Our goal at Renaissance is to provide a rigorous academic program for our students. As an IB student or CWI student, the breadth and depth of mathematics material is part of that rigor. In order to be successful on the IB SL exam/final for m160 in May of 2016, you must be able to complete mathematics problems on a topics that have been covered over a three year period. To aid in your success, you will need to complete the following items over the summer.

□ Syllabus Review (June 4th – July 1st) ◦ Read through the topics that will be on the IB SL Exam ◦ Identify what topics and subtopics have been covered in your notes from SL Year 1

◦ Using an identification technique of your choosing, identify the syllabus topics in your notes. Be sure they are easy to access.

Example: If you are asked to find an example of binomial theorem, you should be able to flip to that section of your notes and find that information within 30 seconds.

□ IA Rubric Overview and paper review (July 1st – August 1st) ◦ Read through the IA Guide and Rubric ◦ Create a list of math topics that you find interesting ▪ This may take some research on your part ▪ 15 things minimum on your list ▪ Try going to the library or researching books on interesting math topics

□ Review Packet for Mrs. Christensen (August 1st – August 25th) ◦ Complete the problems given to you in a packet of problems. Please do not work on these until August.

◦ Show your work! ◦ Take it seriously! ◦ You will be turning this in for a completion grade the first day of school.

Our goal at Renaissance is to help you successfully complete the IB SL Math program and be successful on the IB math test, to also include the CWI m160 exam. More importantly, our goal is to help you improve your math skills so you have a successful future in whatever career you choose. Good luck and we will see you next year! Sincerely, Mrs. Christensen

Course Syllabus: 1.1 Arithmetic sequences and series: Sum of finite arithmetic series: Geometric sequences and series: sum of finite and infinite geometric series. Sigma notation: Applications 1.2 Elementary treatment of exponents and logarithms. Laws of exponents: Laws of logarithms: Change of base.

1.3 The Binomial Theorem expansion of (𝑎 + 𝑏)𝑛 Calculation of binomial coefficients using Pascal’s triangle and (𝑛𝑐

)

2.1 Concepts of functions: Domain, range image: Composite functions: Identity functions: Inverse function. 2.2 The graph of a function: Graphing skills: max, min values, intercepts, horizontal and vertical asymptotes, symmetry and consideration of domain and range: 2.3 Transformations: translations, reflections, vertical and horizontal shifts, stretch factors, composite transformations: 2.4 Quadratics: x and y intercepts, vertex, different equation forms, maximum or minimums. Quadratic formula: 3.1 The unit circle in both degrees and radians: length of an arc, area of sector 3.2 Definitions of sin and cos in terms of the unit circle Definition of tan in terms of sin and cos. Exact values of the

trigonometric rations of: 0,𝜋

6,

𝜋

4,

𝜋

3,

𝜋

2 and their multiples:

3.3 Pythagorean identity of 𝑐𝑜𝑠2𝜗 + 𝑠𝑖𝑛2𝜗 = 1 Double angle identities: the relationship between trig ratios 3.4 The circular functions of sin, cos, and tan their domain, range amplitude and their periodic nature of their graphs.

𝑓(𝑥) = asin(𝑏(𝑥 + 𝑐)) + 𝑑 the transformations of a,b,c and d.

3.5 Solving trigonometric equations in a finite interval. Both graphically and analytically.

3.6 Solutions of triangles: The law of cosine and sine: including the ambiguous case. Area of a triangle: 𝐴 =1

2𝑎𝑏𝑠𝑖𝑛𝐶

4.1 Vectors as displacement in the plane and in three dimensions. Components of a vector: column representations of a vector: Algebraic and geometric approaches to the following: The sum and differences of two vectors. The zero vector and an negative vector Multiplication by a scalar: parallel vectors Magnitude of a vector Unit vectors base vectors i, j, k

Position vectors

4.2 The scalar product of two vectors: Perpendicular and parallel vectors the angel between two vectors

4.3 Vector equations of a line in two and three dimensions: The angle between two lines.

4.4 Distinguishing between coincident and parallel lines. Finding the point of intersections of two lines Determine

whether two lines intersect.

