for students entering IB Math SL Math SL Summer Review Packet Page 5 of 13 IX. Using the point-slope form y – y 1 = m(x – x 1), write an equation for the line 1. with slope –2,

Summer Review Packet

for students entering

IB Math SL

The problems in this packet are designed to help you review topics that are important to your

success in IB Math SL.

Please attempt the problems on your own without any notes and show all work! In addition, do not

use your calculator for these problems. When you come across topics that require a little review,

feel free to look at your old notes, search a website or ask a friend for help. If you want to check

your work with a calculator, that is fine also. You are expected to get each problem correct.

It is recommended that you work with one or more people, but each person must submit his/her own

work. Before you leave school, write down the names, phone numbers, and/or email addresses for at

least two people who are also taking IB Math SL in the fall.

Name ___________________________________ Phone __________________

Email ___________________________________

Name ___________________________________ Phone __________________

Email ___________________________________

Bring the finished packet with you to your IB Math SL class on the first day of school. After you

have an opportunity to ask questions, you will be assessed on these skills during the first week of

school as part of your 1st quarter grade.

Enjoy your summer! I am looking forward to seeing you in September. If you have any questions,

please contact Mrs. Atamas: [email protected].

mailto:[email protected]

IB Math SL Summer Review Packet Page 2 of 13

I. Simplify. Show the work that leads to your answer.

1. 43

42

xx

x

2. 3 8

2

x

x

3. 25

52

x

x

4. 2

2

4 32

16

x x

x

II. Complete the following identities.

1. sin2x + cos2x = __________

2. 1 + tan2x = __________

3. cot2x + 1 = __________

4. cos 2x = __________

5. sin 2x = __________

III. Simplify each expression.

1. 1 1

x h x

2. 2

5

2

10x

x

3.

1 1

3 3x

x

4. 2 2

2 1 8

6 9 1 2 3

x

x x x x x


IV. Solve for z:

1. 4x + 10yz = 0

2. y2 + 3yz – 8z – 4x = 0

V. If f(x) = {(3,5), (2,4), (1,7)} g(x) = 3x

determine each of the following:

h(x) = {(3,2), (4,3), (1,6)} k(x) = x2 + 5

1. (f + h)(1) = 2. (k – g)(5) =

3. (f ◦ h)(3) = 4. (g ◦ k)(7) =

5. f -1(x) = 6. k -1(x) =

7. 1

( )f x =

8. (kg)(x) =

VI.

1. Evaluate ( ) ( )f x h f x

h

and simplify if f(x) = x2 – 2x.

2. Expand (x + y)3

3. Simplify: 𝑥3 2⁄ (𝑥5 2⁄ − 𝑥2 + 𝑥) =

4. Simplify: 𝑥3−𝑥2+𝑥

√𝑥=

5. Find sin 2 if 3

sin5

. How many answers do you expect?


VII. Expand and simplify.

1. 24

0 2n

n

2. 3

31

1

n n

3.

∑1

2𝑘

∞

𝑘=0

=

4.

∑1

𝑛!

∞

𝑘=0

=

VIII. Simplify

1. x

x

2. ln3e

3. (1 ln )xe

4. ln 1

5. ln e7

6. 3

1log

3

7. log 1/2 8

8.

1ln

2

9. 3ln xe

10.

2

1

53

4

12

xy

x y

11. 272/3

12. (5a2/3)(4a3/2)

13. (4a5/3) 3/2

14. ln81 ln3


IX. Using the point-slope form y – y1 = m(x – x1), write an equation for the line

1. with slope –2, containing the point (3, 4) 1. __________________________

2. containing the points (1, -3) and (-5, 2) 2. __________________________

3. with slope 0, containing the point (4, 2) 3. __________________________

4. perpendicular to the line in problem #1,

containing the point (3, 4)

4. __________________________

XI. Without a calculator, determine the exact value of each expression.

1. sin 0 2. sin

2

3. sin

3

4

4. cos 5. cos

7

6

6. cos

3

7. tan 7

4

8. tan

6

9. tan

2

3

10. cos(Sin-1 1

2) 11. Sin-1(sin

7

6

)

X. Given the vectors v = 2i + 5j and w = 3i + 4j, determine

1. 1

2v

2. w – v

3. length of w

4. the unit vector for v


XII. For each function, determine its domain and range.

Function Domain Range

1. 4y x

2. 2 4y x

3. 24y x

4. 2 4y x

XIII. Determine all points of intersection.

1. parabola y = x2 + 3x –4 and line y = 5x + 11 2. y = cos x and y = sin x in the first quadrant

XIV. Solve for x, where x is a real number. Show the work that leads to your solution.

1. x2 + 3x – 4 = 14 2.

3. (x – 5)2 = 9 4. 2x2 + 5x = 8

4

3

10

x

x


Solve for x, where x is a real number. Show the work that leads to your solution.

5. (x + 3)(x – 3) > 0 6. x2 – 2x – 15 0

7. 12x2 = 3x

8. sin 2x = sin x , 0 x 2

9. |x – 3| < 7 10. (x + 1)2(x – 2) + (x + 1)(x – 2)2 = 0

11. 272x = 9x 3 12. log x + log(x – 3) = 1


XV. Graph each function. Give its domain and range.

1. y = sin x

Domain_________________

Range _________________

2. y = ex

Domain_________________

Range _________________

3. y = x

Domain_________________

Range _________________

4. y = 3 x

Domain_________________

Range _________________


Graph each function. Give its domain and range.

