mathematics-logical

Admission Exam Handbook in Mathematics
New Uzbekistan University
Academic Year 2023 – 2024
June 10, 2023
We extend our warmest welcome to all prospective applicants who aspire to join New
Uzbekistan University. At New Uzbekistan University, you will be embraced by a
community of exceptional scholars and innovators, guided by distinguished faculty
members who are leaders in their fields. Our commitment to academic excellence,
coupled with a nurturing environment, fosters intellectual curiosity, critical thinking, and
personal growth.
Within these pages, you will find the essential information and guidelines to navigate the
admission exam in Mathematics, its structure, topics and samples of problems from
previous exams.
As you prepare for your admission exams, we wish you the best of luck, confident that
your hard work and dedication will pave the way for a remarkable future.
— Dr. Bahodir Ahmedov
Vice-Rector for Academic Affairs
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Contents
1 Exam Structure
1.1 Sample Exam Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Probability and Statistics
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Sample Space and Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Probability Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Binomial Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.5 Conditional Probability and Independence . . . . . . . . . . . . . . . . . . . .
2.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Measures of Central Tendency and Dispersion . . . . . . . . . . . . . . . . . .
2.3 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Exercises: Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Polynomials
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Polynomial Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Geometry
4.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Classification by Sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Classification by Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Exercises: Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Functions
5.1 Introduction to Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Types of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Operations on Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Function Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Function Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Series, Powers & Roots
6.1 Introduction to Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Powers and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Logical Thinking
7.1 Introduction to Logical Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Deductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Inductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Logical Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Conditional Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.1 Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
Exam Structure
Exam Questions in Mathematics cover a wide range of topics, including algebra, probability and
statistics, polynomials, geometry, series, powers, and roots. These questions are designed to assess
a student’s mathematical skills and problem-solving abilities, ensuring they are well-prepared for
advanced studies in mathematics.
The questions that match the international level and experiences of admission exams in universities often incorporate the standards set by Cambridge International Assessments. Cambridge
International Assessments is a renowned examining board that provides internationally recognized
qualifications, including the Cambridge International AS and A Level Mathematics.
Algebra questions involve solving equations and inequalities, manipulating algebraic expressions, and analyzing patterns and sequences. These questions may require students to solve equations with multiple variables, factorize polynomials, simplify expressions, or solve systems of equations.
Probability and statistics questions assess a student’s understanding of basic probability concepts, such as calculating probabilities of events and analyzing data sets. Students may be asked
to determine the probability of independent and dependent events, interpret data from tables and
graphs, or calculate measures of central tendency and dispersion.
Polynomial questions can range from factoring and expanding polynomial expressions to solving
polynomial equations. Students may encounter questions that involve finding roots, identifying
factors, or simplifying complex polynomial expressions.
Geometry questions test a student’s knowledge of geometric concepts, including properties of
lines, angles, triangles, circles, and polygons. These questions often require students to apply
theorems and formulas to solve problems related to areas, perimeters, angles, and symmetry.
Series questions involve finding patterns in arithmetic and geometric sequences, determining
the nth term or the sum of a series, and solving problems involving recursive sequences.
Powers and roots questions cover topics such as exponentiation, logarithms, and radicals. Students may be asked to simplify expressions with exponents, solve logarithmic equations, or calculate
roots of numbers.
Including a separate section on logical thinking in admission exams for mathematics at a university is crucial as it assesses a candidate’s ability to approach mathematical problems strategically,
analyze information, identify patterns, draw logical conclusions, and think critically - skills that
are not only fundamental to mathematics but also essential in various academic disciplines and
real-life scenarios, ensuring that prospective students possess the necessary problem-solving and
critical thinking abilities to excel in their mathematical studies and future careers.
1.1
Sample Exam Paper
Every exam paper consist of 20 questions. We ensure that each topic section is represented equally
in every exam paper. For example, there are 3 problems on logical thinking in every exam paper.
Each question worth 5 points meaning that maximum amount of points can be obtained for the
exam is 100 for 20 questions.
Please be familiarized with the sample questions. These questions are taken from the Admission
Exam in May 20, 2023.
1. A drawer contains 5 pairs of black socks, 4 pairs of blue socks, and 3 pairs of brown socks. If
you choose socks at random without looking, what is the minimum number of socks you must
choose to be certain that you have chosen at least one pair of socks of the same color?
A. 3
B. 4
C. 6
D. 8
E. 12
2. If a box contains 15 red balls and 10 green balls, what is the minimum number of balls that
must be drawn from the box to guarantee that at least 5 balls are red?
A. 11
B. 12
C. 15
D. 7
E. 6
3. What is the probability that a randomly chosen integer between 1 and 100 (inclusive) is a
multiple of 3 but not a multiple of 5?
A. 10 %
B. 15 %
C. 20 %
D. 27 %
E. 50 %
4. Expand xy 2 + x
3x2 + xy − y 2
A. x3 y 2 + 3x3 + x2 y 3 + x2 y + 2xy 4 − xy 2
B. x3 y 2 − 3x3 + x2 y 3 + x2 y − 4xy 4 + 3xy 2
C. 3x3 y 2 + 3x3 + x2 y 3 + x2 y − xy 4 − xy 2
D. −x3 y 2 − x2 y 3 + x2 y − xy 4 − 3xy 2
E. −3x3 + x2 y 3 + x2 y − xy 4 − 10xy 2
5. What is the remainder of dividing 2x4 + 3x3 − 2x + 1 by 2x2 − x + 2?
1
A. 11
2 x− 4
1
B. 32 x2 − 11
8 x+ 4
C. 1 − 6x
D. −2x − 7
E. No correct answer.
6. What is the remainder of dividing x4 − 2x3 − 2x + 1 by x2 − 3x + 2?
A. −1 − x
B. 2 − x
C. 1 − 3x
D. −x + 3
E. 2 + x
7. What is the maximal possible area of a triangle whose two sides are equal to 7 and 4?
√
A. 4 7
√
B. 7 2
C. 14
√
D. 14 2
E. 28
8. If the longest side of a triangle is 10 cm and two of its angles are 30 degrees and 60 degrees,
what is the measure of the smallest side of the triangle?
√
A. 4 3
B. 5
C. 4
D. 6
E. cannot be determined
9. A circle is inscribed into a trapezoid with bases of length 4 and 10. If the height of trapezoid
is equal to 6, find the radius of the circle.
