Download Reasoning and Proof Review Questions

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Reasoning and Proof Review Questions
Inductive Reasoning from Patterns
Multiple Choice Questions.
For each question, four alternative choices are given, of which only one is correct. You have to select the correct
alternative and mark it in the appropriate option.
1. What is the next term in the pattern: 1, 4, 9, 16, 25, 36, 49...?
(a)
(b)
(c)
(d)
81
64
121
56
2. What is the next term in the pattern 2, 6, 18, 54, 162?
(a)
(b)
(c)
(d)
486
324
244
None of the above
3. Complete the sequence: 1, -2, 4, -8...
(a)
(b)
(c)
(d)
32
-16
16
-32
4. What is the next number in the sequence: 243, 81, 27, 9...
(a)
(b)
(c)
(d)
0
3
1
1/3
5. Examples of inductive reasoning include _____.
(a)
(b)
(c)
(d)
Visual patterns
Number patterns
Both A, B
None of the above
True or False Questions.
Indicate True or False for the following Statements
6. Deductive reasoning is the process of drawing conclusions based on examples and patterns. ( True/False )
7. Inductive reasoning differs from deductive reasoning in that a general conclusion is drawn by observing
specific examples. ( True/False )
8. Inductive reasoning is the opposite of induction or inductive logic. ( True/False )
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Conjectures and Counterexamples
Multiple Choice Questions.
For each question, four alternative choices are given, of which only one is correct. You have to select the correct
alternative and mark it in the appropriate option.
1. _____ are conjectures disproven through counterexample.
(a)
(b)
(c)
(d)
Pseudo conjectures
False conjectures
Proxy conjectures
None of the above
2. An example that disproves a conjecture is a _____
(a)
(b)
(c)
(d)
A counterexample
Pseudo conjectures
Proxy conjectures
None of the above
3. Find the best counterexample from the given options to show that the following conjecture is false: the
difference of two integers is less than either integer.
(a)
(b)
(c)
(d)
Difference between 7 and 8
Difference between 9 and 2
Difference between 3 and 4
None of the above
4. Which of the following proves that a conjecture is true?
(a)
(b)
(c)
(d)
Deductive reasoning
Counterexample
Both A, B
None of the above
True or False Questions.
Indicate True or False for the following Statements
5. All shapes with four sides of the same length are squares. ( True/False )
6. The counterexample to the statement, "All students are lazy," is, "Any hard working student." ( True/False )
7. If the product of two numbers is even, then the two numbers must be even. The counterexample to this
statement are 7 and 4. ( True/False )
8. 2 is a counterexample to the statement, "All prime numbers are odd numbers." ( True/False )
Deductive Reasoning
Multiple Choice Questions.
For each question, four alternative choices are given, of which only one is correct. You have to select the correct
alternative and mark it in the appropriate option.
1. Determine the right conclusion from the following statements: I. A shape that has more than 2 sides is a
polygon. II. A regular polygon’s sides and angles are all congruent. III. An equilateral triangle has 3 congruent
sides and 3 congruent angles.
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(a)
(b)
(c)
(d)
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All triangles are polygons.
An equilateral triangle is a regular polygon.
A rectangle with sides 2, 2, 4, and 4 is not a regular polygon.
All of the above can be concluded.
2. Using the two equations, determine the relationship between ’p’ and ’q.’ 2p + 52 = p + 3 II. Q − 52 = 1.
Therefore:
(a)
(b)
(c)
(d)
q> p
p>q
p=q
None
3. Which of the following laws would be used to draw a logical conclusion from the statements: If you are not
in London, then you can’t be on the bridge. Ben is in London.
(a)
(b)
(c)
(d)
Law of Contrapositive
Law of Detachment
Both A, B
None of the above
4. Under which of the following laws is a conclusion reached by combining the hypothesis of one statement with
the conclusion of another?
(a)
(b)
(c)
(d)
Law of detachment
Law of syllogism
Law of induction
None of the above
5. Determine the conclusion from the following statements: If it is snowing, then I wear a cap. I am not wearing
a cap.
(a)
(b)
(c)
(d)
It is snowing
It is not snowing
It is winter
None of the above
True or False Questions.
Indicate True or False for the following Statements
6. The difference between inductive and deductive arguments lies in the strength of the evidence that the premises
do not provide for the conclusion, and have nothing to do with the content or subject matter of the argument.
( True/False )
7. ’P,’ ’Q,’ and ’R,’ are collinear points. ’Q’ is equidistant from ’P’ and ’R.’ Therefore, ’Q’ must be the midpoint
between ’P’ and ’R.’ ( True/False )
8. In deductive reasoning, a specific conclusion can be drawn from a general truth. ( True/False )
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If-Then Statements
Multiple Choice Questions.
For each question, four alternative choices are given, of which only one is correct. You have to select the correct
alternative and mark it in the appropriate option.
