Case Study / Scenario-Based MCQs for Sub-Topics of Topic 11: Mathematical Reasoning

Question 1. Consider the following sentences:

(i) The capital of Rajasthan is Jaipur.

(ii) What is your height?

(iii) $x^2 - 4 = 0$

(iv) Every natural number is an integer.

(v) This sentence is false.

Which of these sentences are mathematical statements (propositions)?

(A) (i), (ii), (iii)

(B) (i), (iv), (v)

(C) (i), (iv)

(D) (ii), (iii), (v)

Question 2. A student is told that a sentence is a proposition if it has a definite truth value (True or False). They are given the sentence: "The current Chief Minister of Uttar Pradesh is a woman." To determine if this is a proposition, what must the student ascertain?

(A) Whether the sentence is grammatically correct.

(B) Whether they know who the current Chief Minister is.

(C) Whether the sentence is either definitively True or definitively False.

(D) Whether the sentence is interesting or relevant to mathematics.

Question 3. Your friend makes the statement: "All birds can fly." You know this statement is not universally true (e.g., penguins cannot fly). Based on the definition of a mathematical statement's truth value, how would you classify your friend's statement?

(A) It is a true statement because most birds can fly.

(B) It is a false statement because there is at least one bird that cannot fly.

(C) It is an open sentence because "birds" can refer to different types.

(D) It is not a mathematical statement because it is about animals, not numbers.

Question 4. Consider the sentence "$y$ is taller than 160 cm". This is an open sentence. For this sentence to become a proposition, what needs to be done?

(A) Replace $y$ with a specific person's name (e.g., "Aisha is taller than 160 cm").

(B) Change it to a question (e.g., "Is $y$ taller than 160 cm?").

(C) Add "always" or "never" (e.g., "$y$ is always taller than 160 cm").

(D) Add a logical connective like "and" or "or".

Question 5. A teacher asks the class to provide examples of mathematical statements. Student A says "$2+3=5$". Student B says "Go and solve this problem!". Student C says "Is $\sqrt{4}$ equal to 2?". Student D says "Every square has four sides". Which students gave valid examples of mathematical statements?

(A) Student A and Student B

(B) Student A and Student C

(C) Student A and Student D

(D) Student B and Student C

Question 1. A statement $p$ is given as "The house is red". You need to write the negation of this statement. Which of the following is the correct negation?

(A) The house is blue.

(B) The house is not red.

(C) Is the house red?

(D) The house was red.

Question 2. Consider the statement "All students like ice cream". How would you write the negation of this statement?

(A) All students dislike ice cream.

(B) No student likes ice cream.

(C) Some students like ice cream.

(D) Some students do not like ice cream.

Question 3. Let $p$ be the statement "The weather is pleasant" and $q$ be the statement "I will go for a walk". You want to express the compound statement "The weather is pleasant and I will go for a walk". Which symbol correctly represents this?

(A) $p \lor q$

(B) $p \land q$

(C) $\neg p \land q$

(D) $p \implies q$

Question 4. You are given the compound statement: "The number is even or the number is divisible by 3". How would you identify the simple statements and the connective used?

(A) Simple statements: "The number is even", "the number is divisible by 3". Connective: AND.

(B) Simple statements: "The number is even", "the number is divisible by 3". Connective: OR.

(C) Simple statement: "The number is even or divisible by 3". No connective.

(D) Simple statements: "even", "divisible by 3". Connective: OR.

Question 5. Suppose the statement "$2 > 5$" is given. This statement is false. What is the truth value of the negation of this statement, $\neg(2 > 5)$?

(A) True

(B) False

(C) Cannot be determined

(D) Depends on the numbers used

Question 1. Consider two statements: $p$: "It is cold", $q$: "It is snowing". What is the truth value of the compound statement "$p \land q$" (It is cold and it is snowing) if it is currently cold but not snowing?

(A) True

(B) False

(C) Cannot be determined

(D) Depends on the location

Question 2. Suppose you are offered a choice: "You can have tea or coffee." Using the inclusive OR, under what circumstances is the statement "You have tea or you have coffee" true?

(A) You have tea but not coffee.

(B) You have coffee but not tea.

(C) You have both tea and coffee.

(D) All of the above.

Question 3. You are constructing a truth table for the compound statement $\neg p \lor q$. If $p$ is true and $q$ is false, what is the truth value of $\neg p \lor q$?