A. 3
B. 4
C. 5
D. 6
E. Impossible to determine
10. What is the length of a chord in a circle with radius 6 and central angle of
A. 12
B. 18
√
C. 6 3
D. 6
E. 3
11. Let f (x) = 3x2 − 2x + 1 and g(x) =
A. 3
√
B. 13
√
C. 7
D. −1
√
E. 10
√
x + 1. What is g(f (2))?
π
radians?
3
x+1
12. What is the domain of the function f (x) = √
?
x2 − 9
A. x ∈ (−∞, −3) ∪ (−3, 3) ∪ (3, ∞)
B. x ∈ (−∞, −3) ∪ (3, ∞)
C. x ∈ (−∞, 3) ∪ (3, ∞)
D. x ∈ [−3, 3]
E. x ∈ (3, ∞)
13. What are the roots of the function f (x) = 3x3 − 9x2 + 6x?
A. x = 1, 32
B. x = 0, 1, 2
C. x = 3, 6
D. x = 0, 23
E. x = 0, 12 , 3
14. Find the sum of all the odd numbers between 1 and 49 inclusive.
A. 625
B. 675
C. 600
D. 700
E. 715
15. Find the sum of the first 20 terms of an arithmetic progression where the first term is 5 and
the common difference is 3.
A. 590
B. 620
C. 670
D. 710
E. 700
1
16. Which of the following expressions is equal to (a4 b3 c6 ) 12 ?
A. a3 b4 c2
√
6
B. a2 b3 c2
√ √
4
C. a · bc2
√ √ √
D. 3 a · 4 b · c
√
E. 3 a · bc2
17. Find the value of a4 + b4 if a and b are the roots of the quadratic equation x2 + 3x − 5 = 0.
A. 212
B. 526
C. 311
D. 421
E. 356
18. When three clients are seated in an ice cream shop, the server asks them: “Does everyone want
ice cream?” The first client says “I don’t know.” The second client then says “I don’t know.”
Finally, the third client says “No, not everyone wants ice cream.” Which of the following is
true?
A. The third client wants an ice-cream
B. The first and second clients do not wants an ice-cream
C. The third client does not want an ice-cream
D. The first and the third client wants an ice-cream
E. It is undecidable
19. Emma would like to determine the relative heights of three friends using two facts. First, she
knows that if Sarah is not the tallest of the three, then Rachel is. Second, she knows that
if Rachel is not the shortest, then Leah is the tallest. Is it possible to determine the relative
heights of Sarah, Rachel, and Leah from what Emma knows? If so, who is the tallest and who
is the shortest?
A. It is possible to determine the relative heights, and Sarah is the tallest,
while Rachel is the shortest.
B. It is possible to determine the relative heights, and Rachel is the tallest, while Leah
is the shortest.
C. It is possible to determine the relative heights, and Rachel is the tallest, while Sarah
is the shortest.
D. It is not possible to determine the relative heights from the given information.
E. It is possible to determine the relative heights, and Leah is the tallest, while Rachel
is the shortest.
20. A museum has 15 paintings, each of which was painted by a different artist. The curator knows
that at least one painting is from the 20th century, and given any two paintings, at least one
was painted before the 19th century. Based on these facts, can you determine how many of the
paintings were painted in the 20th century?
A. not possible to determine
B. 8
C. 7
D. 2
E. 1
2
Probability and Statistics
2.1
Introduction
Probability and Statistics are fundamental mathematical disciplines that deal with the analysis
and interpretation of data, as well as the study of randomness and uncertainty. In today’s datadriven world, understanding these topics is essential not only for those pursuing careers in science,
engineering, and technology, but also for those looking to make informed decisions in their everyday
lives. This chapter provides an introduction to the basic concepts and techniques in Probability
and Statistics, with a focus on preparing potential students for university entrance exams.
2.1.1
Probability
Probability is the mathematical study of chance and randomness. It deals with the likelihood of
certain events occurring, given a set of possible outcomes. This is a crucial concept in a wide range
of disciplines, from game theory and finance to engineering and the natural sciences.
Some of the core concepts in Probability include:
• Sample space and events
• Probability axioms
• Conditional probability
• Independent and mutually exclusive events
• Discrete and continuous random variables
• Probability distributions (e.g., uniform, binomial, normal)
2.1.2
Sample Space and Events
Example 1: Consider a single six-sided die. When we roll the die, there are six possible outcomes:
1, 2, 3, 4, 5, or 6. The sample space, denoted by S, is the set of all possible outcomes. In this case,
S = 1, 2, 3, 4, 5, 6.
An event is a subset of the sample space. For example, if we are interested in the event of
rolling an even number, the event E would be the set E = 2, 4, 6.
Exercise 1: If you flip a coin twice, what is the sample space? List all the events in the sample
space.
2.1.3
Probability Axioms
There are three fundamental axioms of probability:
1. The probability of an event is always non-negative: P (E) ≥ 0 for all events E.
2. The probability of the entire sample space is 1: P (S) = 1.
3. If two events are mutually exclusive (i.e., they cannot both occur at the same time), the
probability of their union is the sum of their individual probabilities: P (A∪B) = P (A)+P (B)
for mutually exclusive events A and B.
Example 2: Consider the six-sided die from Example 1. The probability of rolling an even
number can be computed using the probability axioms. Since there are six possible outcomes and
each outcome is equally likely, the probability of rolling an even number (event E) is:
P (E) =
Number of outcomes in E
3
1
= =
Number of outcomes in S
6
2
(1)
Exercise 2: In the context of the coin flipping example from Exercise 1, what is the probability
of flipping two heads in a row?
2.1.4
Binomial Probability
The binomial distribution model deals with finding the probability of success of an event which has
only two possible outcomes in a series of experiments. For example, tossing of a coin always gives a
head or a tail. The probability of finding exactly 3 heads in tossing a coin repeatedly for 10 times
is estimated during the binomial distribution.
Example 3: If you flip a coin twice, what is the probability of getting exactly one head? The
possible outcomes are HH, HT, TH, TT. So, there are 4 possible outcomes and 2 of them (HT and
TH) involve exactly one head. Therefore, P (1 Head in 2 Flips) = 24 = 0.5 or 50%.
