded in Figure 3 is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
2. Find the perimeter of the shaded region, giving your answer to one decimal place.

1. The curve \(C\) has equation

$$y = 3 x ^ { 4 } - 8 x ^ { 3 } - 3$$

1. Find (i) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) (ii) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Verify that \(C\) has a stationary point when \(x = 2\)
3. Determine the nature of this stationary point, giving a reason for your answer.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sector \(A O B\) of a circle with centre \(O\), radius 5 cm and angle \(A O B = 40 ^ { \circ }\) The attempt of a student to find the area of the sector is shown below. $$\begin{aligned} \text { Area of sector } & = \frac { 1 } { 2 } r ^ { 2 } \theta \\ & = \frac { 1 } { 2 } \times 5 ^ { 2 } \times 40 \\ & = 500 \mathrm {~cm} ^ { 2 } \end{aligned}$$

1. Explain the error made by this student.
2. Write out a correct solution.

6. \begin{figure}[h] \end{figure} The shape \(O A B C D E F O\) shown in Figure 1 is a design for a logo.
In the design

• \(O A B\) is a sector of a circle centre \(O\) and radius \(r\)
• sector \(O F E\) is congruent to sector \(O A B\)
• \(O D C\) is a sector of a circle centre \(O\) and radius \(2 r\)
• \(A O F\) is a straight line

Given that the size of angle \(C O D\) is \(\theta\) radians,

1. write down, in terms of \(\theta\), the size of angle \(A O B\)
2. Show that the area of the logo is $$\frac { 1 } { 2 } r ^ { 2 } ( 3 \theta + \pi )$$
3. Find the perimeter of the logo, giving your answer in simplest form in terms of \(r , \theta\) and \(\pi\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a circle with centre \(O\). The points \(A\) and \(B\) lie on the circumference of the circle. The area of the major sector, shown shaded in Figure 1, is \(135 \mathrm {~cm} ^ { 2 }\). The reflex angle \(A O B\) is 4.8 radians. Find the exact length, in cm, of the minor arc \(A B\), giving your answer in the form \(a \pi + b\), where \(a\) and \(b\) are integers to be found.
(4)

6 \includegraphics[max width=\textwidth, alt={}, center]{68f1107f-f188-4698-934e-8fd593b25418-4_442_661_840_260} The diagram shows triangle \(A B C\), with \(A B = x \mathrm {~cm} , A C = y \mathrm {~cm}\) and angle \(B A C = 60 ^ { \circ }\). It is given that the area of the triangle is \(( x + y ) \sqrt { 3 } \mathrm {~cm} ^ { 2 }\).

1. Show that \(4 x + 4 y = x y\). When the vertices of the triangle are placed on the circumference of a circle, \(A C\) is a diameter of the circle.
2. Determine the value of \(x\) and the value of \(y\).

1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 9.5 cm . The angle \(A O B\) is \(25 ^ { \circ }\).

1. Calculate the length of the straight line \(A B\).
2. Find the area of the segment shaded in the diagram.

12 Fig. 12 shows the circle \(( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 25\), the line \(4 y = 3 x - 32\) and the tangent to the circle at the point \(\mathrm { A } ( 5,2 )\). D is the point of intersection of the line \(4 y = 3 x - 32\) and the tangent at A . \begin{figure}[h] \end{figure}

1. Write down the coordinates of C , the centre of the circle.
2. (A) Show that the line \(4 y = 3 x - 32\) is a tangent to the circle.
   (B) Find the coordinates of B , the point where the line \(4 y = 3 x - 32\) touches the circle.
3. Prove that ADBC is a square.
4. The point E is the lowest point on the circle. Find the area of the sector ECB .

14 Fig. 14 shows a circle with centre O and radius \(r \mathrm {~cm}\). The chord AB is such that angle \(\mathrm { AOB } = x\) radians. The area of the shaded segment formed by AB is \(5 \%\) of the area of the circle. \begin{figure}[h] \end{figure}

1. Show that \(x - \sin x - \frac { 1 } { 10 } \pi = 0\). The Newton-Raphson method is to be used to find \(x\).
2. Write down the iterative formula to be used for the equation in part (a).
3. Use three iterations of the Newton-Raphson method with \(x _ { 0 } = 1.2\) to find the value of \(x\) to a suitable degree of accuracy.

1 Fig. 1 shows a sector of a circle of radius 7 cm . The area of the sector is \(5 \mathrm {~cm} ^ { 2 }\). \begin{figure}[h] \end{figure} Find the angle \(\theta\) in radians.

7 The area of a sector of a circle is \(36.288 \mathrm {~cm} ^ { 2 }\). The angle of the sector is \(\theta\) radians and the radius of the circle is \(r \mathrm {~cm}\).

1. Find an expression for \(\theta\) in terms of \(r\). The perimeter of the sector is 24.48 cm .
2. Show that \(\theta = \frac { 24.48 } { r } - 2\).
3. Find the possible values of \(r\).

2 Fig. 2 shows a sector of a circle of radius 8 cm . The angle of the sector is 2.1 radians. \begin{figure}[h] \end{figure}

1. Calculate the length of the arc \(L\).
2. Calculate the area of the sector.

2

1. Write \(65 ^ { \circ }\) in radians, giving your answer in the form \(k \pi\), where \(k\) is a fraction in its lowest terms.
2. Write 0.211 radians in degrees, giving your answer correct to \(\mathbf { 1 }\) decimal place.

7 A wire, 10 cm long, is bent to form the perimeter of a sector of a circle, as shown in the diagram. The radius is \(r \mathrm {~cm}\) and the angle at the centre is \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{20639e13-01cc-4d96-b694-fb3cf1828f4d-07_323_204_342_242} Determine the maximum possible area of the sector, showing that it is a maximum.

12 Explain why the smaller regular hexagon in Fig. C1 has perimeter 6.

13 Show that the larger regular hexagon in Fig. C1 has perimeter \(4 \sqrt { 3 }\).

15 Fig. 15 shows a unit circle and the escribed regular polygon with 12 edges. \begin{figure}[h] \end{figure}

1. Show that the perimeter of the polygon is \(24 \tan 15 ^ { \circ }\).
2. Using the formula for \(\tan ( \theta - \phi )\) show that the perimeter of the polygon is \(48 - 24 \sqrt { 3 }\).

16 On a unit circle, the inscribed regular polygon with 12 edges gives a lower bound for \(\pi\), and the escribed regular polygon with 12 edges gives an upper bound for \(\pi\). Calculate the values of these bounds for \(\pi\), giving your answers:

1. in surd form
2. correct to 2 decimal places. www.ocr.org.uk after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 9 EA.
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4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians. The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).

1. Show that \(\theta = 0.430\) correct to three significant figures.
2. Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
3. The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-004_417_883_1436_557} Calculate, to two significant figures:
   1. the length of the \(\operatorname { arc } A D\);
   2. the area of the shaded region.