UPSC CDS 2 2021 Math Paper

Given, x is a negative number.
1. "There is some natural number k such that kx>0′′.
When a negative number (x) is multiplied by a natural number (k), then the resulting number will be negative.
∴kx>0 is not correct.
2. " x2+x>0 always".
‌ For ‌x=−‌1∕2
⇒x2+x=(−‌1∕2)2+(−‌1∕2)
‌‌=‌1∕4−‌1∕2=−‌1∕4<0
∴x2+x>0‌ is not alwavs correct. ‌
3. " 2x<x<−x "
⇒2x<x
and x<−x
⇒2× (negative number) <( same negative number ) and negative number < positive number [∵−x is positive ]
∴2x<x<−x is always correct.
4. " x2 is always a rational number".
Let x=−(2)1∕4 then, x2=[−(2)1∕4]2
=+21∕4×2=21∕2=√2,
which is irrational.
∴x2 is always a rational number is an incorrect statement.
Hence, 1,2 and 4 are not correct.