Assertion-Reason MCQs for Sub-Topics of Topic 11: Mathematical Reasoning

Question 1. Assertion (A): The sentence "What is your favourite colour?" is not a mathematical statement.

Reason (R): A mathematical statement must be a declarative sentence that is either true or false, but not both.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The sentence "$x + 2 = 5$" is not a proposition.

Reason (R): The truth value of "$x + 2 = 5$" depends on the value of the variable $x$.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The statement "The sum of two odd integers is an odd integer" is false.

Reason (R): The sum of two odd integers is always an even integer.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The sentence "Close the door" is a mathematical statement.

Reason (R): A mathematical statement must be a command.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The truth value of the statement "$2 \times 3 = 6$" is True.

Reason (R): The truth value of a mathematical statement is determined by its consistency with mathematical definitions and axioms.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The sentence "He is a rich person" is not a proposition.

Reason (R): The term "rich" is subjective and lacks a precise, universally agreed-upon definition in this context.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The negation of "All dogs can swim" is "No dogs can swim".

Reason (R): The negation of a universal statement is an existential statement with the predicate negated.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If statement $p$ is true, then its negation $\neg p$ is false.

Reason (R): The negation operator reverses the truth value of a statement.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): "It is raining and it is cold" is a compound statement.

Reason (R): Compound statements are formed by connecting simple statements using logical connectives like "and", "or", etc.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The negation of "$a \geq b$" is "$a < b$".

Reason (R): The negation of "$P$" is "not $P$", which covers all cases where $P$ is false.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The statement "Some apples are green" is the negation of "Some apples are not green".

Reason (R): The negation of "Some P are Q" is "No P are Q" or "All P are not Q".

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The symbol for negation is $\land$.

Reason (R): The symbol $\land$ represents the logical connective "AND".

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): If $p$ is true and $q$ is false, then $p \land q$ is false.

Reason (R): The conjunction $p \land q$ is true only when both $p$ and $q$ are true.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If $p$ is false and $q$ is true, then $p \lor q$ is true (using inclusive OR).

Reason (R): The disjunction $p \lor q$ is false only when both $p$ and $q$ are false.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The truth table for $p \lor q$ has 4 rows.

Reason (R): A truth table for a compound statement with $n$ distinct simple propositions has $2^n$ rows.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The statement "You can have tea or coffee" in a menu implies you cannot have both.

Reason (R): In mathematics, "or" ($\lor$) usually represents the inclusive OR, meaning "one or the other, or both".

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The truth value of $\neg (\neg p)$ is the same as the truth value of $p$.

Reason (R): Applying negation twice cancels out the effect of the negation.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The symbol $\lor$ represents conjunction.

Reason (R): Conjunction corresponds to the English word "AND".

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The statement "If it is raining, then the ground is wet" is false if it is raining and the ground is not wet.

Reason (R): The conditional statement $p \implies q$ is false precisely when the antecedent $p$ is true and the consequent $q$ is false.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If $p$ is false, then the conditional statement $p \implies q$ is always true, regardless of the truth value of $q$.

Reason (R): The truth table for $p \implies q$ shows that when $p$ is false, $p \implies q$ is true.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The statement "$p$ is a sufficient condition for $q$" is equivalent to "$q \implies p$".

Reason (R): "$p$ is sufficient for $q$" means that if $p$ is true, $q$ must be true, which is the definition of $p \implies q$.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The biconditional statement $p \iff q$ is true only when $p$ and $q$ have the same truth value.

Reason (R): $p \iff q$ is logically equivalent to $(p \implies q) \land (q \implies p)$.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The phrase "only if" in a statement "$p$ only if $q$" indicates that $q$ is a necessary condition for $p$.

Reason (R): The logical form of "$p$ only if $q$" is $p \implies q$.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The statement "If $2+2=5$, then the Earth is flat" is false.

Reason (R): A conditional statement is only false when the antecedent is true and the consequent is false.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The contrapositive of a conditional statement $p \implies q$ is $\neg q \implies \neg p$.

Reason (R): A conditional statement and its contrapositive are logically equivalent.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If the converse of a statement is true, then the original statement must also be true.