2.1.5
Conditional Probability and Independence
Conditional probability is the probability of an event occurring, given that another event has
occurred. We denote the conditional probability of event A occurring given that event B has
occurred as P (A|B).
Two events are independent if the occurrence of one event does not affect the probability of
the other event occurring. Mathematically, events A and B are independent if P (A|B) = P (A) or,
equivalently, P (A ∩ B) = P (A)P (B).
Example 4: Suppose you have a deck of 52 playing cards. What is the probability of drawing
a red card (hearts or diamonds) given that you have already drawn a heart?
Solution: Let A be the event of drawing a red card, and let B be the event of having already
drawn a heart. We want to find the conditional probability P (A|B).
In a standard deck, there are 26 red cards (13 hearts and 13 diamonds). Since we have already
drawn a heart, there are now 51 cards left in the deck, with 25 of them being red (12 hearts and
13 diamonds). Therefore, the probability of drawing a red card given that we have already drawn
a heart is:
Number of red cards left
25
P (A|B) =
=
≈ 0.490
(2)
Total number of cards left
51
So, the probability of drawing a red card given that we have already drawn a heart is approximately 0.490.
Exercise 4: If you roll a six-sided die and flip a coin, are the events ”rolling an even number”
and ”flipping a head” independent? Explain your reasoning.
2.2
Statistics
Statistics is the science of collecting, analyzing, and interpreting data. It provides us with the tools
to make sense of the information we gather and draw meaningful conclusions from it.
Key concepts in Statistics include:
• Measures of central tendency (e.g., mean, median, mode)
• Measures of dispersion (e.g., range, variance, standard deviation)
• Probability distributions and their properties
• Estimation and hypothesis testing
• Regression and correlation analysis
2.2.1
Measures of Central Tendency and Dispersion
In Statistics, we use measures of central tendency and dispersion to describe the main features of
a data set.
The mean, often referred to as the average, is calculated by adding up all the numbers in a set
and then dividing by the count of those numbers.
The median is the middle value in a set of numbers. If the set has an odd number of observations,
the median is the middle number. If the set has an even number of observations, the median is the
average of the two middle numbers.
The mode is the number that appears most frequently in a data set. A set may have one mode,
more than one mode, or no mode at all.
Example 5: Given the following test scores: 78, 85, 92, 76, 88, 95, 89, compute the mean, median, and mode.
Solution:
Mean: To calculate the mean, we add up all the test scores and divide by the total number of
scores:
78 + 85 + 92 + 76 + 88 + 95 + 89
603
Mean =
=
≈ 86.1
(3)
7
7
So, the mean test score is approximately 86.1.
Median: To find the median, we first arrange the test scores in ascending order: 76, 78, 85, 88, 89, 92, 95.
Since there are 7 scores, the median is the middle value, which is 88. Mode: The mode is the test
score that occurs most frequently. In this case, all test scores occur only once, so there is no mode.
Therefore, the mean test score is approximately 86.1, the median is 88, and there is no mode.
Exercise 5: Given the following data set: 2, 4, 6, 8, 10, 12, compute the range, variance, and
standard deviation.
2.3
Counting
Counting is fundamental to many areas of mathematics. The basic principle is simple: if we have
a set of objects, we can determine the number of objects in that set. However, when it comes to
arranging, selecting, or ordering these objects, things can get quite complex. Two key concepts in
this area are permutations and combinations.
2.3.1
Permutations
A permutation is an arrangement of objects in a specific order. The number of permutations of n
distinct objects is n!, where ”!” denotes the factorial operation.
Example 1: Consider three books labelled A, B, and C. How many ways can we arrange these
books? Since there are three distinct books, there are 3! = 3 × 2 × 1 = 6 ways to arrange them.
2.3.2
Combinations
A combination is a selection of objects where the order does not matter. The number of combinan!
tions of n objects taken r at a time is given by C(n, r) = r!(n−r)!
.
Example 2: Suppose there are 7 paintings and we want to select 4 to be displayed in an art
gallery. How many different selections can be made? Using the formula for combinations, we find
7!
that there are C(7, 4) = 4!(7−4)!
= 35 ways to make this selection.
2.4
Exercises: Multiple Choice
1. What is the probability that a randomly chosen integer between 1 and 100 (inclusive) is a
multiple of 3 but not a multiple of 5?
A. 10 %
B. 15 %
C. 20 %
D. 27 %
E. 50 %
2. A bag contains 5 red balls, 5 blue balls, and 5 green balls. We draw balls without replacement.
The first ball is red and we take it aside. What is the probability that we draw a red ball
again?
A. 14
B. 25
C. 27
D. 13
E. 15
3. If you flip a fair coin 6 times, what is the probability of getting exactly 3 heads?
5
A. 16
1
B. 64
C. 12
3
D. 32
E. 34
4. Suppose you have a fair 6-sided die with faces numbered 1 through 6. What is the probability
that you roll a prime number?
A. 13
B. 14
C. 12
D. 34
E. Impossible to determine
5. Alice is thinking about a positive integer between 1 and n. For which of the following values
of n the probability of Alice’s number being even is 50%?
A. n=3
B. n=5
C. n=7
D. all of the above
E. none of the above
6. A drawer contains 5 pairs of black socks, 4 pairs of blue socks, and 3 pairs of brown socks. If
you choose socks at random without looking, what is the minimum number of socks you must
choose to be certain that you have chosen at least one pair of socks of the same color?
A. 3
B. 4
C. 6
D. 8
E. 12
7. If a box contains 15 red balls and 10 green balls, what is the minimum number of balls that
must be drawn from the box to guarantee that at least 5 balls are red?
A. 11
B. 12
C. 15
D. 7
E. 6
8. In how many ways can 3 people be selected from a group of 5 people to form a committee?
A. 3
B. 5
C. 7
D. 10
E. 15
9. How many ways are there to arrange the letters in the word “APPLE”?
A. 5
B. 10
C. 24
D. 48
E. 60
10. How many ways are there to arrange 6 books on a shelf?
A. 120
B. 720
C. 30
D. 60
E. 6
3
Polynomials
3.1
Introduction
Polynomials are mathematical expressions involving variables and coefficients, where the variables
are raised to non-negative integer exponents. They are essential building blocks in various branches
of mathematics, including algebra, calculus, and number theory. In this chapter, we will introduce
the concept of polynomials, their roots, and polynomial division. This material is designed to
provide readers with a solid foundation in these concepts.
