1.05a Sine, cosine, tangent: definitions for all arguments

2 \includegraphics[max width=\textwidth, alt={}, center]{5998f4b1-21da-4c25-8b09-91a1cb1eee42-2_565_549_438_797} A non-uniform rod \(A B\) of weight 6 N rests in limiting equilibrium with the end \(A\) in contact with a rough vertical wall. \(A B = 1.2 \mathrm {~m}\), the centre of mass of the rod is 0.8 m from \(A\), and the angle between \(A B\) and the downward vertical is \(\theta ^ { \circ }\). A force of magnitude 10 N acting at an angle of \(30 ^ { \circ }\) to the upwards vertical is applied to the rod at \(B\) (see diagram). The rod and the line of action of the 10 N force lie in a vertical plane perpendicular to the wall. Calculate

1. the value of \(\theta\),
2. the coefficient of friction between the rod and the wall.

5 \includegraphics[max width=\textwidth, alt={}, center]{5998f4b1-21da-4c25-8b09-91a1cb1eee42-3_365_679_264_733} A uniform metal frame \(O A B C\) is made from a semicircular \(\operatorname { arc } A B C\) of radius 1.8 m , and a straight \(\operatorname { rod } A O C\) with \(A O = O C = 1.8 \mathrm {~m}\) (see diagram).

1. Calculate the distance of the centre of mass of the frame from \(O\). A uniform semicircular lamina of radius 1.8 m has weight 27.5 N . A non-uniform object is formed by attaching the frame \(O A B C\) around the perimeter of the lamina. The object is freely suspended from a fixed point at \(A\) and hangs in equilibrium. The diameter \(A O C\) of the object makes an angle of \(22 ^ { \circ }\) with the vertical.
2. Calculate the weight of the frame.

4 A particle is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal. At the instant 3 s after projection the direction of motion of the particle is \(30 ^ { \circ }\) below the horizontal.

1. Find \(V\).
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2. Calculate the distance of the particle from \(O\) at the instant 3 s after projection.

2 A particle is projected from a point on horizontal ground with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The particle strikes the ground 2 s after projection.

1. Find \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{42de91da-d65e-40e7-8de5-f40eda927850-03_67_1571_438_328}
2. Calculate the time after projection at which the direction of motion of the particle is \(20 ^ { \circ }\) below the horizontal.

3 A particle \(P\) is projected with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 4 s after projection the particle passes through the point \(A\), where \(O A = 40 \mathrm {~m}\) and the line \(O A\) makes an angle of \(30 ^ { \circ }\) with the horizontal. Calculate \(V\) and \(\theta\).

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.

9. \begin{figure}[h] \end{figure} Figure 4 shows part of the curve with equation $$y = A \cos ( x - 30 ) ^ { \circ }$$ where \(A\) is a constant. The point \(P\) is a minimum point on the curve and has coordinates \(( 30 , - 3 )\) as shown in Figure 4.

1. Write down the value of \(A\). The point \(Q\) is shown in Figure 4 and is a maximum point.
2. Find the coordinates of \(Q\).

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 \(- \pi < x < \pi\)

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}

9. \begin{figure}[h] \end{figure} Figure 4 shows part of the graph of the curve with equation \(y = \sin x\) Given that \(\sin \alpha = p\), where \(0 < \alpha < 90 ^ { \circ }\)

1. state, in terms of \(p\), the value of
   1. \(2 \sin \left( 180 ^ { \circ } - \alpha \right)\)
   2. \(\sin \left( \alpha - 180 ^ { \circ } \right)\)
   3. \(3 + \sin \left( 180 ^ { \circ } + \alpha \right)\) A copy of Figure 4, labelled Diagram 1, is shown on page 27. On Diagram 1,
2. sketch the graph of \(y = \sin 2 x\)
3. Hence find, in terms of \(\alpha\), the \(x\) coordinates of any points in the interval \(0 < x < 180 ^ { \circ }\) where $$\sin 2 x = p$$
   \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-27_433_1331_296_310}
   \section*{Diagram 1}

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 3 \cos \left( \frac { x } { n } \right) ^ { \circ } \quad x \geqslant 0$$ where \(n\) is a constant.
The curve \(C _ { 1 }\) cuts the positive \(x\)-axis for the first time at point \(P ( 270,0 )\), as shown in Figure 4.