Reason (R): The converse and inverse of a conditional statement are logically equivalent to each other.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The inverse of "If a number is even (p), then it is divisible by 2 (q)" is "If a number is odd, then it is not divisible by 2".

Reason (R): The inverse of $p \implies q$ is formed by negating both the antecedent and the consequent, resulting in $\neg p \implies \neg q$.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): If the contrapositive of a statement is false, then the original statement is also false.

Reason (R): A statement and its contrapositive have the same truth value.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The converse of "If it is Sunday (p), then the market is closed (q)" is "If the market is closed, then it is Sunday".

Reason (R): The converse is formed by swapping the antecedent and the consequent of the original conditional statement.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The inverse of a statement $p \implies q$ is logically equivalent to its converse $q \implies p$.

Reason (R): Both the inverse and the converse are formed by negating and/or swapping parts of the original conditional statement.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The symbol $\forall$ is the existential quantifier.

Reason (R): The existential quantifier is used to state that a property holds for at least one element in the domain.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The negation of "Some birds can fly" is "No birds can fly".

Reason (R): The negation of an existential statement ($\exists x, P(x)$) is a universal statement with the predicate negated ($\forall x, \neg P(x)$).

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The statement "For every integer $n$, $n^2 \geq 0$" is true.

Reason (R): The square of any real number is non-negative, and integers are real numbers.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The statement "There exists a prime number that is even" is true.

Reason (R): The number 2 is a prime number and it is also an even number.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The negation of "Every student passed" is "Every student failed".

Reason (R): Negating "Every" involves changing it to "Some" and negating the predicate.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The statement "Some triangles are equilateral" uses an existential quantifier.

Reason (R): Phrases like "some", "at least one", and "there exists" indicate an existential quantifier.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The statement $p \lor \neg p$ is a tautology.

Reason (R): A tautology is a statement that is always true, regardless of the truth values of its simple components.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The statement $p \land \neg p$ is a contingency.

Reason (R): A contingency is a statement that is sometimes true and sometimes false.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): Two statements $P$ and $Q$ are logically equivalent if and only if $P \iff Q$ is a tautology.

Reason (R): Logical equivalence means the statements have the same truth value in all possible cases.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): To construct a truth table for a statement with 3 simple propositions ($p, q, r$), you need 6 rows.

Reason (R): The number of rows in a truth table is determined by $2^n$, where $n$ is the number of simple propositions.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The statement $(p \land q) \implies p$ is a tautology.

Reason (R): In a conditional statement, if the antecedent is false, the statement is always true.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): If a compound statement is a contradiction, its negation is a tautology.

Reason (R): The truth values in the negation's truth table column are the exact opposite of the original statement's column.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): To prove the statement "If $n$ is odd, then $n^2$ is odd" using direct proof, you assume $n$ is odd and show $n^2$ must be odd.

Reason (R): Direct proof involves assuming the truth of the antecedent and logically deducing the truth of the consequent.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): Proof by contrapositive of $P \implies Q$ is done by proving $Q \implies P$.

Reason (R): The contrapositive of $P \implies Q$ is $\neg Q \implies \neg P$, which is logically equivalent to $P \implies Q$.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): In a proof by contradiction for a statement $P$, you assume $P$ is false and derive a contradiction.

Reason (R): Deriving a contradiction shows that the initial assumption (that $P$ is false) must be incorrect, thus $P$ must be true.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The argument form Modus Ponens is valid.

Reason (R): Modus Ponens states that if $p \implies q$ is true and $p$ is true, then $q$ must be true.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): An argument with true premises must have a true conclusion to be considered valid.

Reason (R): Validity only means that IF the premises are true, THEN the conclusion must be true. It does not guarantee the truth of the premises themselves.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): To disprove a universally quantified statement $\forall x, P(x)$, you need to find just one element $x_0$ such that $\neg P(x_0)$ is true.

Reason (R): A single counterexample is sufficient to show that a universal statement is false.

(A) Both A and R are true, and R is the correct explanation of A.

(B) Both A and R are true, but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.