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

7 The height of the tide in a certain harbour is \(h\) metres at time \(t\) hours. Successive high tides occur every 12 hours. The rate of change of the height of the tide can be modelled by a function of the form \(a \cos ( k t )\), where \(a\) and \(k\) are constants. The largest value of this rate of change is 1.3 metres per hour. Write down a differential equation in the variables \(h\) and \(t\). State the values of the constants \(a\) and \(k\).

1 A golf ball is hit at an angle of \(60 ^ { \circ }\) to the horizontal from a point, O, on level horizontal ground. Its initial speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The standard projectile model, in which air resistance is neglected, is used to describe the subsequent motion of the golf ball. At time \(t \mathrm {~s}\) the horizontal and vertical components of its displacement from O are denoted by \(x \mathrm {~m}\) and \(y \mathrm {~m}\).

1. Write down equations for \(x\) and \(y\) in terms of \(t\).
2. Hence show that the equation of the trajectory is $$y = \sqrt { 3 } x - 0.049 x ^ { 2 }$$
3. Find the range of the golf ball.
4. A bird is hovering at position \(( 20,16 )\). Find whether the golf ball passes above it, passes below it or hits it.

7 Fig. 4 shows a particle projected over horizontal ground from a point O at ground level. The particle initially has a speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal. The particle is a horizontal distance of 44.8 m from O after 5 seconds. Air resistance should be neglected. \begin{figure}[h] \end{figure}

1. Write down an expression, in terms of \(\alpha\) and \(t\), for the horizontal distance of the particle from O at time \(t\) seconds after it is projected.
2. Show that \(\cos \alpha = 0.28\).
3. Calculate the greatest height reached by the particle.

2 A ball is kicked from ground level over horizontal ground. It leaves the ground at a speed of 25 ms 1 and at an angle \(\theta\) to the horizontal such that \(\cos \theta = 0.96\) and \(\sin \theta = 0.28\).

1. Show that the height, \(y \mathrm {~m}\), of the ball above the ground \(t\) seconds after projection is given by \(y = 7 t - 4.9 t ^ { 2 }\). Show also that the horizontal distance, \(x \mathrm {~m}\), travelled by this time is given by \(x = 24 t\).
2. Calculate the maximum height reached by the ball.
3. Calculate the times at which the ball is at half its maximum height. Find the horizontal distance travelled by the ball between these times.
4. Determine the following when \(t = 1.25\).
   (A) The vertical component of the velocity of the ball.
   (B) Whether the ball is rising or falling. (You should give a reason for your answer.)
   (C) The speed of the ball.
5. Show that the equation of the trajectory of the ball is $$y = \frac { 0.7 x } { 576 } ( 240 - 7 x )$$ Hence, or otherwise, find the range of the ball.

4 You should neglect air resistance in this question.
A small stone is projected from ground level. The maximum height of the stone above horizontal ground is 22.5 m .

1. Show that the vertical component of the initial velocity of the stone is \(21 \mathrm {~ms} { } ^ { 1 }\). The speed of projection is \(28 \mathrm {~ms} { } ^ { 1 }\).
2. Find the angle of projection of the stone.
3. Find the horizontal range of the stone.

5 In this question take the value of \(\boldsymbol { g }\) to be \(\mathbf { 1 0 ~ } \mathbf { m ~ s } ^ { \mathbf { 2 } }\). \(\Lambda\) particle \(\Lambda\) is projected over horizontal ground from a point P which is 9 m above a point O on the ground. The initial velocity has horizontal and vertical components of \(10 \mathrm {~ms} ^ { - 1 }\) and \(12 \mathrm {~ms} ^ { - 1 }\) respectively, as shown in Fig. 7. The trajectory of the particle meets the ground at X. Air resistance may be neglected. \begin{figure}[h] \end{figure}

