IB Math SL/HL Summer 2012 Review

Welcome to IB Math SL and IB Math HL. In order to prepare you for the level of work required

in IB Math classes. You need to download and complete the IB Math SL/HL Summer 2012

Review assignment. Some guidelines for this and all future IB math assignments:

Show all work. The answer is only a small part of what I am interested in seeing. I learn

more about you do know and where still might need some clarification by looking at the

process you use to arrive to your answer. In this class and on the External IB Math Exam

in May 2014, you will not receive full credit for simply writing the answer. Marks are

allotted for both the work and the answer. The only time it is acceptable to write just

the answer is when the directions say “write down”

Answers given must be exact (no decimals) when the directions ask for exact answers.

Otherwise, all answers should be exact or expressed to 3 significant figures (3 sf) unless

the directions ask for something else.

Use the IB Math SL, Math HL information booklet. This contains formulas that may be

necessary for you to use. This information booklet will also be provided for your use on

the IB exam, so it is in your best interest to know this booklet inside and out. You will

also need to bring to class.

Complete the packet without the use of calculator. you might need a calculator on a

few questions. Your final external IB exams will consist of two paper for math SL, and

three papers for math HL. You are not allowed to use calculator on paper 1.

The internet has a lot of great sites that explains mathematical concepts such as:

www.khanacademy.com, www.mathworld.wolfram.com, www.purplemath.com and

www.ibostore.org

Recommended calculator: TI-84 Plus Silver edition.

This assignment is due on the second day of school (Tuesday, August 28, 2012). If you

have any questions, you may e-mail me at the address below and I will get back to you

as soon as I can. Remember, I will not give you the answers. This class will challenge you

to think and apply math in a way that may not have been expected of you before.

Good luck, have a safe and enjoyable summer and I will see in August.

Mr. Sawtari

IB Math SL/HL Teacher

asawtari@aisr.org

1. IB Notations

The following notation is used in the examination papers for this course without explanation.

If forms of notation other than those listed in this guide are used on a particular examination

paper, they are defined within the question in which they appear.

Students must always use correct mathematical notation, not calculator notation.

the set of positive integers and zero,

the set of integers,

++ the set of positive integers,

the set of rational numbers

+ the set of positive rational numbers, {xx , x > 0}

the set of real numbers

+ the set of positive real numbers, {xx , x > 0}

the set with elements

the number of elements in the finite set A

the set of all x such that

is an element of

is not an element of

the empty (null) set

the universal set

union

intersection

is a proper subset of

is a subset of

the complement of the set A

{0,1, 2, 3, ...}

{0, 1, 2, 3, ...}

{1, 2, 3, ...}

1 2{ , , ...}x x 1 2, , ...x x

( )n A

{ | }x

U

A

a divides b

, a to the power of , root of a (if then )

, a to the power , square root of a (if then )

the modulus or absolute value of x, that is

is approximately equal to

is greater than

is greater than or equal to

is less than

is less than or equal to

is not greater than

is not less than

the term of a sequence or series

the common difference of an arithmetic sequence

the common ratio of a geometric sequence

the sum of the first n terms of a sequence,

the sum to infinity of a sequence,

the binomial coefficient, , in the expansion of

|a b

1/ nan a

1

n

thn 0a 0n a

1/ 2a a1

20a 0a

x

R,0for

R,0for

xxx

xxx

nu thn

d

r

nS 1 2 ... nu u u

S 1 2 ...u u

n

r

thr 0,1, 2, ...r ( )na b

is a function under which each element of set A has an image in set B

is a function under which is mapped to

the image of under the function

the inverse function of the function

the composite function of and

exponential function of x

logarithm to the base a of x

the natural logarithm of x,

sin, cos, tan the circular functions

the point A in the plane with Cartesian coordinates x and y

the line segment with end points A and B

AB the length of

the line containing points A and B

the angle at A

the angle between and

the triangle whose vertices are A, B and C

probability of event

P(A) probability of the event “not ”

probability of the event given

:f A B f

:f x y f x y

( )f x x f

1f f

f g f g

ex

loga x

ln x elog x

A( , )x y

AB

AB

AB

Â

ˆCAB CA AB

ABC

P( )A A

A

P( | )A B A B

1. A teacher earns an annual salary of 45 000 USD for the first year of her employment

Her annual salary increases by 1750 USD each year.