(A) True

(B) False

(C) Same as $p \land q$

(D) Same as $p \lor q$

Question 4. You need to create a truth table for the statement $(p \land q) \lor r$. How many rows will this truth table have?

(A) 3

(B) 6

(C) 8

(D) 4

Question 5. Consider the statement "The train is late or the bus is delayed". Let $p$ be "The train is late" and $q$ be "The bus is delayed". If both the train is on time and the bus is on time, what is the truth value of the compound statement?

(A) True

(B) False

(C) Cannot be determined

(D) Depends on why they were late

Question 1. Your mother tells you, "If you finish your homework (h), then you can play outside (p)". Suppose you finish your homework, and your mother lets you play outside. Is her statement true or false in this scenario?

(A) True

(B) False

(C) Cannot be determined

(D) Depends on how long you play

Question 2. Consider the statement "If a number is divisible by 4 (p), then it is divisible by 2 (q)". This statement is true. Suppose a number is NOT divisible by 2. Based on the truth table for implication, what can you say about whether the number is divisible by 4?

(A) It must be divisible by 4.

(B) It must not be divisible by 4.

(C) It might be divisible by 4 or not.

(D) The statement $p \implies q$ is irrelevant here.

Question 3. A mathematics definition states: "A triangle is equilateral if and only if all its sides are equal length". Let $p$ be "A triangle is equilateral" and $q$ be "All its sides are equal length". If a triangle has sides of length 3 cm, 3 cm, and 4 cm, is the biconditional statement $p \iff q$ true or false for this triangle?

(A) True, because it is a triangle.

(B) False, because the triangle is not equilateral and its sides are not all equal length.

(C) True, because the statement is a definition.

(D) False, because $p$ is false and $q$ is false.

Question 4. Your friend says, "I will buy a laptop if I get a bonus". Let $p$ be "I get a bonus" and $q$ be "I will buy a laptop". Which of the following correctly represents your friend's statement?

(A) $p \land q$

(B) $p \implies q$

(C) $q \implies p$

(D) $p \iff q$

Question 5. Consider the statement "You pass the exam if and only if you score at least 40 marks". If a student scores 35 marks, what is the truth value of the biconditional statement applied to this student?

(A) True

(B) False

(C) Cannot be determined

(D) Depends on the subject

Question 1. Given the statement "If a shape is a square (s), then it is a rectangle (r)". This statement is true. Which of the following is its converse?

(A) If a shape is a rectangle, then it is a square.

(B) If a shape is not a square, then it is not a rectangle.

(C) If a shape is not a rectangle, then it is not a square.

(D) A shape is a square and it is not a rectangle.

Question 2. Consider the true statement: "If a number is a multiple of 10 (m), then it is a multiple of 5 (f)". Which of the following related statements is also guaranteed to be true?

(A) Converse: If a number is a multiple of 5, then it is a multiple of 10.

(B) Inverse: If a number is not a multiple of 10, then it is not a multiple of 5.

(C) Contrapositive: If a number is not a multiple of 5, then it is not a multiple of 10.

(D) Negation: A number is a multiple of 10 and it is not a multiple of 5.

Question 3. You are given the statement "If it rains (r), then I will carry an umbrella (u)". How would you form the inverse of this statement?

(A) If I carry an umbrella, then it rains.

(B) If it does not rain, then I will not carry an umbrella.

(C) If I do not carry an umbrella, then it does not rain.

(D) It rains and I do not carry an umbrella.

Question 4. Consider the statement "$p \implies q$". Which of the following related statements is logically equivalent to the original statement?

(A) Converse ($q \implies p$)

(B) Inverse ($\neg p \implies \neg q$)

(C) Contrapositive ($\neg q \implies \neg p$)

(D) Negation ($p \land \neg q$)

Question 5. A theorem is stated as "If a triangle is equilateral, then it is isosceles". Which of the following statements, derived from this theorem, is also a theorem (i.e., is true)?

(A) If a triangle is isosceles, then it is equilateral (Converse).

(B) If a triangle is not equilateral, then it is not isosceles (Inverse).

(C) If a triangle is not isosceles, then it is not equilateral (Contrapositive).

(D) A triangle is equilateral and it is not isosceles (Negation).

Question 1. Consider the statement "All students in this class are present today". Which type of quantifier is implicitly used in this statement?

