UPSC NDA I 2019 Mathematics Solved Paper

(D)
R:N → N , xRy ⇒ x‌2 − yxy + 3 y‌2 = 0
⇒( x−y ) ( x−3 y )=0
(i) Reflexive
xRx ⇐ x‌2 − yxx+3 x‌2 = 0⇐ 0 = 0 Identify
⇒ R is reflexive
(ii) Symmetric
For any a∈N (3a,a) ∈ R[(3 a)‌2−4 (3 a) (a)+3 a‌2=0]
But (a , 3 a) ∉ R[a‌2−4 (a) (3 a)+3 (3 a)‌2 = 16 a‌2 ≠ 0 ]
⇒ R is not symmetric
(iii) Transitive
For any a∈N (9 a , 3 a )∈R,(3 a , R) ∈R
But (9a,a) ∉R
⇒ R is not transitive