CBSE Class 11 Mathematics Mathematical Reasoning Assignment Set 01

VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)

Identify which of the following are statements (Q. No 1 to 7)

Question 1. Prime factors of 6 are 2 and 3.
Answer: Statement
In simple words: A sentence that is objectively either true or false is a mathematical statement.

Exam Tip: Check if the sentence has a definitive truth value. Since 6 can be factored into prime numbers 2 and 3, this is a true statement.

Question 2. \(x^2 + 6x + 3 = 0\)
Answer: Not a statement
In simple words: An open sentence containing a variable whose truth depends on the value of x is not a statement.

Exam Tip: Equations with variables are open sentences and do not qualify as statements until the variable is specified.

Question 3. The earth is a planet.
Answer: Statement
In simple words: It is a declarative sentence that is universally true.

Exam Tip: General facts about the universe expressed as declarative sentences are always valid statements.

Question 4. There is no rain without clouds.
Answer: Statement
In simple words: It is an assertive sentence with a definite truth value.

Question 5. All complex numbers are real numbers.
Answer: Statement
In simple words: This is a false mathematical assertion, making it a valid statement.

Exam Tip: A statement does not need to be true; it just needs to be definitively either true or false.

Question 6. Tomorrow is a holiday.
Answer: Not a Statement
In simple words: The truth value depends on when and where it is uttered, so it is not a statement.

Exam Tip: Sentences containing relative time references like 'tomorrow' or 'yesterday' without context are open or ambiguous.

Question 7. Answer this question.
Answer: Not a statement
In simple words: Imperative sentences and commands cannot be true or false.

Exam Tip: Commands, requests, and questions are never statements.

Write negation of the following statements (Q. No 8 to 12)

Question 8. All men are mortal.
Answer: All men are not mortal (or: It is not the case that all men are mortal)
In simple words: The negation denies the universal claim by stating not all men are mortal.

Exam Tip: Use phrases like "It is not the case that" or change the quantifier when negating universal statements.

Question 9. \(\pi\) is not a rational number.
Answer: \(\pi\) is a rational number.
In simple words: Removing the word 'not' gives the negation of the statement.

Exam Tip: The negation of a negated statement is the original affirmative statement.

Question 10. Every one in Spain speaks Spanish.
Answer: Everyone in Spain doesn't speak Spanish (or: There is someone in Spain who does not speak Spanish).
In simple words: To negate 'everyone', we assert that at least one person does not.

Exam Tip: Negation of 'for all' involves 'there exists not'.

Question 11. Zero is a positive number.
Answer: Zero is not a positive number.
In simple words: Adding 'not' forms the direct negation.

Exam Tip: Direct denial is achieved by inserting 'not' into the predicate.

Write the component statements of the following compound statements

Question 12. 7 is both odd and prime number.
Answer: Component 1: 7 is an odd number.
Component 2: 7 is a prime number.
In simple words: The compound statement is split into two independent simple statements joined by 'and'.

Exam Tip: Identify the connective ('and', 'or') and break the sentence into individual complete thoughts.

Question 13. All integers are positive or negative.
Answer: Component 1: All integers are positive.
Component 2: All integers are negative.
In simple words: The component statements are separated at the logical connective 'or'.

Question 14. 36 is a multiple of 4, 6 and 12.
Answer: Component 1: 36 is a multiple of 4.
Component 2: 36 is a multiple of 6.
Component 3: 36 is a multiple of 12.
In simple words: Multiple items combined with 'and' form three distinct component statements.

Exam Tip: Watch out for lists joined by commas and conjunctions.

Question 15. Jack and Jill went up the hill.
Answer: Component 1: Jack went up the hill.
Component 2: Jill went up the hill.
In simple words: The compound subject creates a compound sentence structure.

Exam Tip: Even though it looks like a simple sentence with a compound subject, logically it represents two distinct assertions.

Identify the type ‘Or’ (Inclusive or Exclusive) used in the following statements (Q. No. 16 to 19)

Question 16. Students can take French or Spanish as their third language.
Answer: Exclusive
In simple words: A student typically chooses only one of the two languages, not both.

Exam Tip: 'Or' is exclusive when both options cannot happen simultaneously.

Question 17. To enter in a country you need a visa or citizenship card.
Answer: Inclusive
In simple words: Having either or both documents permits entry.

Exam Tip: 'Or' is inclusive if satisfying either or both conditions fulfills the requirement.

