You can simplify the first statement in following way:

(1) \(x^2-Y^2=2y+1\)

i.e. \(x^2 = Y^2 + 2y+1\)

i.e. \(x^2 = (y+1)^2\)

i.e. \(x=|y+1| = y + 1\)(taking positive root as both x & y are positive integers)

Hence (1) is SUFFICIENT as x & y are consecutive

(2) \(x^2-xy-x=0\)

i.e. \(x(x-y-1)=0\)

i.e. \(x=0\) or \(x=y+1\)

As x is positive integer x<>0, thus \(x=y+1\)

Hence (2) is SUFFICIENT.

Choice (D) is the answer.

_________________

Thanks,

Prashant Ponde

Tough 700+ Level RCs: Passage1 | Passage2 | Passage3 | Passage4 | Passage5 | Passage6 | Passage7

Reading Comprehension notes: Click here

VOTE GMAT Practice Tests: Vote Here

PowerScore CR Bible - Official Guide 13 Questions Set Mapped: Click here

Looking to finance your tuition: Click here