2. Index of Questions and Summaries Question 1 Question 2 (a) Question 2 (b) Question 3 Question 4 Question 5 Question 6 (a) Question 6 (b) Question 7 Question 8 (a) Question 8 (b) Question 10 Question 11 (a) Question 11 (b) Question 12 Log Law summary Exact values trig functions Trig functions in a circle Question 9 (a) Question 9 (b) Exit

3. Read the question… continue

4. Solution Exam time allowed: 3 min Click ‘start’ when you are ready. Examiner's Comments Index Now attempt the problem yourself and time how long it takes… … then check the solution and examiners comments Next Q

5. Read the question… continue

6. Exam time allowed: 3 min Click ‘start’ when you are ready. Now attempt the problem yourself and time how long it takes… … then check the solution and examiners comments Hint: You will need log laws to solve this. Log laws are NOT on the formula sheet, so you need to remember them. To refresh your memory, a summary of log laws is at the following link. Log Laws Solution Examiner's Comments Next Q Index

7. Read the question… continue

8. Exam time allowed: 3 min Click ‘start’ when you are ready. Now attempt the problem yourself and time how long it takes… … then check the solution and examiners comments Hint: You will need exact values of trig functions to simplify the answer. Exact values are NOT on the formula sheet, so you need to remember them. To refresh your memory, a summary is at the following link. Exact Values Solution Examiner's Comments Next Q Index

9. Q3. The diagram shows the graph of a function with domain R (a) For the graph shown, sketch on the same axes the graph of the derivative function. (b) Write down the domain of the derivative function Read the question… continue

10. Exam time allowed: 6 min Click ‘start’ when you are ready. Now attempt the problem yourself and time how long it takes… Q3. The diagram shows the graph of a function with domain R (a) For the graph shown, sketch on the same axes the graph of the derivative function. (b) Write down the domain of the derivative function Solution Examiner's Comments Next Q Index

11. Read the question… continue Q4. A wine glass is being filled with wine at a rate of 8 cm 3 /s. The volume, V, of wine in the glass when the depth of wine in the glass is x cm is given by Find the rate at which the depth of wine in the glass is increasing when the depth is 4 cm .

12. Exam time allowed: 4.5 min Click ‘start’ when you are ready. Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index Q4. A wine glass is being filled with wine at a rate of 8 cm 3 /s. The volume, V, of wine in the glass when the depth of wine in the glass is x cm is given by Find the rate at which the depth of wine in the glass is increasing when the depth is 4 cm .

13. Q5. It is known that 50% of the customers who enter a restaurant order a cup of coffee. If four customers enter the restaurant, what is the probability that more than two of these customers order coffee? (Assume that what any customer orders is independent of what any other customer orders.) Read the question… continue

14. Exam time allowed: 3 min Click ‘start’ when you are ready. Q5. It is known that 50% of the customers who enter a restaurant order a cup of coffee. If four customers enter the restaurant, what is the probability that more than two of these customers order coffee? (Assume that what any customer orders is independent of what any other customer orders.) Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index

15. Q6(a). Two events, A and B, from a given event space, are such that Read the question… continue

16. Exam time allowed: 1.5 min Click ‘start’ when you are ready. Q6(a). Two events, A and B, from a given event space, are such that Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index

17. Q6(b). Two events, A and B, from a given event space, are such that Read the question… continue

18. Exam time allowed: 1.5 min Click ‘start’ when you are ready. Q6(b). Two events, A and B, from a given event space, are such that Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index

19. Q7. If f ( x ) = x cos (3 x ), then f ′ ( x ) = cos (3 x )  3 x sin (3 x ). Use this fact to find an antiderivative of x sin (3 x ) Read the question… continue

20. Exam time allowed: 4.5 min Click ‘start’ when you are ready. Q7. If f ( x ) = x cos (3 x ), then f ′ ( x ) = cos (3 x )  3 x sin (3 x ). Use this fact to find an antiderivative of x sin (3 x ) Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index

21. Read the question… continue Q.8(a) Let Solve the equation

22. Hint: You will need to know how to find solutions around the unit circle to answer this question. To refresh your memory, a summary of the unit circle is at the following link. Exam time allowed: 3 min Click ‘start’ when you are ready. Unit circle Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index Q.8(a) Let Solve the equation

23. Read the question… continue Q8(b). Let Find the smallest possible of x for which g ( x ) is a maximum.

24. Exam time allowed: 3 min Click ‘start’ when you are ready. Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index Q8(b). Let Find the smallest possible of x for which g ( x ) is a maximum.