5.1 Concepts of population, sample, random sample, discrete and continuous data.

Presentation of data: frequency distributions: frequency histograms with equal class intervals: box and whisker

plots: outliers. Group data, use of mid-interval values for calculations: upper and lower interval boundaries; modal class:

5.2 Statistical measure and their interpretations: mean, median, mode, Quartiles, percentiles. Range, IQR, variance

standard deviations: Applications

5.3 Cumulative frequency: graphs median, quartiles, and percentiles 5.4 linear correlations of bivariate data: Pearson’s product moment correlation: scatter diagrams, best fit lines predictions 5.5 Trials; outcomes; equally likely outcomes, sample space: Probability of an event: with complements, Venn diagrams, tree diagrams, and tables of outcomes. 5.6 Combined events: P (A and B) Mutually exclusive events: Conditional Probability using the definition: Independent events: Probabilities with and without replacements. 5.7 Concepts of discrete random variables and their probability distributions. Expected values 6.1 Informal ideas of limit and convergence: Limit notation: Definition of derivative from first principles: Derivative interpreted as gradient functions and rate of change. Tangent and normal and their equations. 6.2 Derivation of 𝑥𝑛, sin 𝑥 , 𝑐𝑜𝑠𝑥 , tan 𝑥 , 𝑒𝑥 , 𝑎𝑛𝑑 ln 𝑥: Differentiation of a sum and a real multiple of these functions. The chain rule, product rule, and quotient rules. The second derivative: extension to higher derivatives.

6.3 Local maximum and minimum points, testing for max and min. Points of inflections: Graphical behavior of functions, including the relationship between he graphs of 𝑓, 𝑓′, 𝑓′′ Optimizations and Applications 6.4 Indefinite integration and anti-differentiation: Indefinite integral: 𝑥𝑛, sin 𝑥 , 𝑐𝑜𝑠𝑥 , tan 𝑥 , 𝑒𝑥 , 𝑎𝑛𝑑 ln 𝑥: The composites of any of the with the linear functions: 6.5 Anti-differentiation with boundary conditions to determine the constant term. Definite integrals, both analytically and using technology. Areas under curve, between curves and the x-axis: Volumes of revolution about the x-axis. 6.6 Kinematic problems involving displacement s and velocity v and accelerations a. Total distance traveled.

Introduction To IA’s

The internally assessed component in this course is a mathematical exploration. This is a short report written by the student based on a topic chosen by him or her, and it should focus on the mathematics of that particular area. The emphasis is on mathematical communication (including formulae, diagrams, graphs and so on), with accompanying commentary, good mathematical writing and thoughtful reflection. A student should develop his or her own focus, with the teacher providing feedback via, for example, discussion and interview. This will allow the students to develop area(s) of interest to them without a time constraint as in an examination, and allow all students to experience a feeling of success.

The final report should be approximately 6 to 12 pages long. It can be either word processed or handwritten. Students should be able to explain all stages of their work in such a way that demonstrates clear understanding. While there is no requirement that students present their work in class, it should be written in such a way that their peers would be able to follow it fairly easily. The report should include a detailed bibliography, and sources need to be referenced in line with the IB academic honesty policy. Direct quotes must be acknowledged.

The purpose of the exploration

The aims of the mathematics SL course are carried through into the objectives that are formally assessed as part of the course, through either written examination papers, or the exploration, or both. In addition to testing the objectives of the course, the exploration is intended to provide students with opportunities to increase their understanding of mathematical concepts and processes, and to develop a wider appreciation of mathematics. These are noted in the aims of the course, in particular, aims 6–9 (applications, technology, moral, social and ethical implications, and the international dimension). It is intended that, by doing the exploration, students benefit from the mathematical activities undertaken and find them both stimulating and rewarding. It will enable students to acquire the attributes of the IB learner profile.