Question 18. \(\sqrt{2}\) is a rational number or an irrational number.
Answer: Exclusive
In simple words: A number cannot be both rational and irrational at the same time.

Exam Tip: Mutually exclusive mathematical categories use the exclusive 'or'.

Question 19. 125 is a multiple of 5 or 8.
Answer: Exclusive
In simple words: 125 is a multiple of 5, but not of 8. Since it cannot be a multiple of both simultaneously in this context of disjoint divisibility, it is exclusive.

Exam Tip: Test if both conditions can be true at once; if not, it is exclusive.

Which of the following statements are true or false. Give Reason. (Question No. 20 to 23)

Question 20. 48 is a multiple of 6, 7 and 8
Answer: False, 48 is not a multiple of 7.
In simple words: For a compound statement with 'and' to be true, all parts must be true. Since 48 is not divisible by 7, the statement is false.

Exam Tip: Check every component of an 'and' statement.

Question 21. \(\pi > 2\) and \(\pi < 3\).
Answer: False, \(\pi\) lies between 3 and 4 (\(\pi \approx 3.14\)).
In simple words: Since \(\pi\) is approximately 3.14, the condition \(\pi < 3\) is false.

Exam Tip: Know the approximate values of common mathematical constants like \(\pi\) and \(e\).

Question 22. Earth is flat or it revolves around the moon.
Answer: False
In simple words: Both component statements are false, making the disjunction false.

Exam Tip: An 'or' statement is false only when all component statements are false.

Question 23. \(\sqrt{2}\) is a rational number or an irrational number.
Answer: True
In simple words: Since \(\sqrt{2}\) is an irrational number, at least one part of the 'or' statement is true.

Question 24. For every integer p, \(\sqrt{p}\) is a real number.
Answer: For every (Universal quantifier)
In simple words: The phrase 'for every' specifies that the statement applies to all integers.

Exam Tip: Universal quantifiers include words like 'every', 'all', and 'for all'.

Question 25. There exists a capital for every country in the world.
Answer: For every, there exists (Universal and Existential quantifiers)
In simple words: 'For every' is the universal quantifier and 'there exists' is the existential quantifier.

Exam Tip: Statements can contain more than one quantifier.

Question 26. There exists a number which is equal to its square.
Answer: There exists (Existential quantifier)
In simple words: 'There exists' indicates that at least one such number (e.g., 0 or 1) can be found.

Exam Tip: Existential quantifiers include 'there exists', 'for some', and 'at least one'.

Write the converse of the following statements (Q. No. 27 to 30)

Question 27. If a number x is even then \(x^2\) is also even.
Answer: If \(x^2\) is even then x is even.
In simple words: The converse of 'If p then q' is 'If q then p'.

Exam Tip: Swap the hypothesis and conclusion to form the converse.

Question 28. If \(3 \times 7 = 21\) then \(3 + 7 = 10\)
Answer: If \(3 + 7 = 10\) then \(3 \times 7 = 21\)
In simple words: Reversing the order of implication yields the converse.

Exam Tip: Do not change the truth values of the statements while writing the converse, just swap their positions.

Question 29. If n is a prime number then n is odd.
Answer: If n is odd then n is a prime number.
In simple words: Swapping hypothesis and conclusion gives the converse statement.

Exam Tip: The converse of a true statement is not necessarily true.

Question 30. Some thing is cold implies that it has low temperature.
Answer: If something has low temperature then it is cold.
In simple words: Reversing the direction of the implication creates the converse.

Question 31. If \(5 > 7\) then \(6 > 7\).
Answer: If \(6 \le 7\) then \(5 \le 7\).
In simple words: The contrapositive of 'If p then q' is 'If ~q then ~p'.

Exam Tip: Negate both parts and swap their positions: \(p \implies q\) becomes \(\sim q \implies \sim p\).

Question 32. x is even number implies that \(x^2\) is divisible by 4.
Answer: If \(x^2\) is not divisible by 4 then x is not even.
In simple words: Negate the conclusion and hypothesis, then reverse them.

Exam Tip: Contrapositive is logically equivalent to the original conditional statement.

Question 33. Check the validity of the statement 'An integer x is even if and only if \(x^2\) is even.'
Answer: Valid
In simple words: We check both directions: (i) If x is even, \(x^2\) is even. (ii) If \(x^2\) is even, x is even. Both are true, so the biconditional statement is valid.

Exam Tip: For 'if and only if' statements, prove both the forward implication and the converse.