1. If a = b and b = c, then
(a)
(b)
(c)
a>c
a<c
a=c
(d)
None o f the above
2. Two lines are perpendicular lines if they intersect to form a _____.
(a)
(b)
(c)
(d)
Obtuse angle
Right Angle
Supplementary angle
None of the above
3. A conclusion is the result of a(n) _____.
(a)
(b)
(c)
(d)
Hypothesis
Premise
Argument
None of the above
4. If x = 10, then 10x is 100.
(a)
(b)
(c)
(d)
TRUE
FALSE
You should meditate
None of the above
5. If a rectangle has four equal side lengths, then it is a _____.
(a)
(b)
(c)
(d)
Square
Rectangle
Polygon
None of the above
6. Give the hypothesis and conclusion of the following statement: If x = 29, then x − 9 = 20.
(a)
(b)
(c)
(d)
No conclusion
The statement is ambiguous.
Hypothesis: x = 29; Conclusion: x - 9 = 20
None of the above
True or False Questions.
Indicate True or False for the following Statements
7. If x = 4, then 3x = 12. ( True/False )
8. A triangle has three sides. ( True/False )
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Converse, Inverse, and Contrapositive
Multiple Choice Questions.
For each question, four alternative choices are given, of which only one is correct. You have to select the correct
alternative and mark it in the appropriate option.
1. Consider the statement, "If two angles are congruent, then they have the same measure". What is the converse
of the statement?
(a)
(b)
(c)
(d)
If two angles have the same measure, then they are congruent.
If two angles are not congruent, then they do not have the same measure.
If two angles do not have the same measure, then they are not congruent.
None of the above
2. In the statement, "If it rains, then they will cancel school",[U+009D] what is the hypothesis?
(a)
(b)
(c)
(d)
They will cancel School
It rains
They will
None of the above
3. Consider the statement, "If it rains, then I do not go outside." Which of the following could be a logical
conclusion that we can draw from this statement?
(a)
(b)
(c)
(d)
If it does not rain, then I can go fishing
If I do not go fishing, then it rains
If I go fishing, then it does not rain
None of the above
4. For any statement where A implies B, not-B _____ implies not-A.
(a)
(b)
(c)
(d)
Always
Not always
Sometimes
None of the above
5. Consider the statement, "All blue objects have color." Its inverse is _____.
(a)
(b)
(c)
(d)
If an object has color, then it is blue
There exists a blue object that does not have the properties of color
If an object is not blue, then it does not have color
None of the above
True or False Questions.
Indicate True or False for the following Statements
6. "If it is cold, then I will wear my woolen clothes." The converse of the preceding statement is, "If I wear my
woolen clothes, then it is not cold." ( True/False )
7. "If something is a bat, then it is a mammal." The contrapositive to this statement is, "If something is a mammal,
then it is not a bat." ( True/False )
8. The converse and inverse of a statement are always true. ( True/False )
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Properties of Equality and Congruence
Multiple Choice Questions.
For each question, four alternative choices are given, of which only one is correct. You have to select the correct
alternative and mark it in the appropriate option.
1. If DE = ED then it is a _____.
(a)
(b)
(c)
(d)
Reflexive property of equality
Symmetric property of equality
Addition Property of equality
None of the above
2. According to the Right Angle Congruence Theorem, all _____ angles are congruent.
(a)
(b)
(c)
(d)
Right
Acute
Obtuse
None of the above
3. If 3b = 18, then
(a)
(b)
(c)
(d)
3b
3
=
18
3
or b = 6. What property makes the statement true?
Division Property of Equality
Substitution property of equality
Symmetric property of equality
Addition Property of equality
4. If two angles form a linear pair, then they are:
(a)
(b)
(c)
(d)
Right angles
Obtuse angles
Supplementery angles
None of the above
5. In a triangle, the largest angle is across from the _____.
(a)
(b)
(c)
(d)
Shortest side
Longest side
Shortest angle
None of the above
6. The sum of interior angles of triangle is _____.
(a)
(b)
(c)
(d)
90 degrees
45 degrees
180 degrees
None of the above
True or False Questions.
Indicate True or False for the following Statements
7. If a = b, then b = a is a symmetric property. ( True/False )
8. If a quadrilateral is a parallelogram, then the opposite sides are perpendicular. ( True/False )
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Two-Column Proofs
Multiple Choice Questions.
For each question, four alternative choices are given, of which only one is correct. You have to select the correct
alternative and mark it in the appropriate option.
1. We need to explicitly state that BD = BD. This idea that something is congruent to itself is called the:
(a)
(b)
(c)
(d)
Reflexive property of equality
Symmetric property of equality
Addition Property of equality
Transitive property
2. In higher-level mathematics, _____ are usually written in paragraph form.
(a)
(b)
(c)
(d)
Proofs
Hypothesis
Axioms
None of the above
3. If 3x + 4 = 16 then x = _____.
(a)
(b)
(c)
(d)
12
4
13
3
4. In parallelogram ABCD, angle ABD is congruent to angle BDC because if we have parallel lines and a
transversal, we know that:
(a)
(b)
(c)
(d)
Alternate interior angles are congruent
same-side interior angles are congruent
corresponding angles are congruent
exterior angles are congruent
True or False Questions.
Indicate True or False for the following Statements
5.
6.
7.
8.
The Vertical Angles Theorem states that vertical angles are congruent. ( True/False )
A statement that is proved is often called a hypothesis. ( True/False )
If a = b, and a + b = 10, and a = 5, then b = 10. ( True/False )
To clarify and emphasize the effectiveness of an argument in geometry, a two-column approach has a list of
statements and the corresponding reasons supporting the truth of these statements is often adapted. ( True/False
)