Question 1 (6 marks)

Consider the three matrices:

.

(a)(1 mark)

Question 2 (6 marks)

Consider the following system of simultaneous equations:

• Write the system in the matrix form ???????? = ????. (1 mark)
• Find the inverse of matrix ????. (2 marks)
• Use your inverse matrix to solve for ????. (2 marks)
• Substitute your solution(s) into the original system to check that they are

(1 mark)

Question 3 (6 marks)

The results of a football season for the first and second grades of four clubs: Norths, Souths, Easts and Wests are shown in the table below.

Teams are awarded 2 points for a win, 1 point for a draw and zero points for a loss.

• Use matrix multiplication to calculate the total number of points accumulated by the first-grade teams during the (2 marks)
• Use matrix multiplication to calculate the total number of points accumulated by the second-grade (2 marks)
• Let us assume that club championship points are determined by multiplying the first grade points by 3 and the second grade points by 2 and then adding them. Use matrix operations to calculate the championship points of each club. Note that your answer should be expressed as a column (2 marks)

Question 4 (5 marks)

A two-metre-high pole casts a 1.8-metre-long shadow. At the same time a nearby tree casts a shadow that is 27 metres in length.

• Draw a diagram to represent both (1 mark)
• Find the angle of elevation to the top of the pole. Round your answer to the nearest (2 marks)
• Use the tangent ratio to calculate the height of the tree to the nearest

(2 marks)

Question 5 (5 marks)

Consider the function

• Draw a graph of the function within the given (2 marks)
• Find the amplitude and the period of the (2 marks)
• From your knowledge of functions, explain why the inverse of this function is not a

function. Note that the inverse is ???? = cos^?1(x/2)

Question 6 (10 marks)

The height, ? in metres of the sea tide varies according to the function:

? = 5 + 2.5 sin(30????)

where ???? is the time in hours.

• Sketch a graph of the function for a 48-hour period. Include a title and label the axes (4 marks)
• Find the maximum tide height and the period of the (2 marks)
• Use your graph to find all the times in the first 24 hours when the tide is 6.7 m

(2 marks)

• By solving the equation 7 = 5 + 2.5 sin(30????), confirm the first timefound in part (c)when the tide is 6.7 m high. (2 marks)

Question 7 (2 marks)

If cos ???? = 0.5, find sin ???? without first finding the angle ????.