(a) Calculate the annual salary for the fifth year of her employment.

(3)

She remains in this employment for 10 years.

(b) Calculate the total salary she earns in this employment during these 10 years.

(3)

(Total 6 marks)

2. The diagram shows the straight lines L1 and L2. The equation of L2 is y = x.

(a) Find

(i) the gradient of L1;

(ii) the equation of L1.

(3)

(b) Find the area of the shaded triangle.

(3)

(Total 6 marks)

3. The equation of the line R1 is 2x + y – 8 = 0. The line R2 is perpendicular to R1.

(a) Calculate the gradient of R2.

(2)

The point of intersection of R1 and R2 is (4, k).

(b) Find

(i) the value of k;

(ii) the equation of R2.

(4)

(Total 6 marks)

4. Let f(x) = x2 – 6x + 8.

(a) Factorise x2 – 6x + 8.

(2)

(b) Hence, or otherwise, solve the equation x2 – 6x + 8 = 0.

(2)

Let g(x) = x + 3.

(c) Write down the solutions to the equation f(x) = g(x).

(2)

(Total 6 marks)

5. Consider the numbers , –5, and the sets , , and . Complete the

following table by placing a tick in the appropriate box if the number is an element of

the set.

6 –5

(Total 6 marks)

,2

126,,3

32

12

6. The length of one side of a rectangle is 2 cm longer than its width.

(a) If the smaller side is x cm, find the perimeter of the rectangle in terms of x.

The perimeter of a square is equal to the perimeter of the rectangle in part (a).

(b) Determine the length of each side of the square in terms of x.

The sum of the areas of the rectangle and the square is 2x2 + 4x +1 (cm2).

(c) (i) Given that this sum is 49 cm2, find x.

(ii) Find the area of the square.

(Total 6 marks)

7. A quadratic function, f(x) = ax2 + bx, is represented by the mapping diagram below.

(a) Use the mapping diagram to write down two equations in terms of a and b.

(2)

(b) Find the value of

(i) a;

(ii) b.

(2)

(c) Calculate the x-coordinate of the vertex of the graph of f(x).

(2)

(Total 6 marks)

8. The diagram below shows the line PQ, whose equation is x + 2y = 12. The line

intercepts the axes at P and Q respectively.

diagram not to scale

(a) Find the coordinates of P and of Q.

(3)

(b) A second line with equation x – y = 3 intersects the line PQ at the point A. Find

the coordinates of A.

(3)

(Total 6 marks)

9. The straight line L passes through the points A(–1, 4) and B(5, 8).

(a) Calculate the gradient of L.

(2)

(b) Find the equation of L.

(2)

The line L also passes through the point P(8, y).

(c) Find the value of y.

(2)

(Total 6 marks)

10. A line joins the points A(2, 1) and B(4, 5).

(a) Find the gradient of the line AB.

(2)

Let M be the midpoint of the line segment AB.

(b) Write down the coordinates of M.

(1)

(c) Find the equation of the line perpendicular to AB and passing through M.

(3)

(Total 6 marks)

11. The following diagram shows the rectangular prism ABCDEFGH. The length is 5

cm, the width is 1 cm, and the height is 4 cm.

Diagram not to scale

(a) Find the length of [DF].

(b) Find the length of [CF].

(Total 8 marks)

C H

BG

DE

A F

12. Consider the expansion of (x + 2)11.

(a) Write down the number of terms in this expansion.

(1)

(b) Find the term containing x2.

(4)

(Total 5 marks)

13. Consider the arithmetic sequence 3, 9, 15, ..., 1353.

(a) Write down the common difference.

