HIGH SCHOOL MATHEMATICS
10 COMPREHENSIVE FINAL
EXAMS
with Complete Step-by-Step Solutions
300 Pages of Exam Practice
Perfect for:
SAT, ACT, AP Calculus, High School Final Exams
Author: RACHID OUSALEM
March 15, 2026
, Copyright © 2024 All Rights Reserved
No part of this book may be reproduced without permission.
,Introduction
* Welcome to Your Exam Success Journey!
Dear Student,
Congratulations! You’ve just picked up the most powerful tool for your high school
mathematics final exams. This book contains 10 complete exams carefully de-
signed to mirror the structure, difficulty, and style of real final examinations.
Why This Book?
• Full Exams – Each exam covers all major topics: Complex Numbers, Sequences,
Functions, Integration, Probability, and 3D Geometry.
• Real Exam Format – Every exam is structured exactly like your actual final exam,
with 5 exercises and a total of 20 points.
• Complete Solutions – Every single question is solved step-by-step with detailed
explanations. No step is skipped!
• Color-Coded – Key concepts in blue, warnings in red, tips in green, and final
answers in gold.
• Professional Graphs – All functions are plotted using TikZ for visual clarity.
• Progressive Difficulty – Exams gradually increase in difficulty, building your
confidence.
How to Use This Book
1. Simulate Exam Conditions: Take each exam in 3 hours without looking at the
solutions.
2. Check Your Answers: Use the detailed solutions to identify your mistakes.
3. Review Weak Topics: Go back to the exercises you got wrong and practice similar
ones.
4. Repeat: Take the next exam after a few days of review.
3
,Exam Structure
Each exam consists of 5 exercises:
Exercise Topic Points
1 Complex Numbers 4
2 Sequences 4
3 Function Study (Exponential/Logarithmic) 5
4 Integration 4
5 Probability or 3D Geometry 3
Total 20
Topics Covered
• Complex Numbers: Modulus, argument, polar form, exponential form, Euler’s
formula, geometric transformations (rotation, translation, dilation), quadratic equa-
tions with negative discriminant.
• Sequences: Arithmetic sequences, geometric sequences, recurrence relations, lim-
its, auxiliary sequences.
• Functions: Exponential functions, logarithmic functions, limits, derivatives, mono-
tonicity, asymptotes, inflection points, tangent lines, curve sketching.
• Integration: Basic integrals, definite integrals, integration by parts, substitution,
area under curves, area between curves.
• Probability: Combinatorics, conditional probability, Bayes’ theorem, binomial dis-
tribution, random variables, expected value, variance.
• 3D Geometry: Vectors, dot product, cross product, equations of lines and planes,
distances.
Acknowledgments
This book would not have been possible without the feedback and encouragement of
hundreds of students and teachers. Thank you for your support!