3.1.1
Polynomials
A polynomial is a mathematical expression of the form:
P (x) = an xn + an−1 xn−1 + ... + a1 x + a0 ,
(4)
where x is a variable, n is a non-negative integer, and an , an−1 , . . . , a1 , a0 are constants called
coefficients. The highest power of x for which the coefficient is non-zero is called the degree of the
polynomial. The coefficient of the highest-degree term, an , is called the leading coefficient.
Some examples of polynomials include:
• P (x) = 2x3 − 5x2 + 3x − 1 (a cubic polynomial with degree 3)
• Q(x) = x2 − 6x + 9 (a quadratic polynomial with degree 2)
• R(x) = 4x − 7 (a linear polynomial with degree 1)
3.1.2
Roots of Polynomials
A root (or zero) of a polynomial is a value of the variable x for which the polynomial evaluates to
zero. In other words, if P (x) is a polynomial and r is a root of P (x), then P (r) = 0. The process
of finding the roots of a polynomial is called solving the polynomial equation.
For example, consider the quadratic polynomial Q(x) = x2 − 6x + 9. To find its roots, we set
Q(x) = 0 and solve for x:
x2 − 6x + 9 = 0.
(5)
This is a quadratic equation, which can be factored as (x − 3)2 = 0. Therefore, the roots are
x = 3, which is a repeated (or double) root.
3.2
Polynomial Division
Polynomial division is the operation of dividing one polynomial by another. This mathematical
process is crucial in understanding more complex algebraic concepts. In this section, we will present
a detailed overview of polynomial division, along with examples and exercises. These materials are
designed to aid readers in familiarizing themselves with this essential technique.
Long division is a method for dividing polynomials that is similar to the long division process
used for dividing numbers. It involves a series of steps in which we repeatedly subtract multiples
of the divisor from the dividend to obtain a remainder.
Example 1: Divide the polynomial P (x) = x3 − 6x2 + 11x − 6 by the polynomial D(x) = x − 2
using long division.
Solution: We set up the long division as follows:
x − 2 x3 −6x2 +11x −6
x3 −2x2
0 −4x2 +11x
−4x2 +8x
0
3x
−6
3x
−6
0
0
After performing the long division, we find that the quotient is Q(x) = x2 − 4x + 8, and the
remainder is 0.
3.3
Exercises
1. Find the roots of the following polynomials:
(a) P (x) = x2 − 5x + 6
(b) Q(x) = x3 − 3x2 − 4x + 12
2. Divide the following polynomials:
(a) P (x) = 2x3 − 5x2 + x + 3 by D(x) = x − 1
(b) Q(x) = x4 − 2x3 + x2 − 3x + 1 by D(x) = x2 − 1
(c) P (x) = x3 + 4x2 − x − 4 by D(x) = x + 1
(d) Q(x) = 3x3 − 2x2 + 5x − 3 by D(x) = x − 1
3.3.1
Multiple Choice Questions
1. Expand xy 2 + x
3x2 + xy − y 2
A. x3 y 2 + 3x3 + x2 y 3 + x2 y + 2xy 4 − xy 2
B. x3 y 2 − 3x3 + x2 y 3 + x2 y − 4xy 4 + 3xy 2
C. 3x3 y 2 + 3x3 + x2 y 3 + x2 y − xy 4 − xy 2
D. −x3 y 2 − x2 y 3 + x2 y − xy 4 − 3xy 2
E. −3x3 + x2 y 3 + x2 y − xy 4 − 10xy 2
2. Expand a2 + ab − 3b (ab + 2a − 1)
A. a3 b + 2a3 + a2 b2 + 2a2 b − a2 − 3ab2 − 7ab + 3b
B. −a3 b + 2a3 + a2 b2 + −a2 − 3ab2 − 7ab − 3b
C. 2a3 b + 2a3 + a2 b2 + −a2 − 3ab2 − 21ab + 3b
D. 3a3 b + 2a3 + 2a2 b − a2 − 3ab2 − 7ab + 9b
E. 2a3 b + 2a3 + a2 b2 + −a2 − 3ab2 − 14ab + 6b
3. What is the product of (x2 − 2)2 and x2 + 2?
A. x4 − 4x2 + 4
B. x2 − 4x + 8
C. x4 − 4
D. x6 − 2x4 − 8
E. x6 − 2x4 − 4x2 + 8
4. Simplify (x2 − 2x)(x + 1) − 2x(x − 3)
A. x3 − 3x2 − 4x
B. x3 − 3x2 + 4x
C. x3 − x2 + 2x − 3
D. x3 − 3x2 + 4x
E. x3 + 8x
5. Simplify (1 + 2x)(1 − 3x) − 2x(x − 2)
A. −4x2 + 5x + 2
B. 8x2 + 2x + 3
C. −8x2 + 3x + 1
D. −2x2 + 2x + 1
E. −x2 + 2x + 3
6. What is the remainder of dividing 2x4 + 3x3 − 2x + 1 by 2x2 − x + 2?
1
A. 11
2 x− 4
1
B. 32 x2 − 11
8 x+ 4
C. 1 − 6x
D. −2x − 7
E. No correct answer.
7. What is the remainder of dividing x4 − 2x3 − 2x + 1 by x2 − 3x + 2?
A. −1 − x
B. 2 − x
C. 1 − 3x
D. −x + 3
E. 2 + x
8. What is the quotient of dividing 2x4 + 3x3 − 2x + 1 by 2x2 − x + 2?
A. x2 − x + 3
B. x2 + x − 2
C. x2 − 2x + 1
D. x2 + 2x
E. x2 − 1
9. What is the quotient of dividing 3x5 + 2x + 1 by x2 + 1?
A. x3 − 2x
B. 3x3 − 3x
C. x3 − x + 1
D. 3x3 + x − 2
E. x3 − 1
10. Find the quotient and the remainder when 3x4 + 2x + 1 is divided by −x + 1.
A. −1 − 3x − x2 − 2x3 and 2 + x
B. −3 − 2x − 3x2 − 4x3 and 2
C. −3x − 7x2 − x3 and 1 + x
D. −2x2 − 3x3 and 1 + x2
E. −5 − 3x − 3x2 − 3x3 and 6
4
Geometry
Geometry is a branch of mathematics that deals with the properties, measurement, and relationships of points, lines, angles, surfaces, and solids. In this chapter, we will focus on some fundamental
geometric shapes: triangles, quadrilaterals, and circles. These shapes form the foundation of more
complex geometric concepts and are essential for readers preparing for university entrance exams.
