0^{' '}. When a negative number (x) is multiplied by a natural number (k), then the resulting number will be negative. ∴ k x > 0 is not correct. 2. " x^{2}+x > 0 always". \text{ For } x=- {1}\/{2} \Rightarrow x^{2}+x=(- {1}\/{2})^{2}+(- {1}\/{2}) = {1}\/{4}- {1}\/{2}=- {1}\/{4} 0 \text{ is not alwavs correct. } 3. " 2 x" >
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.
