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Question 1

SLPaper 2

A research drone is tethered by two cables, $AP$ and $BP$ , attached to points $A$ and $B$ on level ground. The points $A$ and $B$ are 80 m apart. The drone $P$ is directly above the point $P_0$ on the ground. In the horizontal triangle $ABP_0$ , the angle at $P_0$ between the lines $P_0A$ and $P_0B$ is $60^\circ$ .

At a certain instant, the angle of elevation of the drone from $A$ is $28^\circ$ and from $B$ is $35^\circ$ .

1.

Draw a fully-labelled 3-D diagram showing $A, B, P_0, P$ and all known angles.

[2]

2.

Let the horizontal distances $AP_0 = x$ m and $BP_0 = y$ m. Show that $x \tan 28^\circ = y \tan 35^\circ.$

[2]

3.

Using the cosine rule in triangle $ABP_0$ , show that $80^2 = x^2 + y^2 - 2xy \cos 60^\circ.$

[2]

4.

Use your results from part 2 and 3 to find the height $h$ of the drone above the ground. Give your answer to the nearest metre.

[3]

5.

The drone then moves horizontally so that its projection $P_0$ traces out a circular path of radius 80 m centred at $A$ . Calculate the angle swept by $P_0$ in radians and the length of the arc of this path when the bearing of $P_0$ from $A$ changes from $050^\circ$ to $130^\circ$ .

[3]

6.

The tension in each cable is proportional to its length. If the tension in $AP$ is $T_A = k|AP|$ and in $BP$ is $T_B = k|BP|$ , find the ratio $\dfrac{T_A}{T_B}$ at this instant.

[3]

Question 2

SLPaper 2

Given the triangle ABC where $AC=5 \text{ cm}, BC=4 \text{ cm}, AB=6 \text{ cm}$ .

1.

Find $\angle B$ .

[3]

2.

Find area of the triangle $ABC$ .

[3]

Question 3

SLPaper 2

A rescue helicopter $P$ hovers above a mountain valley. It is connected by two winch cables to huts $A$ and $B$ on level ground. The distance between the huts is $90 \text{ m}$ . The helicopter is vertically above point $P_0$ , and in the horizontal triangle $ABP_0$ , $\angle AP_0B = 70^\circ$ .

The angle of elevation of the helicopter is $32^\circ$ from $A$ and $45^\circ$ from $B$ .

1.

Sketch a fully-labelled 3-D diagram showing $A, B, P_0, P$ and all given angles.

[2]

2.

Let $AP_0=x$ m and $BP_0=y$ m. Show that $x\tan32^\circ = y\tan45^\circ.$

[2]

3.

Using the cosine rule in triangle $ABP_0$ , write down an equation relating $x$ and $y$ .

[2]

4.

Use your results to find the height $h$ of the helicopter above the ground, correct to the nearest metre.

[2]

5.

The point $P_0$ later moves horizontally around a circular path of radius $90 \text{ m}$ centred at $A$ . Find the length of the arc, in metres and radians, when its bearing from $A$ changes from $010^\circ$ to $100^\circ$ .

[3]

6.

If tensions are $T_A = k|AP|$ and $T_B = k|BP|$ , find $\dfrac{T_A}{T_B}$ .

[3]

Question 4

SLPaper 2

A cyclist can choose between using two ramps. The equations of the ramps with height $y$ at horizontal distance $x$ are $y = 3x$ and $y = 2x$ .

1.

By finding the angle they make with the ground, find the angle created between the two ramps.

[4]

2.

Explain which ramp the cyclist should take if they are trying to reach a higher level in the shortest horizontal distance.

[1]

Question 5

SLPaper 2

A hot-air balloon $P$ is anchored by two ropes to points $A$ and $B$ on level ground. The distance $AB$ is 75 m. The balloon is vertically above $P_0$ , and $\angle AP_0B = 60^\circ$ in the horizontal triangle $ABP_0$ .

The angles of elevation of the balloon are $28^\circ$ from $A$ and $37^\circ$ from $B$ .

1.

Draw a fully-labelled 3-D diagram showing $A,B,P_0,P$ and all the given angles.

[2]

2.

Let $AP_0=x$ m and $BP_0=y$ m. Show that

x\tan28^\circ = y\tan37^\circ.

[2]

3.

Using the cosine rule in triangle $ABP_0$ , write an equation relating $x$ and $y$ .

[2]

4.

Find the height $h$ of the balloon, to the nearest metre.

[2]

5.

The projection $P_0$ moves horizontally on a circle of radius $75\text{ m}$ centered at $A$ . Find the angle subtended by the arc in radians and the length of the arc in metres when the bearing of $P_0$ from $A$ changes from $045^\circ$ to $115^\circ$ .

[3]

6.

If $T_A = k|AP|$ and $T_B = k|BP|$ , find $\dfrac{T_A}{T_B}$ .

[3]

Question 6

SLPaper 2

1.

For triangle $ABC$ , $\angle A=30^{\circ}$ , $BC=4$ , and $AC=6$ . Find the possible values of angle $B$ and the corresponding lengths of side $AB$ .