4.1
Triangles
A triangle is a three-sided polygon with three angles. The sum of the interior angles of a triangle
is always equal to 180◦ . Triangles can be classified based on their sides and angles.
4.1.1
Classification by Sides
• Equilateral triangle: All three sides are of equal length, and all three angles are equal to 60◦ .
• Isosceles triangle: Two sides are of equal length, and the angles opposite those sides are
equal.
• Scalene triangle: All sides have different lengths, and all angles have different measures.
4.1.2
Classification by Angles
• Acute triangle: All three angles are less than 90◦ .
• Right triangle: One angle is equal to 90◦ .
• Obtuse triangle: One angle is greater than 90◦ .
4.2
Quadrilaterals
A quadrilateral is a four-sided polygon. Quadrilaterals can be classified into several types based on
their sides and angles.
• Parallelogram: A quadrilateral with opposite sides parallel and equal in length.
• Rectangle: A parallelogram with all angles equal to 90◦ .
• Rhombus: A parallelogram with all sides equal in length.
• Square: A rectangle with all sides equal in length (also a special case of a rhombus).
• Trapezoid: A quadrilateral with at least one pair of parallel sides.
4.3
Circles
A circle is a closed curve with all points equidistant from a fixed point called the center. Some
important properties and terms related to circles are:
• Radius: The distance from the center to any point on the circle.
• Diameter: The distance across the circle through the center (twice the radius).
• Circumference: The distance around the circle, given by the formula C = 2πr, where r is the
radius.
• Area: The region enclosed by the circle, given by the formula A = πr2 , where r is the radius.
• Arc: A portion of the circumference of the circle.
• Chord: A line segment joining any two points on the circle.
• Tangent: A line that touches the circle at exactly one point.
4.4
Exercises: Multiple Choice
1. What is the maximal possible area of a triangle whose two sides are equal to 7 and 4?
√
A. 4 7
√
B. 7 2
C. 14
√
D. 14 2
E. 28
2. If the longest side of a triangle is 10 cm and two of its angles are 30 degrees and 60 degrees,
what is the measure of the smallest side of the triangle?
√
A. 4 3
B. 5
C. 4
D. 6
E. cannot be determined
3. If two sides of a triangle have lengths 5 and 7, which of the following numbers cannot be the
length of the third side?
A. 3
B. 5
C. 7
D. 11
E. 13
4. Find the length of the radius of the circumscribed circle of a triangle with side lengths of 10
cm, 24 cm, and 26 cm.
A. 13
B. 12
C. 10
D. 8
E. 6
5. A triangle with sides of length 6 cm, 8 cm, and 10 cm has an altitude drawn to the longest
side. What is the length of this altitude?
A. 4
B. 4.6
C. 4.8
D. 5
E. 5.4
6. A circle is inscribed into a trapezoid with bases of length 4 and 10. If the height of trapezoid
is equal to 6, find the radius of the circle.
A. 3
B. 4
C. 5
D. 6
E. Impossible to determine
√
7. In a rhombus, the diagonals are 10 cm and 20 2 cm. What is the radius of the inscribed circle?
√
5 2
A.
2
√
12 2
B.
5
√
10 2
C.
3
√
D. 10 2
E. 10
2π
radians,
8. A quadrilateral ABCD is inscribed in a circle. If the interior angle A measures
3
what is the measure of the interior angle at C?
A. π/2
B. π/4
C. π/3
D. π/6
E. π
9. What is the length of a chord in a circle with radius 6 and central angle of
A. 12
B. 18
√
C. 6 3
D. 6
E. 3
π
radians?
3
10. If a circle has an area of 16π square units, what is the length of its diameter?
A. 4
B. 8
C. 16
√
D. 4 2
√
E. 4 π
5
Functions
Functions constitute a critical notion in the realm of mathematics, delineating the relationship
between two sets of quantities, commonly referred to as input and output values. In this chapter,
we will acquaint the readers with the concept of functions along with their properties. The content
is carefully curated to cater to individuals concluding their school education and transitioning
towards higher academic pursuits in university settings.
5.1
Introduction to Functions
A function is a mathematical relationship that assigns a unique output value to each input value.
We typically denote a function using the notation f (x), where f is the function name and x is
the input value (also called an argument or independent variable). The output value, which is the
result of applying the function to the input value, is often called the dependent variable.
5.1.1
Domain and Range
The domain of a function is the set of all possible input values for which the function is defined,
while the range is the set of all possible output values. Understanding the domain and range of a
function is crucial for solving problems involving functions.
5.1.2
Types of Functions
There are many types of functions, each with its characteristics and applications. Some of the most
common types of functions include:
• Linear functions: Functions of the form f (x) = mx + b, where m is the slope and b is the
y-intercept.
• Quadratic functions: Functions of the form f (x) = ax2 + bx + c, where a, b, c are constants.
• Polynomial functions: Functions that are the sum of monomials, e.g., f (x) = an xn +
an−1 xn−1 + · · · + a1 x + a0 .
• Rational functions: Functions that are the ratio of two polynomial functions, e.g., f (x) = p(x)
q(x) ,
where p(x) and q(x) are polynomial functions.
• Exponential functions: Functions of the form f (x) = ax , where a > 0 and a ̸= 1.
• Logarithmic functions: Functions of the form f (x) = loga x, where a > 0 and a ̸= 1.
• Trigonometric functions: Functions such as sine, cosine, and tangent that relate to angles in
a right triangle or the unit circle.
5.2
Operations on Functions
Functions can be combined or transformed using various operations, such as addition, subtraction,
multiplication, division, and composition. These operations can be used to create new functions
and analyze the relationships between multiple functions.
