IB Math SL Year 2| Cumulative Year 2 Booklet 1

Name: ___________________________________________

Dear IB Math SL Students,

Congratulations, you have completed ALL of the coursework for IB Math SL! We just finished reviewing Year 1 Material,

so now we will give you an opportunity to continue review, but now with topics that you learned from the second year

of the course. Our final exam coming up will have content from year 1 and year 2, so keep those year 1 review materials

close by and work through this booklet in full to prepare for the final on 4/24 and 4/25, and the IB Exam on 5/2 and 5/3.

In this booklet you will notice a lot of practice problems and very little notes. Pace yourself through this booklet. Some

problems will come easy for you and others may pose a challenge. Be resourceful! You are capable of answering all of

these questions, but if you get stuck remember you can also look back at notes from class, answer keys on the website,

and videos we have shared on EdPuzzle. Don’t forget you also have each other and can help each other prepare the

best you possibly can for your exams.

We enjoyed teaching you the year 2 content this year. As the course winds down, we feel that the best way to teach

you and prepare you for your final and IB exam is to give you opportunities that encourage you to think hard and

persevere. This means that you will experience questions you haven’t seen before which is not something prepare you

for. We can help give you the tools to develop you to attempt and try questions. We promise that the effort that you

put in will impact the result you get out as long as you are persistent and focused in a positive mindset.

Please have this completed and checked by Monday, April 16th. We are here for you if you have any questions, and don’t

forget to use extra help time to work on problems that you may have forgotten! We are so proud of the skills you have

developed this year, and now is your time to show off how much you know!

Good luck!

Ms. Devine and Ms. DeGrazia

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1. The vectors a = and b = are perpendicular to each other.

a) Find the value of .

b) Given that c = a + 2b, find c.

2. a) A line passes through the points and .

(i) Show that .

(ii) Write down a vector equation for .

a) A line has equation . The lines and intersect at a point .

Show that the coordinates of are .

#2 continued on next page…

VECTORS NON-CALCULATOR

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b) A point lies on line so that and . Find .

3. Let and .

a) Find

(i) ; (ii) ; (iii) .

b) Find the angle between and .

VECTORS CALCULATOR

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4. The following diagram shows a right-angled triangle, , where .

a) Show that .

b) Find .

5. Let , where .

a) Find .

b) Find .

TRIG (Solving Equations & Graphing) NON-CALCULATOR

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6. Note: In this question, distance is in millimetres.

a) Let , for . Show that .

b) The graph of passes through the origin. Let be any point on the graph of with -coordinate ,

where . A straight line passes through all the points .

Find the coordinates of and of .

c) Find the equation of .

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7. The following diagram shows part of the graph of .

The point (

) is a maximum point and the point (

) is a minimum point.

Find the value of:

a) p

b) r

c) q

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8. Let . The graph of f passes through the point .

a) Find the value of .

b) Find the minimum value of .

c) Let . The graph of g is translated to the graph of by the vector .

Write down the value of and of .

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9. The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an

anticlockwise (counter-clockwise) direction.

a) The diameter of the wheel is 8 metres. The centre of the

wheel, A, is 2 metres above the water level. After t seconds,

the height of the bucket above the water level is given by

.

Show that .

b) The wheel turns at a rate of one rotation every 30 seconds. Show that .

c) In the first rotation, there are two values of t when the bucket is descending at a rate of .

Find these values of t .

d) In the first rotation, there are two values of t when the bucket is descending at a rate of .

Determine whether the bucket is underwater at the second value of t .

TRIG (Solving Equations & Graphing) CALCULATOR

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10. Let and .

a. Write down

(i) ; (ii) .

b. Let . Find the exact value of .

11. Let . Find .

CALCULUS NON-CALCULATOR

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12. The following diagram shows part of the graph of a quadratic function f .The x-intercepts are at and

, and the y-intercept is at .

a. Write down in the form

b. Find another expression for in the form .

c. Show that can also be written in the form .

d. A particle moves along a straight line so that its velocity, , at time t seconds is given by

, for .

(i) Find the value of t when the speed of the particle is greatest.

(ii) Find the acceleration of the particle when its speed is zero.

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13. In this question, you are given that , and . The displacement of an object from a fixed

point, O is given by for .

a. Find .

b. In this interval, there are only two values of t for which the object is not moving. One value is .

Find the other value.

c. Show that between these two values of t .

14. Let , for . Show that .

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15. Fred makes an open metal container in the shape of a cuboid, as shown in the following diagram.

The container has height , width and length . The volume is . Let be the outside surface area

of the container.

a. Show that .

b. Find .

c. Given that the outside surface area is a minimum, find the height of the container.

d. Fred paints the outside of the container. A tin of paint covers a surface area of and costs $20. Find

the total cost of the tins needed to paint the container.

