Exercise: 1-E
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.
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.
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.
Answer: b. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
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