5.2.1
Function Arithmetic
Given two functions f (x) and g(x), we can perform arithmetic operations as follows:
• (f + g)(x) = f (x) + g(x)
• (f − g)(x) = f (x) − g(x)
• (f · g)(x) = f (x) · g(x)
(x)
• (f /g)(x) = fg(x)
, provided that g(x) ̸= 0
5.2.2
Function Composition
Function composition is the process of applying one function to the output of another function.
Given two functions f (x) and g(x), the composition of f and g is denoted as (f ◦ g)(x) and is
defined as (f ◦ g)(x) = f (g(x)). This operation involves first applying the function g(x) to the
input value x and then applying the function f (x) to the result.
5.3
Inverse Functions
An inverse function is a function that ”undoes” the action of the original function. If f (x) is a
function, its inverse is denoted as f −1 (x), and the relationship between the two functions is such
that f (f −1 (x)) = x and f −1 (f (x)) = x. Not all functions have an inverse, but when they do, the
inverse function is unique.
5.4
Exercises
√
Determine the domain and range of the following functions: (a) f (x) = x − 2
1
(b) g(x) = x+3
Perform the following function arithmetic: (a) f (x) = x2 and g(x) = 2x − 1, find (f + g)(x)
√
(b) h(x) = x and k(x) = x3 , find (h · k)(x)
Given f (x) = 3x − 5 and g(x) = x2 − 1, find the composition (f ◦ g)(x). Find the inverse
function of f (x) = 2x + 1.
5.4.1
Multiple Choice Questions
1. Let f (x) = 3x2 − 2x + 1 and g(x) =
√
x + 1. What is g(f (2))?
A. 3
√
B. 13
√
C. 7
D. −1
√
E. 10
x+1
2. What is the domain of the function f (x) = √
?
x2 − 9
A. x ∈ (−∞, −3) ∪ (−3, 3) ∪ (3, ∞)
B. x ∈ (−∞, −3) ∪ (3, ∞)
C. x ∈ (−∞, 3) ∪ (3, ∞)
D. x ∈ [−3, 3]
E. x ∈ (3, ∞)
3. What are the roots of the function f (x) = 3x3 − 9x2 + 6x?
A. x = 1, 32
B. x = 0, 1, 2
C. x = 3, 6
D. x = 0, 23
E. x = 0, 12 , 3
4. For which values of x the function f (x) = x2 − 5x + 6 is increasing?
A. (2.5, ∞)
B. (−∞, 3)
C. (−∞, 2) ∪ (3, ∞)
D. (−2, −3)
E. (2, 2.5) ∪ (3, ∞)
2x − 1
. What is the inverse of f (x)?
x+3
x+3
A. f −1 (x) = 2x−1
5. Let f (x) =
x−3
B. f −1 (x) = 2x+1
x+1
C. f −1 (x) = 2x−3
D. f −1 (x) = 3x+1
2−x
x−1
E. f −1 (x) = 2x+3
6. Let f (x) = x2 − 3x + 2 and g(x) = −x + 1. For what values of x is f (x) ≤ g(x)?
A. x ≥ 1
B. 1 ≤ x < 2
C. x ≤ −1
D. x = 1
E. x = 3
7. Which of the following functions is odd?
A. x2 + 2
B. x2 + cos(x)
1
C.
2x
D. ex + e−x
√
E. x2 + 1
8. Which of the following is a possible equation of a function that passes through the point (1, 4)
and (2, 6)?
A. f (x) = x + 3
B. f (x) = 2x + 2
C. f (x) = x2 + 3x + 2
x+2
D. f (x) =
x+1
E. f (x) = 4x2
√
9. For which of the following values of x is the function f (x) =
A. x = 3
B. x = 10
C. x = 100
D. x = 13
E. x = 5
10. For what value of x is the function f (x) =
A. x = −1
B. x = −5
C. x = 13
D. x = −3
E. x = 1
x−1
equal to 3?
x+3
x−4
x−2 undefined?
6
Series, Powers & Roots
Series, powers, and roots hold significant importance in the field of mathematics. Obtaining a
solid understanding of these concepts is essential for individuals advancing to higher-level academic
pursuits. In this chapter, we will elucidate these topics, providing a comprehensive insight into
both arithmetic and geometric series. This discussion will be supplemented with carefully designed
examples and exercises, aimed at promoting a profound understanding in this area.
6.1
Introduction to Series
A series is the sum of the terms in a sequence. There are many types of series, but in this chapter,
we will focus on arithmetic and geometric series.
6.1.1
Arithmetic Series
An arithmetic series is the sum of the terms in an arithmetic sequence. An arithmetic sequence is
a list of numbers where each term is obtained by adding a constant difference, called the common
difference (d), to the previous term. The sum of the first n terms of an arithmetic series can be
calculated using the following formula:
n
(a1 + an )
(6)
2
where Sn is the sum of the first n terms, a1 is the first term, and an is the nth term.
Example: Find the sum of the first 10 terms of the arithmetic series 4, 7, 10, . . .
In this example, a1 = 4, the common difference d = 3, and n = 10. To find the 10th term, we
use the formula an = a1 + (n − 1)d, so a10 = 4 + (10 − 1)(3) = 31. Now, we can find the sum using
the formula (1):
S10 = 10
2 (4 + 31) = 5 · 35 = 175
Sn =
6.1.2
Geometric Series
A geometric series is the sum of the terms in a geometric sequence. A geometric sequence is a list
of numbers where each term is obtained by multiplying the previous term by a constant factor,
called the common ratio (r). The sum of the first n terms of a geometric series can be calculated
using the following formula:
a1 (rn − 1)
(7)
r−1
where Sn is the sum of the first n terms, a1 is the first term, and r is the common ratio.
Example: Find the sum of the first 5 terms of the geometric series 2, 6, 18, . . .
In this example, a1 = 2, the common ratio r = 3, and n = 5. We can find the sum using the
formula (2):
5 −1)
S5 = 2(33−1
= 2(243−1)
= 242
2
Sn =
6.2
Powers and Roots
Powers and roots are essential mathematical concepts that deal with repeated multiplication and
the inverse operation of finding the root of a number.
6.2.1
Powers
A power is the result of multiplying a number by itself a certain number of times. The number
being multiplied is called the base, and the number of times it is multiplied is called the exponent.
We write a power as ab , where a is the base and b is the exponent.