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16. Let , for . The graph of has a maximum point at P.

a. Find the -coordinate of P.

b. The -coordinate of P is . Find , expressing your answer as a single logarithm.

17. The following diagram shows part of the graph of . The graph has a local maximum at , where

, and a local minimum at , where .

a. On the following axes, sketch the graph of .

b. Write down the following in order from least to

greatest: .

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18. Let , for .

a. Show that ( )

√

b. Find .

c. Consider another function . Let R be a point on the graph of . The -coordinate of R is 1. The equation of

the tangent to the graph at R is . Write down .

d. Find .

e. Let . Find the equation of the tangent to the graph of at the point where .

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19. A function has its derivative given by , where is a constant.

a. Find .

b. The graph of has a point of inflexion when . Show that .

c. Find .

d. Find the equation of the tangent to the curve of at , giving your answer in the form .

e. Given that , explain why the graph of has a local maximum when .

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20. Let .

a. Write down .

b. The tangent to the graph of f at the point has gradient m .

(i) Show that .

(ii) Find b .

c. Hence, write down the equation of this tangent.

21. Consider .

a. Find .

b. Find the gradient of the curve of at .

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22. Consider the functions , and . The following table gives some values associated with these

functions.

a. Write down the value of , of , and of .

b. The following diagram shows parts of the graphs of and .

There is a point of inflexion on the graph of at P, when .

Explain why P is a point of inflexion.

c. Given that ,find the -coordinate of P.

d, Find the equation of the normal to the graph of at P.

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23. Let . Line L is the normal to the graph of f at the point (4, 2) .

a. Show that the equation of L is .

b. Point A is the x-intercept of L . Find the x-coordinate of A.

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24. The following diagram shows the graph of , the derivative of . The graph of has a local minimum at A, a

local maximum at B and passes through . The point lies on the graph of the function, .

a. Write down the gradient of the curve of at P.

b. Find the equation of the normal to the curve of at P.

c. Determine the concavity of the graph of when and justify your answer.

25. The following diagram shows the graph of a function . There is a

local minimum point at , where .

a. The derivative of is given by .

Find the -coordinate of .

b. The -intercept of the graph is at ( ). Find an expression for .

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26. A function f is defined for . The graph of f is given below. The graph has a local maximum when

, and local minima when , .

a. Write down the x-intercepts of the graph of the

derivative function, .

b. Write down all values of x for which is positive.

c. At point D on the graph of f , the x-coordinate is . Explain why at D.

27. Consider , , where p is a constant.

a. Find .

b. There is a minimum value of when . Find the value of .

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28. Let , , .The graph of f is shown below. The region between and

is shaded.

a. Show that .

b. Given that , find the x-values of the coordinates of all points of inflexion.

c. It is given that . Find the area of the shaded region, giving your answer in

the form .

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29. The diagram shows part of the graph of . The x-intercepts are at points A and C. There is a minimum

at B, and a maximum at D.

a. (i) Write down the value of at C.

(i) Hence, show that C corresponds to a

minimum on the graph of f , i.e. it has

the same x-coordinate.

b. Which of the points A, B, D corresponds to

a maximum on the graph of f ?

c. Show that B corresponds to a point of inflexion on the graph of f .

30. Find .

31. Find .

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32. Find

a) .

b) Find , given that and .

33. The following diagram shows the graph of , for , and the line .

Let be the region enclosed by the graph of , the -axis and the line

. Find the area of .

34. Let . Given that , find .

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35. Let .Use the quotient rule to

a. Show that .

b. Find .

c. The following diagram shows part of the graph of . The shaded

region is enclosed by the graph of , the -axis, and the lines

and . This region has an area of . Find the

value of .

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36. Let .

a. Find .

b. The following diagram shows part of the graph of . The shaded region is enclosed by the graph of , the

-axis and the lines and .

Find the volume of the solid formed when is revolved about the -axis.

37. Let . Find the value of .

38. The graph of a function h passes through the point . Given that , find .

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39. A rocket moving in a straight line has velocity km s–1 and displacement km at time seconds. The velocity

is given by . When , . Find an expression for the displacement of the rocket in

terms of .

40. The graph of , for , is shown below.

The region enclosed by the curve of f and the x-axis is rotated

about the x-axis. Find the volume of the solid formed.

41. The graph of the function passes through the point . The gradient function of f is given as

. Find .

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42. Let , .

a. Show that .

b. Let the line L be the normal to the curve of f at . Find the equation of L .

c. The graph of f and the line L intersect at the point (0, 1) and at a second point P.

(i) Find the x-coordinate of P.

(ii) Find the area of the region enclosed by the graph of f and the line L .

