If m and n are direct natural numbers, then which of the following is/are integer/integers (1) m/n + n/m (2) mn × (m/n + n/m) × (m2 + n2)-1 (3) mn/ (m2 + n2) Select the correct answer using the code given below. ?

1 and 2
2 only
2 and 3
3 only

LohaHridoy

4 views

2025-01-04 04:42

1 Answers

AhmedNazir

Option 2 : 2 only

Statement I:

m/n + n/m = (m² + n²) /mn

Let this be an integer k

⇒ m² + n² = kmn

⇒ (m² + n² – 2mn) = (k – 2) × mn

⇒ (m – n)² = (k – 2) × mn

⇒ m – n = √

⇒ √ for a integer, requires a specific combination of (m, n)

For example, if m = 1 and n = 2

⇒ ½ + 2 = 5/2 not an integer

Statement II:

mn × (m/n + n/m) × (m² + n²)^-1

⇒ mn × × (m² + n²)^-1 = 1

⇒ It is an integer

Statement III:

It is the reciprocal of the expression in statement I

When m = 1 and n = 2

The value will be 2/5 which is not an integer

∴ Only statement 2 is correct.

4 views Answered 2022-09-04 04:29

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