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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].
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
IB Math SL Summer Review Packet Page 3 of 13
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?
IB Math SL Summer Review Packet Page 4 of 13
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
IB Math SL Summer Review Packet Page 5 of 13
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
IB Math SL Summer Review Packet Page 6 of 13
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
IB Math SL Summer Review Packet Page 7 of 13
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
IB Math SL Summer Review Packet Page 8 of 13
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 _________________
IB Math SL Summer Review Packet Page 9 of 13
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
IB Math SL Summer Review Packet Page 10 of 13
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.
IB Math SL Summer Review Packet Page 11 of 13
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
IB Math SL Summer Review Packet Page 12 of 13
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 ⃗.
IB Math SL Summer Review Packet Page 13 of 13
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.