CALCULUS CALCULATOR

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43. Note: In this question, distance is in metres and time is in seconds.

a. A particle P moves in a straight line for five seconds. Its acceleration at time is given by ,

for . Write down the values of when .

b. Hence or otherwise, find all possible values of for which the velocity of P is decreasing.

c. When , the velocity of P is .Find an expression for the velocity of P at time .

d. Find the total distance travelled by P when its velocity is increasing.

44. Let and .The graphs of and intersect at and , where .

a. Find the value of and of .

b. Hence, find the area of the region enclosed by the graphs of and .

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45. A particle moves in a straight line. Its velocity after seconds is given by

After seconds, the particle is 2 m from its initial position. Find the possible values of .

46. All lengths in this question are in metres.

Let , for . Mark uses as a model to create a barrel. The region enclosed

by the graph of , the -axis, the line and the line is rotated

360° about the -axis. This is shown in the following diagram.

a. Use the model to find the volume of the barrel.

b. The empty barrel is being filled with water. The volume of water in the

barrel after minutes is given by . How long will it take for the barrel to be half-full?

47. Let and , for .Find the area of the region enclosed by the graphs

of f and g .

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48. A gradient function is given by . When , . Find the value of y when .

49. Let , where , and . Find the equation of the normal to

the graph of at .

50. Consider the curve . Let P be the point on the curve where .

a. Write down the gradient of the curve at P.

b. The normal to the curve at P cuts the x-axis at R. Find the coordinates of R.

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51. The following diagram shows the graph of , for .

a. The graph of has a minimum point at and a maximum point at

.

(i) Find the value of .

(ii) Show that .

(iii) Find the value of .

b. The graph of is obtained from the graph of by a translation of . The maximum point on the graph of has

coordinates .

(i) Write down the value of .

(i) Find .

c. The graph of changes from concave-up to concave-down when .

(i) Find .

(ii) Hence or otherwise, find the maximum positive rate of change of .

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52. Let for .Point is on the curve of . Find the equation of the tangent to the

curve of at .

53. Let . Part of the graph of f is shown

below.The y-intercept is at (0, 13) .

a. Show that .

b. Given that (correct to 3 significant figures), find

the value of k.

c.

(i) Using your value of k , find .

(ii) Hence, explain why f is a decreasing function.

(i) Write down the equation of the horizontal asymptote of the graph f .

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54. a. Let . Find .

b. Find .

55. The following diagram shows the graph of .

a. The points A, B, C, D and E lie on the graph of f . Two of

these are points of inflexion. Identify the two points of

inflexion.

b.

(i) Find .

(ii) Show that .

c. Find the x-coordinate of each point of inflexion.

d. Use the second derivative to show that one of these points is a point of inflexion.

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56. a. Let , for .

Find .

b. On the grid below, sketch the graph of .

57. Let and .

a. Find .

b. Find .

c. Let . Find .

58. a. Let .There are two points of inflexion on the graph of f . Write down the x-

coordinates of these points.

b. Let . Explain why the graph of g has no points of inflexion.

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59. A farmer wishes to create a rectangular enclosure, ABCD, of area 525 m2, as shown below.

The fencing used for side AB costs per metre. The fencing for the other three sides costs per metre. The farmer

creates an enclosure so that the cost is a minimum. Find this minimum cost.

60. a. The diagram below shows part of the graph of the gradient function, .On the grid below, sketch a

graph of , clearly indicating the x-intercept.

b. Complete the table, for the graph of .

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61. a. Let , for . The graph of passes through the point , where .Find

the value of .

b. The following diagram shows part of the graph of The region enclosed by the

graph of , the -axis and the lines and is rotated 360° about the -

axis. Find the volume of the solid formed.

62. a. A particle P moves along a straight line. Its velocity after seconds is given by ,

for . The following diagram shows the graph of .

Write down the first value of at which P changes direction.

b. Find the total distance travelled by P, for .

c. A second particle Q also moves along a straight line. Its velocity, after seconds is given by for

. After seconds Q has travelled the same total distance as P. Find .

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63. a. Let and , for .

Solve .

b. Find the area of the region enclosed by the graphs of and .

64. The acceleration, , of a particle at time t seconds is given by

The particle is at rest when .Find the velocity of the particle when .

65. a. Let , . Find .

b. i. Sketch the graph of h for and

including any asymptotes.

ii. Write down the equations of the asymptotes.

iii. Write down the x-intercept of the graph of h .

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66. The velocity of a particle in ms−1 is given by , for

.

On the grid below, sketch the graph of .

b. Find the total distance travelled by the particle in the first five seconds.

c. Write down the positive -intercept.

67. a. Let , for . Sketch the graph of f .

b.

(i) Write down the x-coordinate of the maximum point on the graph of f .

(ii) Write down the interval where f is increasing.

c. Show that .

d. Find the interval where the rate of change of f is increasing.