25. Q9(a). The graph of is shown. The normal to the graph of f where it crosses the y -axis is also shown. (a) Find the equation of the normal to the graph of f where it crosses the y -axis . Read the question… continue

26. Exam time allowed: 3 min Click ‘start’ when you are ready. Q9(a). The graph of is shown. The normal to the graph of f where it crosses the y -axis is also shown. (a) Find the equation of the normal to the graph of f where it crosses the y -axis . Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index

27. Q9(b). Find the exact area of the shaded region. Read the question… continue

28. Exam time allowed: 4.5 min Click ‘start’ when you are ready. Q9(b). Find the exact area of the shaded region. Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index

29. Q10. The area of the region bounded by the curve with equation, where k is a positive constant, the x -axis and the line with equation x = 9, is 27. Find k Read the question… continue

30. Exam time allowed: 4.5 min Click ‘start’ when you are ready. Q10. The area of the region bounded by the curve with equation, where k is a positive constant, the x -axis and the line with equation x = 9, is 27. Find k Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index

31. Q.11 Background information: There is a daily flight from Paradise Island to Melbourne. The probability of the flight departing on time, given that there is fine weather on the island, is 0.8, and the probability of the flight departing on time, given that the weather on the island is not fine, is 0.6. In March the probability of a day being fine is 0.4. Q11(a). Find the probability that on a particular day in March the flight from Paradise Island departs on time. Read the question… continue

32. Exam time allowed: 3 min Click ‘start’ when you are ready. Q.11 Background information: There is a daily flight from Paradise Island to Melbourne. The probability of the flight departing on time, given that there is fine weather on the island, is 0.8, and the probability of the flight departing on time, given that the weather on the island is not fine, is 0.6. In March the probability of a day being fine is 0.4. Q11(a). Find the probability that on a particular day in March the flight from Paradise Island departs on time. Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index

33. Q.11 Background information: There is a daily flight from Paradise Island to Melbourne. The probability of the flight departing on time, given that there is fine weather on the island, is 0.8, and the probability of the flight departing on time, given that the weather on the island is not fine, is 0.6. In March the probability of a day being fine is 0.4. Q11(b). Find the probability that on a particular day in March the weather is fine on Paradise Island, given that the flight departs on time. Read the question… continue

34. Exam time allowed: 3 min Click ‘start’ when you are ready. Q.11 Background information: There is a daily flight from Paradise Island to Melbourne. The probability of the flight departing on time, given that there is fine weather on the island, is 0.8, and the probability of the flight departing on time, given that the weather on the island is not fine, is 0.6. In March the probability of a day being fine is 0.4. Q11(b). Find the probability that on a particular day in March the weather is fine on Paradise Island, given that the flight departs on time. Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Next Q Index

35. Q.12 P is the point on the line 2 x + y  10 = 0 such that the length of OP, the line segment from the origin O to P, is a minimum. Find the coordinates of P and its minimum length. Read the question… continue

36. Exam time allowed: 6 min Click ‘start’ when you are ready. Q.12 P is the point on the line 2 x + y  10 = 0 such that the length of OP, the line segment from the origin O to P, is a minimum. Find the coordinates of P and its minimum length. Now attempt the problem yourself and time how long it takes… Solution Examiner's Comments Index

37. 1. For example log e (2) + log e (3) = log e (2  3) = log e (6) log e ( x ) + log e (3 x ) = log e ( x  3 x ) = log e (3 x 2 ) log e ( a ) + log e ( b ) = log e ( a  b ) 2. For example log e (8)  log e (2) = log e (8  2) = log e (4) For example Log laws that you need to know 3. log e ( a ) n = n × log e ( a ) log e (2) 3 = 3 log e (2) For example log e (49) = log e (7) 2 = 2 log e (7) Q2(a) Index log e ( a )  log e ( b ) = log e ( a  b ) = log e ( ) log e (4 x 2 )  log e ( x ) = log e ( ) = log e (4 x )

38. There are two basic triangles that provide a quick way of ‘reading off’ exact values of trig functions. Exact values of trig functions that you need to know This triangle lets you ‘read off’ exact values for sine, cos and tan of 45  . This triangle lets you ‘read off’ exact values for sine, cos and tan of 30  and 60  . Q2(b) Index or 45 0 1 1 2 or 60 0 or 30 0 1

39. Angles around the circle go from 0 to 360 degrees. ‘ CAST’ tells us the sign of trig functions in each quadrant. In any problem, find the basic solution,  . Then find the other solutions around the circle, using the CAST diagram. Q8(a) Index C A S T 1 st quadrant 2 nd quadrant 3 rd quadrant 4 th quadrant A ll trig functions are positive in the 1 st quadrant S ine is positive in the 2 nd quadrant T an is positive in the 3 rd quadrant C os is positive in the 4 th quadrant 1 st quadrant 2 nd quadrant 3 rd quadrant 4 th quadrant 0 0 2  or 360 0  or 180 0 0 0 2  or 360 0  or 180 0 C A S T The basic solution,  , is in the 1 st quadrant. Solutions in the 2 nd quadrant are (    ). Solutions in the 3 rd quadrant are (  +  ). Solutions in the 4 th quadrant are (2    ).