1. Calculate the specd of projection \(u \mathrm {~ms} ^ { - 1 }\) and the angle of projection \(\theta ^ { \circ }\).
2. Show that, \(t\) seconds after projection, the height of particle A above the ground is \(9 + 12 t - 5 t ^ { 2 }\). Write down an expression in terms of \(t\) for the horizontal distance of the particle from O at this time.
3. Calculate the maximum height of particle \(\Lambda\) above the point of projection.
4. Calculate the distance OX . \(\wedge\) second particle, \(B\), is projected from \(O\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(60 ^ { \circ }\) to the horizontal. The trajectories of A and B are in the same vertical plane. Particles A and B are projected at the same time.
5. Show that the horizontal displacements of A and B are always cqual.
6. Show that, \(t\) seconds after projection, the height of particle B above the ground is \(10 \sqrt { 3 } t - 5 t ^ { 2 }\).
7. Show that the particles collide 1.7 seconds after projection (correct to two significant figures).

The diagram shows a sector \(CAB\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D\), \(E\), and \(F\), where \(COD\) is a straight line and angle \(ACD\) is \(\theta\) radians.

1. Find \(C D\) in terms of \(r\) and \(\sin \theta\).
   It is now given that \(r = 4\) and \(\theta = \frac { 1 } { 6 } \pi\).
2. Find the perimeter of sector \(C A B\) in terms of \(\pi\).
3. Find the area of the shaded region in terms of \(\pi\) and \(\sqrt { 3 }\).

5 In this question you must show detailed reasoning. Fig. 5 shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 10\).
The points \(( 1,0 )\) and \(( 7,0 )\) lie on the circle. The point C is the centre of the circle. \begin{figure}[h] \end{figure} Find the area of the part of the circle below the \(x\)-axis.

4 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-5_487_789_251_639} The diagram shows the triangle \(A O B\), in which angle \(A O B = 0.8\) radians, \(O A = 7 \mathrm {~cm}\) and \(O B = 10 \mathrm {~cm}\). \(C D\) is the arc of a circle with centre \(O\) and radius \(O C\). The area of the triangle \(A O B\) is twice the area of the sector COD

1. Find the length \(O C\).
2. Find the perimeter of the region \(A B C D\).

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\). The radius of the circle is 6 cm and the angle \(A O B\) is 1.2 radians.

1. Find the area of the sector \(O A B\).
2. Find the perimeter of the sector \(O A B\).

3 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 20 cm . The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-3_453_499_429_804} The length of the \(\operatorname { arc } A B\) is 28 cm .

1. Show that \(\theta = 1.4\).
2. Find the area of the sector \(O A B\).
3. The point \(D\) lies on \(O A\). The region bounded by the line \(B D\), the line \(D A\) and the arc \(A B\) is shaded. \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-3_440_380_1372_806} The length of \(O D\) is 15 cm .
   1. Find the area of the shaded region, giving your answer to three significant figures.
      (3 marks)
   2. Use the cosine rule to calculate the length of \(B D\), giving your answer to three significant figures.
      (3 marks)

6. \includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-2_577_970_799_360} The diagram shows triangle \(A B C\) in which \(A C = 14 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(\angle A B C = 1.7\) radians.

1. Find the size of \(\angle A C B\) in radians. The point \(D\) lies on \(A C\) such that \(B D\) is an arc of a circle, centre \(C\).
2. Find the perimeter of the shaded region bounded by the arc \(B D\) and the straight lines \(A B\) and \(A D\).

7 An arrow is fired from a point \(A\) with a velocity of \(25 \mathrm {~ms} ^ { - 1 }\), at an angle of \(40 ^ { \circ }\) above the horizontal. The arrow hits a target at the point \(B\) which is at the same level as the point \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-4_195_1093_1594_511}

1. State two assumptions that you should make in order to model the motion of the arrow.
   (2 marks)
2. Show that the time that it takes for the arrow to travel from \(A\) to \(B\) is 3.28 seconds, correct to three significant figures.
3. Find the distance between the points \(A\) and \(B\).
4. State the magnitude and direction of the velocity of the arrow when it hits the target.
5. Find the minimum speed of the arrow during its flight.

