Exercise: 1-F
Assertion – Reason Questions
Q1: Assertion (A): Rational numbers are closed under addition and multiplication but not under subtraction.
Reason (R): For any two rational numbers a and b, a + b is a rational number, a – b is a rational number, a × b is a rational number.
Step-by-step Analysis:
✅ Closure under addition:
Let a = 2⁄3, b = 4⁄5 ⇒ a + b = 22⁄15 ∈ ℚ ✅
✅ Closure under multiplication:
a × b = 8⁄15 ∈ ℚ ✅
✅ Closure under subtraction:
a – b = -2⁄15 ∈ ℚ ✅
🔍 So, subtraction is also closed for rational numbers. Assertion is **false**.
Reason says all 3 operations (addition, subtraction, multiplication) are closed ⇒ Reason is **true**.
Correct Answer: d. Assertion (A) is false but Reason (R) is true.
Q2: Assertion (A): The negative (additive inverse) of -5 is 5 and the negative of 5 is -5.
Reason (R): The negative of a negative rational number is negative and the negative of a positive rational number is negative.
Step-by-step Analysis:
✅ Additive inverse of -5 = 5 ✅
✅ Additive inverse of 5 = -5 ✅
Assertion is **true**.
🔍 Reason says: “negative of negative is negative” → ❌ False
Actually, -(-5) = 5, which is **positive**.
So, Reason is **false**.
Correct Answer: c. Assertion (A) is true but Reason (R) is false.
Q3: Assertion (A): For any rational numbers x, y, z, we have, x × (y + z) = x × y + x × z
Reason (R):Rational numbers are associative under addition and multiplication.
Step-by-step Analysis:
✅ Let x = 1⁄2, y = 2⁄3, z = 1⁄6
LHS = x × (y + z) = 1⁄2 × (5⁄6) = 5⁄12
RHS = (x × y) + (x × z) = 1⁄2 × 2⁄3 + 1⁄2 × 1⁄6 = 1⁄3 + 1⁄12 = 5⁄12 ✅
This is **Distributive Property**, not Associative.
Reason talks about associativity, not distributivity.
✔️ Both Assertion and Reason are **true**, but Reason is **not** correct explanation.
Correct Answer: b. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