(1)

(b) Find the number of terms in the sequence.

(3)

(c) Find the sum of the sequence.

(2)

(Total 6 marks)

14. Find the sum of the arithmetic series

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

(Total 4 marks)

15. (a) Find log2 32.

(1)

(b) Given that log2 can be written as px + qy, find the value of p and of q.

(4)

(Total 5 marks)

16. Find the exact solution of the equation 92x = 27(1–x).

(Total 6 marks)

y

x

8

32

17. Let f(x) = 7 – 2x and g(x) = x + 3.

(a) Find (g ° f)(x).

(2)

(b) Write down g–1(x).

(1)

(c) Find (f ° g–1)(5).

(2)

(Total 5 marks)

18. The following diagram shows part of the graph of f, where f (x) = x2 − x − 2.

(a) Find both x-intercepts.

(4)

(b) Find the x-coordinate of the vertex.

(2)

(Total 6 marks)

19. The equation of a curve may be written in the form y = a(x – p)(x – q). The curve

intersects the x-axis at A(–2, 0) and B(4, 0). The curve of y = f (x) is shown in the

diagram below.

(i) Write down the value of p and of q.

(ii) Given that the point (6, 8) is on the curve, find the value of a.

4

2

–2

–4

–6

–4 –2 0 2 4 6 x

y

A B

(iii) Write the equation of the curve in the form y = ax2 + bx + c.

(Total 6 marks)

20. Solve the equation 2cos x = sin 2x, for 0 ≤ x ≤ 3π.

(Total 7 marks)

21. The following diagram shows triangle ABC.

diagram not to scale

AB = 7 cm, BC = 9 cm and = 120°.

(a) Find AC.

(3)

(b) Find .

(3)

(Total 6 marks)

CBA

CAB

22. The following diagram shows the triangle ABC.

diagram not to scale

The angle at C is obtuse, AC = 5 cm, BC = 13.6 cm and the area is 20 cm2.

(a) Find .

(4)

(b) Find AB.

(3)

(Total 7 marks)

23. The circle shown has centre O and radius 3.9 cm.

diagram not to scale

BCA

Points A and B lie on the circle and angle AOB is 1.8 radians.

(a) Find AB.

(3)

(b) Find the area of the shaded region.

(4)

(Total 7 marks)

24. (a) Given that 2 sin2 θ + sinθ – 1 = 0, find the two values for sin θ.

(4)

(b) Given that 0° ≤ θ ≤ 360° and that one solution for θ is 30°, find the other two

possible values for θ.

(2)

(Total 6 marks)

25. The diagram below shows the probabilities for events A and B, with P(A′) = p.

(a) Write down the value of p.

(1)

(b) Find P(B).

(3)

(c) Find P(A′ | B).

(3)

(Total 7 marks)

26. The letters of the word PROBABILITY are written on 11 cards as shown below.

Two cards are drawn at random without replacement.

Let A be the event the first card drawn is the letter A.

Let B be the event the second card drawn is the letter B.

(a) Find P(A).

(1)

(b) Find P(B│A).

(2)

(c) Find P(A ∩ B).

(3)

(Total 6 marks)

27. The following diagram is a box and whisker plot for a set of data.

The interquartile range is 20 and the range is 40.

(a) Write down the median value.

(1)

(b) Find the value of

(i) a;

(ii) b.

(4)

(Total 5 marks)

28. A box contains 100 cards. Each card has a number between one and six written on it.

The following table shows the frequencies for each number.

Number 1 2 3 4 5 6

Frequency 26 10 20 k 29 11

(a) Calculate the value of k.

(2)

(b) Find

(i) the median;

(ii) the interquartile range.

(5)

(Total 7 marks)

29. Solve the equation log3(x + 17) – 2 = log3 2x.

(Total 5 marks)

30. (a) Show that = tan θ.

(2)

(b) Hence find the value of cot in the form a + , where a, b .

(3)

(Total 5 marks)

2cos1

2sin

8

π2b