Question 1
SLPaper 1The diameter of a spherical planet is km.
Write down the radius of the planet.
The volume of the planet can be expressed in the form where and .
Find the value of and the value of .
Question 2
HLPaper 1By using the substitution or otherwise, find an expression for in terms of , where is a non-zero real number.
Question 3
SLPaper 2The following diagram shows quadrilateral ABCD.
Find .
Find
Question 4
SLPaper 2A farmer is placing posts at points , , and in the ground to mark the boundaries of a triangular piece of land on his property.
From point , he walks due west 230 metres to point . From point , he walks 175 metres on a bearing of to reach point .
This is shown in the following diagram.
The farmer wants to divide the piece of land into two sections. He will put a post at point , which is between and . He wants the boundary to divide the piece of land such that the sections have equal area.
This is shown in the following diagram.
Find the distance from point to point .
Find the area of this piece of land.
Find .
Find the distance from point to point .
Question 5
HLPaper 2The voltage in a circuit is given by the equation
where is measured in seconds.
The current in this circuit is given by the equation
The power in this circuit is given by .
The average power in this circuit from to is given by the equation
where .
Write down the maximum and minimum value of .
Write down two transformations that will transform the graph of onto the graph of .
Sketch the graph of for , showing clearly the coordinates of the first maximum and the first minimum.
Find the total time in the interval 0 ≤ ≤ 0.02 for which
Find .
With reference to your graph of , explain why for all .
Question 6
SLPaper 1The following diagram shows a right triangle ABC. Point D lies on AB such that CDbisects AĈB.
AĈD = and AC = 14 cm
Given that , find the value of .
Find the value of .
Hence or otherwise, find .
Question 7
SLPaper 1A buoy is floating in the sea and can be seen from the top of a vertical cliff. A boat is travelling from the base of the cliff directly towards the buoy.
The top of the cliff is 142 m above sea level. Currently the boat is 100 metres from the buoy and the angle of depression from the top of the cliff to the boat is 64°.
Question 8
SLPaper 1The following diagram shows triangle , with , and . Given that , find the area of the triangle. Give your answer in the form where .
Question 9
HLPaper 1[ \begin{{align*}} \text{{A sector of a circle with radius }} r \text{{ cm , where }} r > 0 \text{{, is shown on the following diagram.}} \ \text{{The sector has an angle of 1 radian at the centre.}} \end{{align*}} ]
[ \text{{}} ]
[ \text{{Let the area of the sector be }} A \text{{ cm}}^2 \text{{ and the perimeter be }} P \text{{ cm. Given that }} A = P \text{{, find the value of }} r\text{{.}} ]
A sector of a circle with radius ({r}) cm, where (r > 0), is shown on the following diagram. The sector has an angle of 1 radian at the centre.
Let the area of the sector be (A) cm^2 and the perimeter be (P) cm. Given that (A = P), find the value of (r).
Question 10
SLPaper 2The following diagram shows the quadrilateral ABCD. AB = 6.73 cm, BC = 4.83 cm, BĈD = 78.2° and CD = 3.80 cm.
Find .
The area of triangle ABD is . Find the possible values of .