UPSC CDS 2 2021 Math Paper

Given inequalities:
1. ‌

a2−b2

a2+b2

>‌

a−b

a+b

where a>b>0
i.e., a and b are positive and a>b.
⇒‌

(a−b)(a+b)

a2+b2

>‌

(a−b)

(a+b)

⇒‌

a+b

a2+b2

>‌

1

a+b

[∵(a−b) is a positive number ]
⇒(a+b)2>a2+b2
⇒a2+b2+2ab>a2+b2
⇒2ab>0
which is true. ‌‌[∵a>0 and b>0]
2. ‌

a3+b3

a2+b2

>‌

a2+b2

a+b

only when a>b>0
Since, a and b are positive.
⇒(a+b)(a3+b3)
⇒a4+ab3+ba3+b4
⇒‌‌>a4+b4+2a2b2
⇒ab(b2+a2)−2a2b2>0
⇒‌‌ab[b2+a2−2ab]>0
⇒‌‌ab(a−b)2>0
∴ If a and b are positive then the given inequalities forms into ab(a−b)2>0, which is true but here, it is not necessary that a>b. The inequality is true for b>a>0 also.
i.e., a>b>0 is not the only condition.
Hence, option (a) is correct.
2. ‌

a3+b3

a2+b2

>‌

a2+b2

a+b

only when a>b>0
Since, a and b are positive.
∴ If a and b are positive then the given inequalities forms into ab(a−b)2>0, which is true but here, it is not necessary that a>b.
The inequality is true for b>a>0 also.
i.e., a>b>0 is not the only condition.
Hence, option (a) is correct.