1. 1. State the value of \(n\)
   2. State the period of \(C _ { 1 }\) The point \(Q\), shown in Figure 4, is a minimum point of \(C _ { 1 }\)
2. State the coordinates of \(Q\). The curve \(C _ { 2 }\) has equation \(y = 2 \sin x ^ { \circ } + k\), where \(k\) is a constant.
   The point \(R \left( a , \frac { 12 } { 5 } \right)\) and the point \(S \left( - a , - \frac { 3 } { 5 } \right)\), both lie on \(C _ { 2 }\) Given that \(a\) is a constant less than 90
3. find the value of \(k\).

1. \begin{figure}[h] \end{figure} Figure 1 shows the position of three stationary fishing boats \(A , B\) and \(C\), which are assumed to be in the same horizontal plane. Boat \(A\) is 10 km due north of boat \(B\). Boat \(C\) is 8 km on a bearing of \(065 ^ { \circ }\) from boat \(B\).

1. Find the distance of boat \(C\) from boat \(A\), giving your answer to the nearest 10 metres.
2. Find the bearing of boat \(C\) from boat \(A\), giving your answer to one decimal place.

7. Given that $$x = 6 \sin ^ { 2 } 2 y \quad 0 < y < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { A \sqrt { \left( B x - x ^ { 2 } \right) } }$$ where \(A\) and \(B\) are integers to be found.
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9. \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 5 shows the curve with equation $$y = \frac { 1 + 2 \cos x } { 1 + \sin x } \quad - \frac { \pi } { 2 } < x < \frac { 3 \pi } { 2 }$$ The point \(M\), shown in Figure 5, is the minimum point on the curve.

1. Show that the \(x\) coordinate of \(M\) is a solution of the equation $$2 \sin x + \cos x = - 2$$
2. Hence find, to 3 significant figures, the \(x\) coordinate of \(M\).

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

A curve \(C\) has equation $$x = \sin ^ { 2 } 4 y \quad 0 \leqslant y \leqslant \frac { \pi } { 8 } \quad 0 \leqslant x \leqslant 1$$ The point \(P\) with \(x\) coordinate \(\frac { 1 } { 4 }\) lies on \(C\)

1. Find the exact \(y\) coordinate of \(P\)
2. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\)
3. Hence show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be written in the form $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { q + r ( x + s ) ^ { 2 } } }$$ where \(q , r\) and \(s\) are constants to be found. Using the answer to part (c),
4. 1. state the \(x\) coordinate of the point where the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is a minimum,
   2. state the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at this point.

7. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\).] Two ships, \(P\) and \(Q\), are moving with constant velocities.
The velocity of \(P\) is \(( 9 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)

1. Find the direction of motion of \(P\), giving your answer as a bearing to the nearest degree. When \(t = 0\), the position vector of \(P\) is \(( 9 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) and the position vector of \(Q\) is \(( \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively.
2. Find an expression for
   1. \(\mathbf { p }\) in terms of \(t\),
   2. \(\mathbf { q }\) in terms of \(t\).
3. Hence show that, at time \(t\) hours, $$\overrightarrow { Q P } = ( 8 + 5 t ) \mathbf { i } + ( 6 - 10 t ) \mathbf { j }$$
4. Find the values of \(t\) when the ships are 10 km apart.

6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively] Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a particle \(P\) of mass 0.5 kg . \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\).
Given that the resultant force of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is in the same direction as \(- 2 \mathbf { i } - \mathbf { j }\),

1. show that \(p - 2 q = - 16\) Given that \(q = 3\)
2. find the magnitude of the acceleration of \(P\),
3. find the direction of the acceleration of \(P\), giving your answer as a bearing to the nearest degree. XXXXXXXXXXIXITEINTIIS AREA XX女X女X女X女X DO NOT WIRIE IN THS AREA.

1. \hspace{0pt} [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]

Two speedboats, \(A\) and \(B\), are each moving with constant velocity.

• the velocity of \(A\) is \(40 \mathrm { kmh } ^ { - 1 }\) due east
• the velocity of \(B\) is \(20 \mathrm { kmh } ^ { - 1 }\) on a bearing of angle \(\alpha \left( 0 ^ { \circ } < \alpha < 90 ^ { \circ } \right)\), where \(\tan \alpha = \frac { 4 } { 3 }\) The boats are modelled as particles.
  1. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) in \(\mathrm { km } \mathrm { h } ^ { - 1 }\)

At noon

• the position vector of \(A\) is \(20 \mathbf { j } \mathrm {~km}\)
• the position vector of \(B\) is \(( 10 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\)

At time \(t\) hours after noon

• the position vector of \(A\) is \(\mathbf { r k m }\), where \(\mathbf { r } = 20 \mathbf { j } + 40 t \mathbf { i }\)
• the position vector of \(B\) is \(\mathbf { s }\) km
• Find an expression for \(\mathbf { s }\) in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\).
• Show that at time \(t\) hours after noon,

$$\overrightarrow { A B } = [ ( 10 - 24 t ) \mathbf { i } + ( 12 t - 15 ) \mathbf { j } ] \mathrm { km }$$

Show that the boats will never collide.

