Question 1: State whether the following sentences are mathematically acceptable statements. Give reasons for your answers.

(i) The sum of all interior angles of a triangle is 180°.
(ii) Today is a very cold day.
(iii) For any real number x, x² ≥ 0.

Solution:

(i) Yes, it is a mathematically acceptable statement because it is always true for any Euclidean triangle.

(ii) No, it is not a mathematically acceptable statement because 'very cold' is subjective and relative; it varies from person to person and location to location.

(iii) Yes, it is a mathematically acceptable statement because the square of any real number is always non-negative (either zero or positive), making it unambiguously true.

Question 2: Disprove the following conjecture by finding a counterexample: "For every positive integer n, the expression n² + n + 11 is a prime number."

Solution:

To disprove a conjecture, we need to find at least one positive integer 'n' for which n² + n + 11 is not a prime number.

Let us test values of n:

• For n = 1: 1² + 1 + 11 = 13 (Prime)
• For n = 2: 2² + 2 + 11 = 17 (Prime)
• For n = 10: 10² + 10 + 11 = 121

Notice that 121 = 11 × 11, which is a composite number (not prime).

Conclusion: n = 10 serves as a counterexample. Hence, the statement is false.

Question 3: Write the converse of the statement: "If a number is divisible by 10, then it is divisible by 5." State whether the converse is true or false, giving a reason.

Solution:

Given Statement: If a number is divisible by 10, then it is divisible by 5.

Converse Statement: "If a number is divisible by 5, then it is divisible by 10."

Validity of Converse: The converse is False.

Reason / Counterexample: Consider the number 15. It is divisible by 5 (15 / 5 = 3), but it is NOT divisible by 10 (15 / 10 = 1.5). Thus, divisibility by 5 does not guarantee divisibility by 10.

Question 4: Prove by contradiction that √2 is an irrational number.

Solution:

Step 1: Assumption
Let us assume on the contrary that √2 is a rational number.

Step 2: Representation
Then √2 can be written in the form a/b, where 'a' and 'b' are co-prime integers (having no common factor other than 1) and b ≠ 0.

√2 = a/b

Squaring both sides:
2 = a²/b² ⇒ a² = 2b² --- (Equation 1)

Step 3: Deduction 1
Since 2 divides a², by Theorem, 2 must also divide 'a'.
So, we can write a = 2c for some integer 'c'.

Step 4: Substitution
Substitute a = 2c in Equation 1:
(2c)² = 2b² ⇒ 4c² = 2b² ⇒ b² = 2c²

Step 5: Deduction 2
This means 2 divides b², which implies 2 must also divide 'b'.

Step 6: Contradiction
From Deductions 1 and 2, 'a' and 'b' have at least 2 as a common factor. But this contradicts our initial assumption that 'a' and 'b' are co-prime (have no common factor other than 1).

Conclusion: Our assumption that √2 is rational was wrong. Therefore, √2 is an irrational number. Hence proved.

Question 5: Using deductive reasoning, prove that if two lines intersect each other, then vertically opposite angles are equal.

Solution:

Given: Two lines AB and CD intersect at point O.

To Prove:
(i) ∠AOC = ∠BOD
(ii) ∠AOD = ∠BOC

Proof:
1. Ray OA stands on line CD.
Therefore, ∠AOC + ∠AOD = 180° (Linear pair axiom) --- (Equation 1)

2. Ray OD stands on line AB.
Therefore, ∠AOD + ∠BOD = 180° (Linear pair axiom) --- (Equation 2)

3. From Equation 1 and Equation 2:
∠AOC + ∠AOD = ∠AOD + ∠BOD

Subtracting ∠AOD from both sides:
∠AOC = ∠BOD

4. Similarly, by considering ray OB on line CD, we get:
∠AOD = ∠BOC

Hence, by deductive reasoning from the linear pair axiom, vertically opposite angles are equal. Hence proved.