Question 1
SLPaper 2The Voronoi diagram below shows four supermarkets represented by points with coordinates, , and . The vertices , , are also shown. All distancesare measured in kilometres.
The equation of is and the equation of is .
The coordinates of are and the coordinates of are .
A town planner believes that the larger the area of the Voronoi cell , the more people willshop at supermarket .
Find the midpoint of .
Find the equation of .
Find the coordinates of .
Determine the exact length of .
Given that the exact length ofis , find the size of in degrees.
Hence find the area of triangle .
State one criticism of this interpretation.
Question 2
HLPaper 1The rate, , of a chemical reaction at a fixed temperature is related to the concentration oftwo compounds, and , by the equation
, where,, .
A scientist measures the three variables three times during the reaction and obtains thefollowing values.
Find , and .
Question 3
SLPaper 2Consider the function, where x > 0 and k is a constant.
The graph of the function passes through the point with coordinates (4 , 2).
P is the minimum point of the graph of f (x).
Find the value of k.
Using your value of k , find f ′(x).
Use your answer to part (b) to show that the minimum value of f(x) is −22 .
Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.
Question 4
SLPaper 1Let , where p≠ 0. FindFind the number of roots for the equation .
Justify your answer.
Question 5
SLPaper 2Let ,be a periodic function with
The following diagram shows the graph of.
There is a maximum point at A. The minimum value of is −13 .
A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, so that it moves back and forth vertically.
The distance, d centimetres, of the centre of the ball from O at time t seconds, is given by
Find the coordinates of A.
For the graph of , write downthe amplitude.
For the graph of , write down the period.
Hence, write in the form.
Find the maximum speed of the ball.
Find the first time when the ball’s speed is changing at a rate of 2 cm s−2.
Question 6
SLPaper 2Let for.
Let.
The function can be written in the form .
The range of is ≤ ≤ . Find and .
Find the range of .
Find the value of and of .
Find the period of .
The equationhas two solutions where≤≤.Find both solutions.
Question 7
SLPaper 2A wind turbine is designed so that the rotation of the blades generates electricity. The turbine is built on horizontal ground and is made up of a vertical tower and three blades.
The point is on the base of the tower directly below point at the top of the tower. The height of the tower, , is . The blades of the turbine are centred at and are each of length . This is shown in the following diagram.
The end of one of the blades of the turbine is represented by point on the diagram. Let be the height of above the ground, measured in metres, where varies as the blade rotates.
Find the
The blades of the turbine complete rotations per minute under normal conditions, moving at a constant rate.
The height, , of point can be modelled by the following function. Time, , is measuredfrom the instant when the blade first passes and is measured in seconds.
Looking through his window, Tim has a partial view of the rotating wind turbine. The positionof his window means that he cannot see any part of the wind turbine that is more than above the ground. This is illustrated in the following diagram.
maximum value of .
minimum value of .
Find the time, in seconds, it takes for the blade to make one complete rotation under these conditions.
Calculate the angle, in degrees, that the blade turns through in one second.
Write down the amplitude of the function.
Find the period of the function.
Sketch the function for , clearly labelling the coordinates of the maximumand minimum points.
Find the height of above the ground when .
Find the time, in seconds, that point is above a height of , during eachcomplete rotation.
At any given instant, find the probability that point is visible from Tim’s window.
The wind speed increases. The blades rotate at twice the speed, but still at a constant rate.
At any given instant, find the probability that Tim can see point from his window.Justify your answer.
Question 8
SLPaper 1A factory produces shirts. The cost, C, in Fijian dollars (FJD), of producing x shirts can be modelled by
C(x) = (x− 75)2 + 100.
The cost of production should not exceed 500 FJD. To do this the factory needs to produce at least 55 shirts and at most s shirts.
Find the cost of producing 70 shirts.
Find the value of s.
Find the number of shirts produced when the cost of production is lowest.
Question 9
SLPaper 1Let f(x) = ax2 − 4x − c. A horizontal line, L , intersects the graph of f at x = −1 and x = 3.
The equation of the axis of symmetry is x = p. Find p.
Hence, show that a = 2.
Question 10
SLPaper 2The graph of the quadratic functionintersects the -axis at .
The vertex of the function is .
The equation has two solutions. The first solution is .
Let be the tangent at .
Find the value of .
Write down the equation for the axis of symmetry of the graph.
Use the symmetry of the graph to show that the second solution is .
Write down the -intercepts of the graph.
On graph paper, draw the graph of for and.Use a scale of to represent unit on the -axis and to represent units on the -axis.
Write down the equation of .
Draw the tangent on your graph.
Given and , state whether the function, , is increasing or decreasing at . Give a reason for your answer.