The specific purposes of the exploration are to:

develop students’ personal insight into the nature of mathematics and to develop their ability to ask their own questions about mathematics

provide opportunities for students to complete a piece of mathematical work over an extended period of time enable students to experience the satisfaction of applying mathematical processes independently provide students with the opportunity to experience for themselves the beauty, power and usefulness of

mathematics encourage students, where appropriate, to discover, use and appreciate the power of technology as a

mathematical tool enable students to develop the qualities of patience and persistence, and to reflect on the significance of their

work provide opportunities for students to show, with confidence, how they have developed mathematically.

A. Communication

B. Mathematical

Representation

C. Personal

Engagement

D. Reflection E. Use of Mathematics

0 The exploration does

not reach the standard

described by the

descriptors below.

The exploration does

not reach the standard

described by the

descriptors below.

The exploration

does not reach the

standard described

by the descriptors

below.

The exploration

does not reach the

standard

described by the

descriptors below.

The exploration does not reach

the standard described by the

descriptors below.

1 The exploration has

some coherence.

There is some

appropriate

mathematical

presentation.

There is evidence

of limited or

superficial

personal

engagement.

There is evidence

of limited or

superficial

reflection.

Some relevant mathematics is

used.

2 The exploration has

some coherence and

shows some

organization.

The mathematical

presentation is mostly

appropriate.

There is evidence

of some personal

engagement.

There is evidence

of meaningful

reflection.

Some relevant mathematics is

used. Limited understanding is

demonstrated

3 The exploration is

coherent and well

organized.

The mathematical

presentation is

appropriate

throughout.

There is evidence

of significant

personal

engagement.

There is

substantial

evidence of

critical reflection.

Relevant mathematics

commensurate with the level of

the course is used. Limited

understanding is demonstrated

4 The exploration is

coherent, well

organized, concise and

complete.

There is abundant

evidence of

outstanding

personal

engagement.

Relevant mathematics

commensurate with the level of

the course is used. The

mathematics explored is

partially correct. Some

knowledge and understanding

are demonstrated.

5 Relevant mathematics

commensurate with the level of

the course is used. The

mathematics explored is mostly

correct. Good knowledge and

understanding are

demonstrated.

Relevant mathematics

commensurate with the level of

the course is used. The

mathematics explored is

correct. Thorough knowledge

and understanding are

demonstrated.

Score /4 /3 /4 /3 /6

Summer 2015 Review Due at the start of the year 2015 Name:_______________________

1. Find the sum of the arithmetic series

17 + 27 + 37 +...+ 417.

(Total 4 marks)

2. Each day a runner trains for a 10 km race. On the first day she runs 1000 m, and then increases the distance

by 250 m on each subsequent day.

(a) On which day does she run a distance of 10 km in training?

(b) What is the total distance she will have run in training by the end of that day? Give your answer

exactly.

(Total 4 marks)

3. The following table shows four series of numbers. One of these series is geometric, one of the series is

arithmetic and the other two are neither geometric nor arithmetic.

(a) Complete the table by stating the type of series that is shown.

Series Type of series

(i) 1 11 111 1111 11111 …

(ii) 1 …

(iii) 0.9 0.875 0.85 0.825 0.8 …

(iv)

(b) The geometric series can be summed to infinity. Find this sum.

(Total 6 marks)

4. Find the sum of the infinite geometric series

(Total 4 marks)

5. Portable telephones are first sold in the country Cellmania in 1990. During 1990, the number of units sold is

160. In 1991, the number of units sold is 240 and in 1992, the number of units sold is 360.

In 1993 it was noticed that the annual sales formed a geometric sequence with first term 160, the 2nd and 3rd

terms being 240 and 360 respectively.

(a) What is the common ratio of this sequence?

(1)

Assume that this trend in sales continues.

(b) How many units will be sold during 2002?

(3)

(c) In what year does the number of units sold first exceed 5000?

(4)

4

3

16

9

64

27

6

5

5

4

4

3

3

2

2

1

...8116

278

94

32

Between 1990 and 1992, the total number of units sold is 760.

(d) What is the total number of units sold between 1990 and 2002?

(2)

During this period, the total population of Cellmania remains approximately 80 000.

(e) Use this information to suggest a reason why the geometric growth in sales would not continue.

