UPSC CDS 1 2021 Math Paper

Given:
1. If x is directly proportional to z and y is directly proportional to z, then (x2−y2) is directly proportional to z2.
2. If x is inversely proportional to z and y is inversely proportional to z, then ( xy ) is inversely proportional to z2
Calculation
Case I.
According to the question
x∝z and y∝z
⇒x=zk and y=rz where k and r are constants ⇒x2=z2k2 and y2=r2z2 ⇒x2−y2=z2(k2−r2)
⇒x2−y2∝z2
Hence, Case I is correct.
Case II.
x∝1∕z and y∝1∕z
⇒xz=k and yz=r where k and r are constants
⇒xz.yz=kr
⇒xyz2=kr
⇒xy=kr∕z2
⇒xy∝1∕z2
Hence, Case II is correct.