RACHID OUSALEM
Date
4 RACHID OUSALEM
,Contents
Introduction 3
1 Exam 1: Foundations 9
Solutions to Exam 1 11
2 Exam 2: Advanced Functions & Integration 15
Solutions to Exam 2 19
3 Exam 3: Logarithmic & Exponential Functions 29
Solutions to Exam 3 33
4 Exam 4: Advanced Functions & Integration 43
Solutions to Exam 4 47
5 Exam 4: Advanced Functions & 3D Geometry 49
Solutions to Exam 4 53
6 Exam 5: Logarithmic Functions & Analytic Geometry 63
Solutions to Exam 5 67
7 Exam 6: Advanced Functions & 3D Geometry 75
Solutions to Exam 6 79
8 Exam 7: Inverse Trig Functions & Advanced Geometry 87
Solutions to Exam 7 91
9 Exam 8: Radical Functions & Space Geometry 99
Solutions to Exam 8 103
10 Exam 9: Exponential & Logarithmic Mastery 111
Solutions to Exam 9 115
5
, CONTENTS CONTENTS
11 Exam 10: Trigonometric Functions & Space Geometry 121
Solutions to Exam 10 125
12 Exam 11: Advanced Exponential & Analytic Geometry 131
Solutions to Exam 11 135
13 Exam 12: Logarithmic Mastery & Space Geometry 141
Solutions to Exam 12 145
14 Exam 13: Advanced Exponential & Conic Sections 151
Solutions to Exam 13 155
15 Exam 14: Inverse Trig & Space Geometry 161
Solutions to Exam 14 165
16 Exam 15: Exponential-Logarithmic Mastery & Conics 169
Solutions to Exam 15 173
17 Exam 16: Cubic Root & Complex Sequences 177
Solutions to Exam 16 181
18 Exam 17: Fourth Root & Analytic Geometry 185
Solutions to Exam 17 189
19 Exam 18: nth Root & Complex Geometry 193
Solutions to Exam 18 197
Solutions to Exam 20 205
20 Exam 1: Advanced Functions Integration 213
21 Exam 2: Advanced Functions Integration 217
22 Exam 3: Advanced Functions Integration 221
23 Exam 4: Advanced Functions Integration 225
24 Exam 5: Advanced Functions Integration 229
25 Exam 6: Advanced Functions Integration 233
26 Exam 7: Advanced Functions Integration 237
27 Exam 8: Advanced Functions Integration 241
6 RACHID OUSALEM
,CONTENTS CONTENTS
28 Exam 9: Advanced Functions Integration 245
29 Exam 10: Advanced Functions Integration 249
30 Exam 11: Advanced Functions Integration 253
31 Exam 12: Advanced Functions Integration 257
32 Exam 13: Advanced Functions Integration 261
33 Exam 14: Advanced Functions Integration 265
34 Exam 15: Advanced Functions Integration 269
35 Exam 16: Advanced Functions Integration 273
36 Exam 17: Advanced Mathematics 277
37 Exam 18: Advanced Functions Integration 279
38 Exam 19: Analysis Discrete Mathematics 283
39 Exam 20: Comprehensive Mathematics Assessment 285
40 Exam 19: Advanced Functions Integration 287
41 Exam 21: Calculus and Discrete Structures 291
42 Exam 22: Comprehensive Mathematics 293
43 Exam 23: Advanced Mathematics 295
44 Exam 24: Comprehensive Assessment 297
45 Exam 25: Mastery of Integration and Analysis 299
46 Exam 26: Comprehensive Study with Auxiliary Functions 301
47 Exam 27: Comprehensive Study of Space and Functions 303
48 Exam 28: Comprehensive Mathematical Assessment 305
49 Exam 29: Advanced Complex Geometry Probability 307
50 Exam 30: Decade 3 Comprehensive Finale 309
51 Exam 31: Advanced Functions Integration 311
52 Exam 32: Advanced Functions Integration 315
7 RACHID OUSALEM
,CONTENTS CONTENTS
8 RACHID OUSALEM
,Chapter 1
Exam 1: Foundations
EXAM 1
Duration: 3 hours — Total: 20 points
Calculator permitted unless stated otherwise
Exercise 1: Complex Numbers (4 points)
1.1. Solve in C the equation: z 2 − 4z + 8 = 0. Let z1 be the solution with positive
imaginary part and z2 the other solution.
1.2. Write z1 and z2 in exponential form reiθ .
1.3. In the complex plane, consider points A(z1 ), B(z2 ), and C(2).
z1 −2
(a) Calculate z2 −2
. What can you conclude about triangle ABC?
(b) Let D be the image of A by the rotation with center O and angle π2 . Find the
affix zD of D.
(c) Show that O, B, and D are collinear.