(1)

(Total 11 marks)

6. Solve the equation log9 81 + log9 + log9 3 = log9 x.

(Total 4 marks)

7. (a) Given that log3 x – log3 (x – 5) = log3 A, express A in terms of x.

(b) Hence or otherwise, solve the equation log3 x – log3 (x – 5) = 1.

(Total 6 marks)

8. Solve the equation 9x–1

=

(Total 4 marks)

9. Find the exact solution of the equation 92x

= 27(1–x)

.

(Total 6 marks)

10. A group of ten leopards is introduced into a game park. After t years the number of leopards, N, is modelled

by N = 10 e0.4t

.

(a) How many leopards are there after 2 years?

(b) How long will it take for the number of leopards to reach 100? Give your answers to an appropriate

degree of accuracy.

Give your answers to an appropriate degree of accuracy.

(Total 4 marks)

11. If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for

(a) log2 5;

(b) loga 20.

(Total 4 marks)

12. Find the coefficient of x5

in the expansion of (3x – 2)8.

(Total 4 marks)

9

1

.31

2x

13. Given that = p + where p and q are integers, find

(a) p;

(b) q.

(Total 6 marks)

14. Find the coefficient of a3

b4 in the expansion of (5a + b)

7.

(Total 4 marks)

15. Consider the binomial expansion

(a) By substituting x = 1 into both sides, or otherwise, evaluate

(b) Evaluate .

(Total 4 marks)

16. Let f (x) = 2x, and g (x) = , (x 2).

Find

(a) (g f ) (3);

(b) g–1

(5).

(Total 6 marks)

17. The diagram represents the graph of the function

f : x (x – p)(x – q).

(a) Write down the values of p and q.

(b) The function has a minimum value at the point C. Find the x-coordinate of C.

(Total 4 marks)

18. (a) Express f (x) = x2 – 6x + 14 in the form f (x) = (x – h)

2 + k, where h and k are to be determined.

(b)Hence, or otherwise, write down the coordinates of the vertex of the parabola with equation

(Total 4 marks)

373 7q

.3

4

2

4

1

41)1( 4324 xxxxx

.3

4

2

4

1

4

8

9

7

9

6

9

5

9

4

9

3

9

2

9

1

9

2–x

x

x

y

C

212

–

19. The diagram shows the graph of the function y = ax2 + bx + c.

Complete the table below to show whether each expression is positive, negative or zero.

Expression positive negative zero

a

c

b2 – 4ac

b

(Total 4 marks)

20. The function f is defined by

Evaluate f –1

(5).

(Total 4 marks)

21. The function f is given by f (x) = x2 – 6x + 13, for x 3.

(a) Write f (x) in the form (x – a)2

+ b.

(b) Find the inverse function f –1

.

(c) State the domain of f –1

.

(Total 6 marks)

22. O is the centre of the circle which has a radius of 5.4 cm.

The area of the shaded sector OAB is 21.6 cm2. Find the length of the minor arc AB.

(Total 4 marks)

y

x

.2

3,2–3: xxaxf

O

A B

23. The following diagram shows a triangle ABC, where is 90, AB = 3, AC = 2 and is .

(Total 6 marks)

24. (a) Express 2 cos2

x + sin x in terms of sin x only.

(b) Solve the equation 2 cos2

x + sin x = 2 for x in the interval 0 x , giving your answers exactly.

(Total 4 marks)

25. Solve the equation 3 sin2

x = cos2

x, for 0° x 180°.

(Total 4 marks)

26. If A is an obtuse angle in a triangle and sin A = , calculate the exact value of sin 2A.

(Total 4 marks)

27. Given that sin x = , where x is an acute angle, find the exact value of

(a) cos x;

(b) cos 2x.

(Total 6 marks)

28. A triangle has sides of length 4, 5, 7 units. Find, to the nearest tenth of a degree, the size of the largest angle.

(Total 4 marks)

29. The points P, Q, R are three markers on level ground, joined by straight paths PQ, QR, PR as shown in

the diagram. QR = 9 km, = 35°, = 25°.