6.2.2
Roots
A root is the inverse operation of raising a number to a power. The nth root of a number a is
√
written as n a, and it is the number that, when raised to the power of n, equals a. The most
common root is the square root (n = 2), but other roots, such as the cube root (n = 3), can also
be calculated.
Example: Find the square root and cube root of√the number 64.
√
In this example, we want to find the square root 64 and the cube root 3 64. The square root
of 64 is 8 because 82 = 64. The cube root of 64 is 4 because 43 = 64.
6.3
Exercises
Find the sum of the first 15 terms of the arithmetic series 3, 10, 17, . . . Find the sum of the first 6
terms of the geometric series 5, −10, 20, . . . Calculate the following powers: (a) 25
(b) (−3)4
√
Calculate
the following roots: (a) 81
√
(b) 3 27
6.3.1
Multiple Choice Questions
1
1. Which of the following expressions is equal to (a4 b3 c6 ) 12 ?
A. a3 b4 c2
√
6
B. a2 b3 c2
√ √
4
C. a · bc2
√ √ √
D. 3 a · 4 b · c
√
E. 3 a · bc2
2. Which of the following expressions is equivalent to (2xy 2 )3 ?
A. 8x3 y 6
B. 6x3 y 6
C. 12x3 y 5
D. 8x2 y 3
E. 12x2 y 5
3. Which of the following expressions is equivalent to (4x2 y 3 )−2 ?
1
A.
16x4 y 6
16
B. 4 6
x y
x2 y 2
16
16
D. 2 3
x y
1
E. 4 6
x y
C.
4. Find the value of a4 + b4 if a and b are the roots of the quadratic equation x2 + 3x − 5 = 0.
A. 212
B. 526
C. 311
D. 421
E. 356
5. Find the sum of all roots of the polynomial (x2 + x + 1)(x − 1)(x − 2)(x + 1)(x + 3).
A. 5
B. -1
C. 0
D. 1
E. -2
6. Find the product of all roots of the polynomial x4 + x3 − 7x2 − x + 6.
A. -1
B. 6
C. -12
D. 1
E. 7
7. Find the sum of all the odd numbers between 1 and 49 inclusive.
A. 625
B. 675
C. 600
D. 700
E. 715
8. Find the sum of the first 20 terms of an arithmetic progression where the first term is 5 and
the common difference is 3.
A. 590
B. 620
C. 670
D. 710
E. 700
9. What is the sum of the first 5 terms of the sequence −4, 8, −16, 32, . . .?
A. -14
B. -28
C. -24
D. -34
E. -44
10. What is the sum of the first 10 terms of the sequence 1, 12 , 14 , 18 , . . .?
A. 510
256
B. 1023
512
C. 513
256
511
D. 1024
E. 1023
1024
7
Logical Thinking
Logical thinking constitutes a pivotal ability for individuals transitioning towards advanced academic engagements in universities. In this chapter, we will illuminate the principles of logical
thinking and reasoning. The material is supplemented with examples and exercises meticulously
crafted to foster the development of logical reasoning abilities among the readers. Acquiring these
skills will prove instrumental for their forthcoming mathematics courses and achieving academic
success in higher education settings.
7.1
Introduction to Logical Thinking
Logical thinking is the process of using reasoning and critical thinking skills to analyze information and draw valid conclusions. It involves the use of various principles and techniques, such as
deduction, induction, logical connectives, and conditional statements, to evaluate statements and
make decisions based on the given information.
7.2
Deductive Reasoning
Deductive reasoning is a method of reasoning in which a conclusion follows necessarily from the
given premises. In other words, if the premises are true, the conclusion must be true as well.
Deductive reasoning is often used to analyze statements and derive conclusions based on given
facts.
7.2.1
Example 1
All mammals have hair. A dog is a mammal. Therefore, a dog has hair.
In this example, the conclusion (a dog has hair) follows necessarily from the premises (all
mammals have hair and a dog is a mammal). If the premises are true, the conclusion must be true
as well.
7.2.2
Example 2
If it is raining, then the ground is wet. It is raining. Therefore, the ground is wet.
In this example, the conclusion (the ground is wet) follows necessarily from the premises (if it
is raining, then the ground is wet and it is raining). If the premises are true, the conclusion must
be true as well.
7.3
Inductive Reasoning
Inductive reasoning is a method of reasoning in which a conclusion is likely to be true based on the
given evidence or observations, but it is not guaranteed. Inductive reasoning is often used to make
predictions or generalizations based on a limited set of observations.
7.3.1
Example 1
Every time you’ve seen a swan, it has been white. Therefore, you conclude that all swans are white.
In this example, the conclusion (all swans are white) is likely to be true based on the observations
(every time you’ve seen a swan, it has been white), but it is not guaranteed to be true.
7.3.2
Example 2
You have observed that the sun rises every morning. Based on this observation, you conclude that
the sun will rise tomorrow.
In this example, the conclusion (the sun will rise tomorrow) is likely to be true based on the
observations (the sun rises every morning), but it is not guaranteed to be true.
7.4
Logical Connectives
Logical connectives are words or phrases that are used to link statements or propositions together.
Some common logical connectives include ”and,” ”or,” ”not,” ”if. . . then,” and ”if and only if.”
Understanding these connectives and their properties is essential for logical thinking and reasoning.
7.4.1
Example 1
Statement 1: I will go to the store.
Statement 2: I will buy milk.
Compound statement using ”and”: I will go to the store and buy milk.
In this example, the compound statement is true only if both Statement 1 and Statement 2 are
true.
7.4.2
Example 2
Statement 1: It is snowing.
Statement 2: It is raining.
Compound statement using ”or”: It is snowing or raining.
In this example, the compound statement is true if either Statement 1 or Statement 2 is true.
7.5
Conditional Statements
Conditional statements are statements that take the form ”if. . . then.” In a conditional statement,
if the ”if” part (called the antecedent) is true, then the ”then” part (called the consequent) must
also be true. Understanding conditional statements is important for logical thinking and reasoning,
as they are often used to express relationships between different statements or propositions.
7.5.1
Example 1
If it is raining, then the ground is wet.
In this example, the antecedent is ”it is raining,” and the consequent is ”the ground is wet.” If
it is true that it is raining, then it must also be true that the ground is wet.
7.5.2
Example 2
If a number is divisible by 2, then it is even.