Exercise 2: Numerical Sequences (4 points)
Let (un ) be the sequence defined by u0 = 2 and for all n ∈ N:
1
un+1 = un + 2
3
2.1. Calculate u1 , u2 , and u3 .
2.2. Prove by induction that un < 3 for all n ∈ N.
2.3. Show that (un ) is strictly increasing.
2.4. Let vn = un − 3.
(a) Show that (vn ) is a geometric sequence. Determine its ratio and first term.
(b) Express vn then un in terms of n.
(c) Calculate limn→+∞ un .
9
, EXAM 1 EXAM 1
Exercise 3: Logarithmic Function (5 points)
Let f be the function defined on (0, +∞) by:
f (x) = ln x − x + 2
3.1. Calculate limx→0+ f (x) and limx→+∞ f (x). Interpret geometrically.
3.2. Calculate f ′ (x) and study the sign of f ′ (x). Construct the variation table of
f.
3.3. Show that the equation f (x) = 0 has a unique solution α in the interval (2, 3).
3.4. Determine the equation of the tangent (T ) to the curve Cf at the point with
abscissa x = 1.
3.5. Sketch the curve Cf , showing the asymptotes and the tangent (T ).
Exercise 4: Integral Calculus (4 points)
Consider the function g(x) = xe−x defined on R.
4.1. Calculate the following integrals:
R1
(a) I = 0 g(x) dx (using integration by parts)
R1
(b) J = 0 x2 e−x dx
4.2. Let h(x) = (x + 1)e−x . Show that h′ (x) = −xe−x .
4.3. Calculate the area of the region bounded by Cg , the x-axis, and the lines x = 0
and x = 1.
Exercise 5: Probability (3 points)
An urn contains 5 red balls and 3 blue balls. We draw 3 balls simultaneously.
5.1. Calculate the total number of possible outcomes.
5.2. Calculate the probability of the following events:
(a) A: ”The three balls are red”
(b) B: ”Exactly two balls are red”
(c) C: ”At least one ball is blue”
5.3. Let X be the random variable equal to the number of red balls drawn.
(a) Determine the probability distribution of X.
(b) Calculate E(X) and V ar(X).
10 RACHID OUSALEM
10 COMPREHENSIVE FINAL
EXAMS
with Complete Step-by-Step Solutions
300 Pages of Exam Practice
Perfect for:
SAT, ACT, AP Calculus, High School Final Exams
Author: RACHID OUSALEM
March 15, 2026
, Copyright © 2024 All Rights Reserved
No part of this book may be reproduced without permission.
,Introduction
* Welcome to Your Exam Success Journey!
Dear Student,
Congratulations! You’ve just picked up the most powerful tool for your high school
mathematics final exams. This book contains 10 complete exams carefully de-
signed to mirror the structure, difficulty, and style of real final examinations.
Why This Book?
• Full Exams – Each exam covers all major topics: Complex Numbers, Sequences,
Functions, Integration, Probability, and 3D Geometry.
• Real Exam Format – Every exam is structured exactly like your actual final exam,
with 5 exercises and a total of 20 points.
• Complete Solutions – Every single question is solved step-by-step with detailed
explanations. No step is skipped!
• Color-Coded – Key concepts in blue, warnings in red, tips in green, and final
answers in gold.
• Professional Graphs – All functions are plotted using TikZ for visual clarity.
• Progressive Difficulty – Exams gradually increase in difficulty, building your
confidence.
How to Use This Book
1. Simulate Exam Conditions: Take each exam in 3 hours without looking at the
solutions.
2. Check Your Answers: Use the detailed solutions to identify your mistakes.
3. Review Weak Topics: Go back to the exercises you got wrong and practice similar
ones.
4. Repeat: Take the next exam after a few days of review.
3
,Exam Structure
Each exam consists of 5 exercises:
Exercise Topic Points
1 Complex Numbers 4
2 Sequences 4
3 Function Study (Exponential/Logarithmic) 5
4 Integration 4
5 Probability or 3D Geometry 3
Total 20
Topics Covered
• Complex Numbers: Modulus, argument, polar form, exponential form, Euler’s
formula, geometric transformations (rotation, translation, dilation), quadratic equa-
tions with negative discriminant.