Diagram not to scale

BCA CAB

135

3

1

35° 25°9 km

P

Q R

(a) Show that sin = .

(b) Show that sin 2 = .

(c) Find the exact value of

cos 2.

3

5

9

54

QRPRQP

(a) Find the length PR.

(3)

(b) Tom sets out to walk from Q to P at a steady speed of 8 km h–1

. At the same time, Alan sets out to jog

from R to P at a steady speed of a km h–1

. They reach P at the same time. Calculate the value of a.

(7)

(c) The point S is on [PQ], such that RS = 2QS, as shown in the diagram.

Find the length QS.

(6)

(Total 16 marks)

30. The diagram shows a triangle ABC in which AC = 7 , BC = 6, = 45°.

(a) Use the fact that sin 45° = to show that sin = .

(2)

The point D is on (AB), between A and B, such that sin = .

(b) (i) Write down the value of + .

(ii) Calculate the angle BCD.

(iii) Find the length of [BD].

(6)

(c) Show that = .

(2)

(Total 10 marks)

P

Q R

S

2

2CBA

A

B C6

45°

722

Diagramnot to scale

2

2CAB

7

6

CDB7

6

CDB CAB

BAC of Area

BDC of Area

Δ

Δ

BA

BD

31. A formula for the depth d meters of water in a harbour at a time t hours after midnight is

where P and Q are positive constants. In the following graph the point (6, 8.2) is a minimum point and the

point (12, 14.6) is a maximum point.

(a) Find the value of

(i) Q;

(ii) P.

(3)

(b) Find the first time in the 24-hour period when the depth of the water is 10 metres.

(3)

(c) (i) Use the symmetry of the graph to find the next time when the depth of the water is 10 metres.

(ii) Hence find the time intervals in the 24-hour period during which the water is less than 10 metres

deep.

(4)

Verify the trig Identities:

32. 𝒔𝒊𝒏𝟐𝝑 + 𝒄𝒐𝒔𝟐𝝑 + 𝒕𝒂𝒏𝟐𝝑 = 𝒔𝒆𝒄𝟐𝝑

33. 𝒔𝒊𝒏𝝏

𝒄𝒐𝒔𝝏 𝒕𝒂𝒏𝝏= 𝟏

34. 𝒔𝒊𝒏𝜶 𝒄𝒔𝒄𝜶 − 𝒄𝒐𝒔𝟐𝜶 = 𝒔𝒊𝒏𝟐𝜶

Know the unit circle!

35. The probability distribution of a discrete random variable X is given by

P(X = x) = , x {1, 2, k}, where k > 0.

(a) Write down P(X = 2).

(b) Show that k = 3.

(c) Find E(X).

(Total 7 marks)

,240,6

cos

ttQPd

0 6 12 18 24

15

10.

5

d

t

(6, 8.2)

(12, 14.6)

14

2x

36. A random variable X is distributed normally with a mean of 20 and variance 9.

(a) Find P(X ≤ 24.5).

(3)

(b) Let P(X ≤ k) = 0.85.

(i) Represent this information on the following diagram.

(ii) Find the value of k.

(5)

(Total 8 marks)

37. A box holds 240 eggs. The probability that an egg is brown is 0.05.

(a) Find the expected number of brown eggs in the box.

(2)

(b) Find the probability that there are 15 brown eggs in the box.

(2)

(c) Find the probability that there are at least 10 brown eggs in the box.

(3)

(Total 7 marks)

38. Two fair 4-sided dice, one red and one green, are thrown. For each die, the faces are labelled 1, 2, 3, 4. The

score for each die is the number which lands face down.

(a) List the pairs of scores that give a sum of 6.

(3)

The probability distribution for the sum of the scores on the two dice is shown below.

Sum 2 3 4 5 6 7 8

Probability p q r

(b) Find the value of p, of q, and of r.

(3)

Fred plays a game. He throws two fair 4-sided dice four times. He wins a prize if the sum is 5 on three or

more throws.

(c) Find the probability that Fred wins a prize.

(6)

(Total 12 marks)

16

3

16

4

16

3

16

1