In this example, the antecedent is ”a number is divisible by 2,” and the consequent is ”it is
even.” If it is true that a number is divisible by 2, then it must also be true that the number is
even.
7.6
Exercises
Now that we have introduced some of the main concepts in logical thinking and reasoning, let’s
practice with some exercises. These exercises will help you become more familiar with the material
and better prepared for the types of questions you might encounter in your entrance exams.
1. Identify whether the following statements use deductive or inductive reasoning:
(a) All birds can fly. Penguins are birds. Therefore, penguins can fly.
(b) Every time you’ve seen a cat, it has had a tail. Therefore, you conclude that all cats have
tails.
(c) If a shape has four sides and all sides are equal in length, then it is a square. A shape
has four sides and all sides are equal in length. Therefore, it is a square.
2. Determine the truth value of the following compound statements:
(a) Statement 1: The Earth is round. Statement 2: The Moon is made of cheese. Compound
statement using ”and”: The Earth is round and the Moon is made of cheese.
(b) Statement 1: 5 is an odd number. Statement 2: 6 is an even number. Compound
statement using ”or”: 5 is an odd number or 6 is an even number.
3. Write the converse, inverse, and contrapositive of the following conditional statements:
(a) If it is hot outside, then I will wear shorts.
(b) If a number is prime, then it has exactly two factors.
7.6.1
Multiple Choice Questions
1. When three clients are seated in an ice cream shop, the server asks them: “Does everyone want
ice cream?” The first client says “I don’t know.” The second client then says “I don’t know.”
Finally, the third client says “No, not everyone wants ice cream.” Which of the following is
true?
A. The third client wants an ice-cream
B. The first and second clients do not wants an ice-cream
C. The third client does not want an ice-cream
D. The first and the third client wants an ice-cream
E. It is undecidable
2. Three friends are at a pizza restaurant. The waiter asks them, “Does everyone want pizza?”
The first friend says, “I don’t know.” The second friend says, “I don’t know.” The third friend
says, “No, not everyone wants pizza.” Which of the following is true?
A. The third friend wants a pizza
B. The first and second friends do not want a pizza
C. The third friend does not want a pizza
D. The first and the third friends wants a pizza
E. It is undecidable
3. Three friends are at a movie theater. The ticket attendant asks them, ”Does everyone want to
see the movie?” The first friend says, ”I don’t know.” The second friend says, ”I don’t know.”
The third friend says, ”No, not everyone wants to see the movie.” Which of the following is
true?
A. The third friend wants to see the movie
B. The first and second friend do not want to see the movie
C. The third friend does not want to see the movie
D. The first and the third friend want to see the movie
E. It is undecidable
4. Three coworkers are at a café. The barista asks them, “Does everyone want coffee?” The first
coworker says, “I don’t know.” The second coworker says, “I don’t know.” The third coworker
says, ”No, not everyone wants coffee.” Which of the following is true?
A. The third coworker wants a coffee
B. The first and second coworkers do not want a coffee
C. The third coworker does not want a coffee
D. The first and the third coworker want a coffee
E. It is undecidable
5. Three siblings are at an amusement park. The ride operator asks them, “Does everyone want
to ride the roller coaster?” The first sibling says, “I don’t know.” The second sibling says, “I
don’t know.” The third sibling says, ”No, not everyone wants to ride the roller coaster.” Which
of the following is true?
A. The third sibling wants to ride
B. The first and second siblings do not want to ride
C. The third sibling does not want to ride
D. The first and the third siblings wants to ride
E. It is undecidable
6. Emma would like to determine the relative heights of three friends using two facts. First, she
knows that if Sarah is not the tallest of the three, then Rachel is. Second, she knows that
if Rachel is not the shortest, then Leah is the tallest. Is it possible to determine the relative
heights of Sarah, Rachel, and Leah from what Emma knows? If so, who is the tallest and who
is the shortest?
A. It is possible to determine the relative heights, and Sarah is the tallest,
while Rachel is the shortest.
B. It is possible to determine the relative heights, and Rachel is the tallest, while Leah
is the shortest.
C. It is possible to determine the relative heights, and Rachel is the tallest, while Sarah
is the shortest.
D. It is not possible to determine the relative heights from the given information.
E. It is possible to determine the relative heights, and Leah is the tallest, while Rachel
is the shortest.
7. A teacher is trying to determine the ages of three students. The teacher knows that if John
is not the youngest, then Tom is. The teacher also knows that if Tom is not the oldest, then
Mary is the youngest. Can the teacher determine the ages of John, Tom, and Mary? If so, who
is the oldest and who is the youngest?
A. It is possible to determine the ages, and Mary is the oldest, while Tom is the youngest.
B. It is possible to determine the ages, and John is the oldest, while Tom is the youngest.
C. It is possible to determine the ages, and Tom is the oldest, while John is
the youngest.
D. It is not possible to determine the ages from the given information.
E. It is possible to determine the ages, and Mary is the oldest, while John is the youngest.
8. Sarah is trying to determine the order of weight of three boxes, A, B, and C. Sarah knows that
if Box B is not the heaviest, then Box A is. She also knows that if Box A is not the lightest,
then Box C is the heaviest. Can Sarah determine the order of weight of the boxes? If so, which
box is the heaviest and which is the lightest?
A. It is possible to determine the order of weight, and Box A is the heaviest, while Box
C is the lightest.
B. It is possible to determine the order of weight, and Box B is the heaviest, while Box
C is the lightest.
C. It is possible to determine the order of weight, and Box B is the heaviest,
while Box A is the lightest.
D. It is not possible to determine the order of weight from the given information.
E. It is possible to determine the order of weight, and Box C is the heaviest, while Box
B is the lightest.
9. A museum has 15 paintings, each of which was painted by a different artist. The curator knows
that at least one painting is from the 20th century, and given any two paintings, at least one
was painted before the 19th century. Based on these facts, can you determine how many of the
paintings were painted in the 20th century?
A. not possible to determine
B. 8
C. 7
D. 2
E. 1
10. A university has 20 professors, each of whom teaches a different course. The dean knows that at
least one professor teaches a science course, and given any two professors, at least one teaches a
humanities course. Based on these facts, can you determine how many professors teach science
courses?
A. not possible to determine
B. 11
C. 10
D. 9
E. 1