[7]

Question 7

SLPaper 2

A satellite dish $D$ is positioned vertically above a point $D_0$ on the ground. Two control posts, $C_1$ and $C_2$ , are 40 m apart. The bearing of $C_2$ from $C_1$ is $090^\circ$ . The bearing of the dish $D_0$ from $C_1$ is $010^\circ$ . At a certain instant, the angle of elevation of the dish from $C_1$ is $60^\circ$ and from $C_2$ is $45^\circ$ .

1.

Draw a fully-labelled 3D diagram showing $C_1,C_2,D_0,D$ and all known angles.

[2]

2.

Let the horizontal distances $C_1 D_0 = x$ m and $C_2 D_0 = y$ m. Show that

x\tan60^\circ = y\tan45^\circ .

[2]

3.

Using the cosine rule in triangle $C_1C_2D_0$ , express $y$ in terms of $x$ and known constants.

[3]

4.

Combine your results from parts 2 and 3 to find the height $h$ of the dish. Give your answer to the nearest tenth of a metre.

[3]

5.

The dish's projection $D_0$ describes a circular path of radius $x$ m. The path subtends an angle of $2.5$ radians. Calculate the arc length in metres.

[3]

6.

Find the ratio of the horizontal distance from $C_2$ to the 3D distance $|C_2 D|$ at this instant.

[3]

Question 8

SLPaper 2

A rescue beacon $B$ is positioned vertically above a point $B_0$ on the ground. Two observers, $O_1$ and $O_2$ , are 500 m apart on level ground. The bearing of $O_2$ from $O_1$ is $045^\circ$ . The bearing of the beacon $B_0$ from $O_1$ is $075^\circ$ . At a certain instant, the angle of elevation of the beacon from $O_1$ is $15^\circ$ and from $O_2$ is $25^\circ$ .

1.

Draw a fully-labelled 3-D diagram showing $O_1,O_2,B_0,B$ and all known angles.

[2]

2.

Let the horizontal distances $O_1 B_0 = x$ m and $O_2 B_0 = y$ m. Show that $x\tan15^\circ = y\tan25^\circ .$

[2]

3.

Using the cosine rule in triangle $O_1O_2B_0$ , express $y$ in terms of $x$ and known constants.

[3]

4.

Combine your results from parts 2 and 3 to find the height $h$ of the beacon, given that $O_1B_0 > 500\text{ m}$ . Give your answer to the nearest metre.

[5]

5.

In a different scenario, the beacon's projection $B_0$ moves along a circular path of radius 600 m centered at $O_1$ . Calculate, in metres, the arc length of the path when the bearing from $O_1$ to $B_0$ changes from $075^\circ$ to $135^\circ$ .

[3]

6.

Using the values found in part 4, let the length of the tether cable attached at $O_1$ be $L_1 = |O_1B|$ and the length of the cable attached at $O_2$ be $L_2 = |O_2B|$ . Find the ratio $\dfrac{L_1}{L_2}$ at that instant.

[3]

Question 9

SLPaper 2

Given the triangular farm $ABC$ with $AB=12$ , $AC=20$ and $\angle BAC=35^{\circ}$ . Now there lies a point $X$ on $AC$ such that $AX:XC$ is $1:4$ as shown in the figure. The fencing has to be installed around its perimeter.

1.

Find the total length of the fencing required.

[3]

2.

Now $B$ connects to $X$ making a new farm. Calculate the area of triangle $BXC$ .

[4]

Question 10

SLPaper 2

Two observation towers, $A$ and $B$ , stand on level ground 250 m apart. Tower $A$ is 40 m high, tower $B$ is 70 m high. A drone $D$ is observed simultaneously from the tops of both towers. From $A$ , the angle of elevation of $D$ is $25^\circ$ ; from $B$ , the angle of elevation is $21.3^\circ$ . The horizontal bearing of $B$ from $A$ is $090^\circ$ . The bearing of the drone from $A$ is $042^\circ$ , and from $B$ is $330^\circ$ .

1.

Draw a clearly labelled 3-D diagram showing the towers, the ground separation, the bearings, and the line of sight from each tower to $D$ .

[2]

2.

Let the point vertically below the drone be $P$ on the ground. Find the horizontal distances $AP$ and $BP$ .

[3]

3.

Hence find the height of the drone above the ground and its straight-line 3-D distance from the top of tower $A$ .

[4]

4.

The drone now circles tower $A$ at constant height $146.1$ m, describing a horizontal circle of radius 200 m. A camera on $A$ can rotate through a maximum of $160^\circ$ before needing to reset. Determine the length of visible path of the drone from $A$ before the camera must reset, giving your answer to the nearest metre.

[3]

5.

During this circular motion, the drone begins a climb such that its height increases proportionally to the square of the horizontal angle turned: $h(\theta) = 146.1 + 2.5\theta^2 \quad (0 \le \theta \le 2.7925),$ where $\theta$ is in radians. Find, correct to three significant figures,

the total 3-D distance flown by the drone over this visible path, and
the greatest angle of elevation of the drone from the top of tower A.

[6]

SL 3.2—2d and 3d trig, sine rule, cosine rule, area Questionbank