• Sequences: Arithmetic sequences, geometric sequences, recurrence relations, lim-
its, auxiliary sequences.
• Functions: Exponential functions, logarithmic functions, limits, derivatives, mono-
tonicity, asymptotes, inflection points, tangent lines, curve sketching.
• Integration: Basic integrals, definite integrals, integration by parts, substitution,
area under curves, area between curves.
• Probability: Combinatorics, conditional probability, Bayes’ theorem, binomial dis-
tribution, random variables, expected value, variance.
• 3D Geometry: Vectors, dot product, cross product, equations of lines and planes,
distances.
Acknowledgments
This book would not have been possible without the feedback and encouragement of
hundreds of students and teachers. Thank you for your support!
RACHID OUSALEM
Date
4 RACHID OUSALEM
,Contents
Introduction 3
1 Exam 1: Foundations 9
Solutions to Exam 1 11
2 Exam 2: Advanced Functions & Integration 15
Solutions to Exam 2 19
3 Exam 3: Logarithmic & Exponential Functions 29
Solutions to Exam 3 33
4 Exam 4: Advanced Functions & Integration 43
Solutions to Exam 4 47
5 Exam 4: Advanced Functions & 3D Geometry 49
Solutions to Exam 4 53
6 Exam 5: Logarithmic Functions & Analytic Geometry 63
Solutions to Exam 5 67
7 Exam 6: Advanced Functions & 3D Geometry 75
Solutions to Exam 6 79
8 Exam 7: Inverse Trig Functions & Advanced Geometry 87
Solutions to Exam 7 91
9 Exam 8: Radical Functions & Space Geometry 99
Solutions to Exam 8 103
10 Exam 9: Exponential & Logarithmic Mastery 111
Solutions to Exam 9 115
5
, CONTENTS CONTENTS
11 Exam 10: Trigonometric Functions & Space Geometry 121
Solutions to Exam 10 125
12 Exam 11: Advanced Exponential & Analytic Geometry 131
Solutions to Exam 11 135
13 Exam 12: Logarithmic Mastery & Space Geometry 141
Solutions to Exam 12 145
14 Exam 13: Advanced Exponential & Conic Sections 151
Solutions to Exam 13 155
15 Exam 14: Inverse Trig & Space Geometry 161
Solutions to Exam 14 165
16 Exam 15: Exponential-Logarithmic Mastery & Conics 169
Solutions to Exam 15 173
17 Exam 16: Cubic Root & Complex Sequences 177
Solutions to Exam 16 181
18 Exam 17: Fourth Root & Analytic Geometry 185
Solutions to Exam 17 189
19 Exam 18: nth Root & Complex Geometry 193
Solutions to Exam 18 197
Solutions to Exam 20 205
20 Exam 1: Advanced Functions Integration 213
21 Exam 2: Advanced Functions Integration 217
22 Exam 3: Advanced Functions Integration 221
23 Exam 4: Advanced Functions Integration 225
24 Exam 5: Advanced Functions Integration 229
25 Exam 6: Advanced Functions Integration 233
26 Exam 7: Advanced Functions Integration 237
27 Exam 8: Advanced Functions Integration 241
6 RACHID OUSALEM
,CONTENTS CONTENTS
28 Exam 9: Advanced Functions Integration 245
29 Exam 10: Advanced Functions Integration 249
30 Exam 11: Advanced Functions Integration 253
31 Exam 12: Advanced Functions Integration 257
32 Exam 13: Advanced Functions Integration 261
33 Exam 14: Advanced Functions Integration 265
34 Exam 15: Advanced Functions Integration 269
35 Exam 16: Advanced Functions Integration 273
36 Exam 17: Advanced Mathematics 277
37 Exam 18: Advanced Functions Integration 279
38 Exam 19: Analysis Discrete Mathematics 283