Find the distance between the boats when the bearing of \(B\) from \(A\) is \(225 ^ { \circ }\)

8. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a particle \(A\) of mass \(2 m\). The other end of the string is attached to a particle \(B\) of mass \(3 m\). Particle \(A\) is held at rest on a rough plane which is inclined to horizontal ground at an angle \(\alpha\), where \(\tan \alpha = \frac { 5 } { 12 }\) The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. Particle \(B\) hangs vertically below \(P\) with the string taut, at a height \(h\) above the ground, as shown in Figure 4. The part of the string between \(A\) and \(P\) lies along a line of greatest slope of the plane. The two particles, the string and the pulley all lie in the same vertical plane.
The coefficient of friction between \(A\) and the plane is \(\frac { 11 } { 36 }\) The particle \(A\) is released from rest and begins to move up the plane.

1. Show that the frictional force acting on \(A\) as it moves up the plane is \(\frac { 22 m g } { 39 }\)
2. Write down an equation of motion for \(B\).
3. Show that the acceleration of \(A\) immediately after its release is \(\frac { 1 } { 3 } g\) In the subsequent motion, \(A\) comes to rest before it reaches the pulley.
4. Find, in terms of \(h\), the total distance travelled by \(A\) from when it was released from rest to when it first comes to rest again.

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5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is at rest on a rough plane which is inclined to the horizontal at \(30 ^ { \circ }\). The particle is held in equilibrium by a force of magnitude 8 N , acting at an angle of \(40 ^ { \circ }\) to the plane, as shown in Figure 2. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding up the plane, find the value of \(\mu\).

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 2 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a smooth light pulley which is fixed at the top of a rough plane. The plane is inclined to horizontal ground at an angle \(\alpha\), where tan \(\alpha = \frac { 3 } { 4 }\). Initially \(P\) is held at rest on the inclined plane with the part of the string from \(P\) to the pulley parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below the pulley at a height of 0.5 m above the ground, as shown in Figure 3. The coefficient of friction between \(P\) and the plane is \(\mu\). The system is released from rest, with the string taut, and \(Q\) strikes the ground before \(P\) reaches the pulley. The speed of \(Q\) at the instant when it strikes the ground is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. For the motion before \(Q\) strikes the ground, find the tension in the string.
2. Find the value of \(\mu\).

   END

8. \begin{figure}[h] \end{figure} A rough plane is inclined at \(30 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.5 kg is held at rest on the plane by a horizontal force of magnitude 5 N , as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The particle is on the point of moving up the plane.

1. Find the magnitude of the normal reaction of the plane on \(P\).
2. Find the coefficient of friction between \(P\) and the plane. The force of magnitude 5 N is now removed and \(P\) accelerates from rest down the plane.
3. Find the speed of \(P\) after it has travelled 3 m down the plane.

1. The diagram shows the curve with equation \(y = \ln ( 2 + \cos x ) , x \geq 0\).
The smallest value of \(x\) for which the curve meets the \(x\)-axis is \(a\) as shown.

1. Find the value of \(a\).
2. Use Simpson's rule with four strips of equal width to estimate the area of the region bounded by the curve in the interval \(0 \leq x \leq a\) and the coordinate axes.

4 The \(n\)th term, \(t _ { n }\), of a sequence is given by $$t _ { n } = \sin ( \theta + 180 n ) ^ { \circ }$$ Express \(t _ { 1 }\) and \(t _ { 2 }\) in terms of \(\sin \theta ^ { \circ }\).

9 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 \sin x & \text { for } - \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi \\ \mathrm {~g} ( x ) = 4 - 2 x ^ { 2 } & \text { for } x \in \mathbb { R } . \end{array}$$

1. State the range of f and the range of g .
2. Show that \(\operatorname { gf } ( 0.5 ) = 2.16\), correct to 3 significant figures, and explain why \(\mathrm { fg } ( 0.5 )\) is not defined.
3. Find the set of values of \(x\) for which \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\) is not defined.