(A) Existential Quantifier ($\exists$)

(B) Universal Quantifier ($\forall$)

(C) Negation ($\neg$)

(D) Conjunction ($\land$)

Question 2. A biologist states, "There exists a species of fish that can walk on land". Which type of quantifier is used in this statement?

(A) Universal Quantifier ($\forall$)

(B) Existential Quantifier ($\exists$)

(C) Both quantifiers

(D) No quantifier

Question 3. A teacher claims, "Every student passed the test". You know this might not be true. How would you state the negation of this claim?

(A) Every student did not pass the test.

(B) No student passed the test.

(C) Some students passed the test.

(D) Some students did not pass the test.

Question 4. Consider the statement "Some numbers are positive". How would you write the negation of this statement?

(A) Some numbers are not positive.

(B) All numbers are positive.

(C) No number is positive (or All numbers are not positive).

(D) There exists a number that is not positive.

Question 5. The statement "For every integer $n$, $n+1 > n$" uses the universal quantifier. What is the truth value of this statement over the domain of integers?

(A) True

(B) False

(C) Depends on the integer $n$

(D) Cannot be determined

Question 6. The statement "There exists a number $x$ such that $x^2 = -1$" is false in the domain of real numbers. What is the truth value of its negation in the domain of real numbers?

(A) True

(B) False

(C) Cannot be determined

(D) Depends on the number $x$

Question 1. You construct a truth table for a compound statement and find that the final column contains only 'True' values. How would you classify this statement?

(A) Contradiction

(B) Contingency

(C) Tautology

(D) Valid Argument

Question 2. You construct a truth table for the statement $p \land \neg p$. What will the final column of the truth table contain?

(A) All True values

(B) All False values

(C) A mix of True and False values

(D) Cannot be determined without more information

Question 3. Consider two statements $P$ and $Q$. You find that in every possible truth assignment for the simple propositions within $P$ and $Q$, $P$ and $Q$ have the same truth value. How would you describe the relationship between $P$ and $Q$?

(A) They are both tautologies.

(B) They are both contradictions.

(C) They are logically equivalent.

(D) They are contingencies.

Question 4. You are given the statement $(p \lor q) \land \neg p$. You construct its truth table. What type of statement will it be classified as (assuming it's not always True or always False)?

(A) Tautology

(B) Contradiction

(C) Contingency

(D) Logical Equivalence

Question 5. Which of the following is a correct method to check if two statements $P$ and $Q$ are logically equivalent?

(A) Check if $P \land Q$ is a tautology.

(B) Check if $P \lor Q$ is a tautology.

(C) Check if $P \implies Q$ is a tautology.

(D) Check if the biconditional $P \iff Q$ is a tautology.

Question 1. You want to prove the statement "If an integer $n$ is odd, then $n+1$ is even". Which proof method would be the most straightforward approach?

(A) Proof by Contradiction

(B) Proof by Contrapositive

(C) Direct Proof

(D) Proof by Counterexample

Question 2. To prove the statement "If $n^2$ is even, then $n$ is even" using proof by contrapositive, what statement would you actually prove?

(A) If $n$ is even, then $n^2$ is even.

(B) If $n$ is odd, then $n^2$ is odd.

(C) If $n^2$ is odd, then $n$ is odd.

(D) Assume $n^2$ is even and derive a contradiction.

Question 3. You are asked to prove the statement "There is no largest prime number". Which proof technique is commonly used and most suitable for this type of existence/non-existence proof?

(A) Direct Proof

(B) Proof by Contrapositive

(C) Proof by Contradiction

(D) Proof by Induction

Question 4. Consider the argument: "If it is raining (p), I will use an umbrella (q). It is raining (p). Therefore, I will use an umbrella (q)." This argument form is known as Modus Ponens. Is this a valid argument form?

(A) Yes

(B) No

(C) Only if the premises are true.

(D) Only if the conclusion is true.

Question 5. You are trying to validate the argument: Premises: $P_1, P_2, \dots, P_n$. Conclusion: $C$. You construct a truth table and find a row where all premises $P_1, \dots, P_n$ are true, but the conclusion $C$ is false. What does this tell you about the argument?

(A) The argument is valid.

(B) The argument is sound.

(C) The argument is invalid.

(D) The premises must be false.

Question 6. What is the primary purpose of a mathematical proof?

(A) To make a statement more believable.

(B) To demonstrate that a statement is true under specific conditions.

(C) To provide an absolute, logically certain verification of the truth of a statement based on axioms and logic.

(D) To convince someone that a statement is correct using examples.