39 Exam 20: Comprehensive Mathematics Assessment 285
40 Exam 19: Advanced Functions Integration 287
41 Exam 21: Calculus and Discrete Structures 291
42 Exam 22: Comprehensive Mathematics 293
43 Exam 23: Advanced Mathematics 295
44 Exam 24: Comprehensive Assessment 297
45 Exam 25: Mastery of Integration and Analysis 299
46 Exam 26: Comprehensive Study with Auxiliary Functions 301
47 Exam 27: Comprehensive Study of Space and Functions 303
48 Exam 28: Comprehensive Mathematical Assessment 305
49 Exam 29: Advanced Complex Geometry Probability 307
50 Exam 30: Decade 3 Comprehensive Finale 309
51 Exam 31: Advanced Functions Integration 311
52 Exam 32: Advanced Functions Integration 315
7 RACHID OUSALEM
,CONTENTS CONTENTS
8 RACHID OUSALEM
,Chapter 1
Exam 1: Foundations
EXAM 1
Duration: 3 hours — Total: 20 points
Calculator permitted unless stated otherwise
Exercise 1: Complex Numbers (4 points)
1.1. Solve in C the equation: z 2 − 4z + 8 = 0. Let z1 be the solution with positive
imaginary part and z2 the other solution.
1.2. Write z1 and z2 in exponential form reiθ .
1.3. In the complex plane, consider points A(z1 ), B(z2 ), and C(2).
z1 −2
(a) Calculate z2 −2
. What can you conclude about triangle ABC?
(b) Let D be the image of A by the rotation with center O and angle π2 . Find the
affix zD of D.
(c) Show that O, B, and D are collinear.
Exercise 2: Numerical Sequences (4 points)
Let (un ) be the sequence defined by u0 = 2 and for all n ∈ N:
1
un+1 = un + 2
3
2.1. Calculate u1 , u2 , and u3 .
2.2. Prove by induction that un < 3 for all n ∈ N.
2.3. Show that (un ) is strictly increasing.
2.4. Let vn = un − 3.
(a) Show that (vn ) is a geometric sequence. Determine its ratio and first term.
(b) Express vn then un in terms of n.
(c) Calculate limn→+∞ un .
9
, EXAM 1 EXAM 1
Exercise 3: Logarithmic Function (5 points)
Let f be the function defined on (0, +∞) by:
f (x) = ln x − x + 2
3.1. Calculate limx→0+ f (x) and limx→+∞ f (x). Interpret geometrically.
3.2. Calculate f ′ (x) and study the sign of f ′ (x). Construct the variation table of
f.
3.3. Show that the equation f (x) = 0 has a unique solution α in the interval (2, 3).
3.4. Determine the equation of the tangent (T ) to the curve Cf at the point with
abscissa x = 1.
3.5. Sketch the curve Cf , showing the asymptotes and the tangent (T ).
Exercise 4: Integral Calculus (4 points)
Consider the function g(x) = xe−x defined on R.
4.1. Calculate the following integrals:
R1
(a) I = 0 g(x) dx (using integration by parts)
R1
(b) J = 0 x2 e−x dx
4.2. Let h(x) = (x + 1)e−x . Show that h′ (x) = −xe−x .
4.3. Calculate the area of the region bounded by Cg , the x-axis, and the lines x = 0
and x = 1.
Exercise 5: Probability (3 points)
An urn contains 5 red balls and 3 blue balls. We draw 3 balls simultaneously.
5.1. Calculate the total number of possible outcomes.
5.2. Calculate the probability of the following events:
(a) A: ”The three balls are red”
(b) B: ”Exactly two balls are red”
(c) C: ”At least one ball is blue”
5.3. Let X be the random variable equal to the number of red balls drawn.
(a) Determine the probability distribution of X.
(b) Calculate E(X) and V ar(X).
10 RACHID OUSALEM