5. y = ln x

Domain_________________

Range _________________

6. y = |x + 3| 2

Domain_________________

Range _________________

7.

Domain_________________

Range _________________

8.

Domain_________________

Range _________________

1y

x

2 if 0

2 if 0 3

4 if 3

x x

y x x

x


XVI. Compute each of the following limits:

1. cos

limx

x

x

2. 3

1

1lim

1x

x

x

XVII.

Let

2

1 , if 2

4 , if 2 2

2

3 , if 2

xx

xf x x

x

x

Compute the following limits:

a) limx

f x

b) 2

limx

f x

c) 2

limx

f x

d) 2

limx

f x

e) 2

limx

f x

f) limx

f x

XVIII. Write each sum using summation notation, assuming the suggested pattern continues.

1. 2 + 5 + 8 + 11 + ... + 29 =

2. 1 + 2 + 6 + 24 + 120 + 720 =

3. 6 12 + 24 48 + ... =

4. 1 1 + 1 1 + ... =

5. 1 +1

4+

1

9+

1

25+ ⋯ = 6. 0.1 + 0.01 + 0.001 + 0.0001 + ... =

XIX. Remember you are not using a calculator.

1. In a triangle ABC, angles A and C measure 45

and 30 degrees respectively. Side BC is 14

centimeters long. Sketch a diagram and find

a) AB

b) Area of the triangle ABC

2. Find the area of the

shaded region.


XX. The Binomial Theorem.

1. Find the coefficient of x5 in the expansion of (3x – 2)8.

2. Use the binomial theorem to complete this expansion. (3x + 2y)4 = 81x4 + 216x3 y +...

3. Determine the constant term in the expansion of (𝑥 −2

𝑥2)9.

XXI. Vectors

1. ABCD is a rectangle and O is the midpoint of [AB].

Express each of the following vectors in terms of 𝑂𝐶⃗⃗⃗⃗ ⃗ and 𝑂𝐷⃗⃗⃗⃗⃗⃗

(a) 𝐶𝐷⃗⃗⃗⃗ ⃗

(b) 𝑂𝐴⃗⃗⃗⃗ ⃗

(c) 𝐴𝐷⃗⃗ ⃗⃗ ⃗

2. The quadrilateral OABC has vertices with coordinates O(0, 0), A(5, 1), B(10, 5) and C(2, 7).

(a) Find the vectors 𝑂𝐵⃗⃗ ⃗⃗ ⃗ and 𝐴𝐶⃗⃗⃗⃗ ⃗.

(b) Find the cosine of the angle between the diagonals of the quadrilateral OABC.

A B

CD

O


O

–1

–2

–3

–4

–2 –1

4

3

2

1

1 2 3 4 5 6

y

x

3. The vectors 𝑖 and 𝑗 are unit vectors along the x-axis and y-axis respectively.

The vectors �⃗� = −𝑖 + 2𝑗 and 𝑣 = 3𝑖 + 5𝑗 are given.

(a) Find �⃗� + 2𝑣 in terms of 𝑖 and 𝑗 .

A vector �⃗⃗� has the same direction as �⃗� + 2𝑣 , and has a magnitude of 26.

(b) Find �⃗⃗� in terms of 𝑖 and 𝑗 .

4. Find a vector equation of the line passing through (–1, 4) and (3, –1). Give your answer in the

form r = p + td, where t .

5. The triangle ABC is defined by the following information

𝑂𝐴⃗⃗⃗⃗ ⃗ = (2

−3), 𝐴𝐵⃗⃗⃗⃗ ⃗ = (

34), 𝐴𝐵⃗⃗⃗⃗ ⃗ ∙ 𝐵𝐶⃗⃗⃗⃗ ⃗ = 0, 𝐴𝐶⃗⃗⃗⃗ ⃗ is parallel to (

01)

(a) On the grid below, draw an accurate diagram of triangle ABC.

(b) Write down the vector 𝑂𝐶⃗⃗⃗⃗ ⃗.


XXII. The following vector problem could be challenging. I hope you will figure it out.

Points P and Q have position vectors −5i +11j −8k and −4i + 9 j − 5k respectively, and both lie on a line L1.

(a) (i) Find .𝑃𝑄⃗⃗⃗⃗ ⃗.

(ii) Hence show that the equation of L1 can be written as r = (−5 + s) i + (11− 2s) j + (−8 + 3s) k.

The point R (2, y1, z1) also lies on L1.

(b) Find the value of y1 and of z1.

The line L2 has equation r = 2i + 9 j +13k + t (i + 2 j + 3k).

(c) The lines L1 and L2 intersect at a point T. Find the position vector of T.

(d) Find the cosine of the angle between the lines L1 and L2.

for students entering IB Math SL Math SL Summer Review Packet Page 5 of 13 IX. Using the point-slope form y – y 1 = m(x – x 1), write an equation for the line 1. with slope –2,

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