4 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-05_510_606_1745_274} The angle \(A O B\) is \(\theta\) radians. The arc length \(A B\) is 15 cm and the area of the sector is \(45 \mathrm {~cm} ^ { 2 }\).

1. Find the values of \(r\) and \(\theta\).
2. Find the area of the segment bounded by the arc \(A B\) and the chord \(A B\).

8 The diagram shows a sector of a circle \(O A B\). \(C\) is the midpoint of \(O B\).
Angle \(A O B\) is \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-10_700_963_536_534} 8

1. Given that the area of the triangle \(O A C\) is equal to one quarter of the area of the sector \(O A B\), show that \(\theta = 2 \sin \theta\) 8
2. Use the Newton-Raphson method with \(\theta _ { 1 } = \pi\), to find \(\theta _ { 3 }\) as an approximation for \(\theta\). Give your answer correct to five decimal places.
   8
3. Given that \(\theta = 1.89549\) to five decimal places, find an estimate for the percentage error in the approximation found in part (b).
   Turn over for the next question

3 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 2 \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-03_374_455_1187_790} The angle \(A O B\) is \(\theta\) radians and the perimeter of the sector is 6
Find the value of \(\theta\) Circle your answer.
[0pt] [1 mark]
1 \(\sqrt { 3 }\) 2
3

10 The diagram shows a sector of a circle \(O A B\). \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-16_758_796_360_623} The point \(C\) lies on \(O B\) such that \(A C\) is perpendicular to \(O B\).
Angle \(A O B\) is \(\theta\) radians.
10

1. Given the area of the triangle \(O A C\) is half the area of the sector \(O A B\), show that $$\theta = \sin 2 \theta$$ 10
2. Use a suitable change of sign to show that a solution to the equation $$\theta = \sin 2 \theta$$ lies in the interval given by \(\theta \in \left[ \frac { \pi } { 5 } , \frac { 2 \pi } { 5 } \right]\)

   10 The Newton-Raphson method is used to find an approximate solution to the equation

   \(\theta = \sin 2 \theta\)

   10 (c) (i) Using \(\theta _ { 1 } = \frac { \pi } { 5 }\) as a first approximation for \(\theta\) apply the Newton-Raphson method twice

   to find the value of \(\theta _ { 3 }\) Give your answer to three decimal places.
   10 (c) (ii) Explain how a more accurate approximation for \(\theta\) can be found using the Newton-Raphson method.
   10 (c) (iii) Explain why using \(\theta _ { 1 } = \frac { \pi } { 6 }\) as a first approximation in the Newton-Raphson method
   [0pt] [2 marks] does not lead to a solution for \(\theta\).

1. A plot of land \(O A B\) is in the shape of a sector of a circle with centre \(O\).

Given

• \(O A = O B = 5 \mathrm {~km}\)
• angle \(A O B = 1.2\) radians
  1. find the perimeter of the plot of land.
     (2)

\begin{figure}[h] \end{figure} A point \(P\) lies on \(O B\) such that the line \(A P\) divides the plot of land into two regions \(R _ { 1 }\) and \(R _ { 2 }\) as shown in Figure 2. Given that $$\text { area of } R _ { 1 } = 3 \times \text { area of } R _ { 2 }$$

show that the area of \(R _ { 2 } = 3.75 \mathrm {~km} ^ { 2 }\)

Find the length of \(A P\), giving your answer to the nearest 100 m .

4 A sector \(A O B\) of a circle has radius \(r \mathrm {~cm}\) and the angle \(A O B\) is \(\theta\) radians. The perimeter of the sector is 40 cm and its area is \(100 \mathrm {~cm} ^ { 2 }\).

1. Write down equations for the perimeter and area of the sector in terms of \(r\) and \(\theta\).
2. Use your equations to show that \(r ^ { 2 } - 20 r + 100 = 0\) and hence find the value of \(r\) and of \(\theta\).