Consider the following1. \(\rm \sqrt{-a} × \sqrt{-b} = \sqrt{ab}\)2.

1.	Consider the following1. \(\rm \sqrt{-a} × \sqrt{-b} = \sqrt{ab}\)2. i4m+3 = iWhich of the above statement is/are correct?1. Only 12. Only 23. Both 1and 24. Neither 1 nor 2
Answer» Correct Answer - Option 4 : Neither 1 nor 2 Concept: For any two real numbers a and b, the result \(\rm \sqrt{a} × \sqrt{b} = \sqrt{ab}\)is true only when at least one of the given numbers is either zero or positive. i = \(\rm \sqrt{-1}\), i² = -1, i3 = -i, i⁴ = 1 Calculation: 1. We know, \(\rm \sqrt{a} × \sqrt{b} = \sqrt{ab}\) only when a, b ≥ 0 \(\rm \sqrt{-a} × \sqrt{-b} =\rm \sqrt{-1}\sqrt{a} × \sqrt{-1}\sqrt{b} \) = \(\rm \sqrt{ab}\) × (i × i) ....(∵ i = \(\rm \sqrt{-1}\)) = -\(\rm \sqrt{ab}\) ....(∵ i² = -1) 2. i4m+3 i4m+3 = i^4mi³ = i³....(∵ i^4m = 1) = -i ....(∵ i³ = -i) So, both the statements are not correct. Hence, option (4) is correct.

Consider the following1. \(\rm \sqrt{-a} × \sqrt{-b} = \sqrt{ab}\)2. i4m+3 = iWhich of the above statement is/are correct?1. Only 12. Only 23. Both 1and 24. Neither 1 nor 2

Answer» Correct Answer - Option 4 : Neither 1 nor 2

Concept:

For any two real numbers a and b, the result \(\rm \sqrt{a} × \sqrt{b} = \sqrt{ab}\)is true only when at least one of the given numbers is either zero or positive.

i = \(\rm \sqrt{-1}\),

i² = -1,

i3 = -i,

i⁴ = 1

Calculation:

1. We know, \(\rm \sqrt{a} × \sqrt{b} = \sqrt{ab}\) only when a, b ≥ 0

\(\rm \sqrt{-a} × \sqrt{-b} =\rm \sqrt{-1}\sqrt{a} × \sqrt{-1}\sqrt{b} \)

= \(\rm \sqrt{ab}\) × (i × i) ....(∵ i = \(\rm \sqrt{-1}\))

= -\(\rm \sqrt{ab}\) ....(∵ i² = -1)

2. i4m+3

i4m+3 = i^4mi³

= i³....(∵ i^4m = 1)

= -i ....(∵ i³ = -i)

So, both the statements are not correct.

Hence, option (4) is correct.

Discussion

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