If a is between 0 and 1, which of the following statements is (are) true?
(i) $$a^2$$ - 1> 0
(ii) $$a^2$$ + 1 > 0
(iii) $$a^2$$ - a > 0
Given 0<a<1..............................................(1)
Let us go with the verification of options
Option (i):
squaring on both sides of (1), we get
0<$$a^2$$<1
Subtracting '1' in the above equation, we get
-1<$$a^2$$-1<0....................(2)
$$\Rightarrow a^2$$-1< 0, Hence option (i) is false.
Option (ii):
Squaring on both sides of (1), we get
0<$$a^2$$<1
Adding '1' on both sides of above equation, we get
1<$$a^2$$+1<2....................(3)
$$\Rightarrow a^2$$+1>0, Hence Option (ii) is true.
Option (iii):
As 0<a<1, $$a^2<a$$.............(4)
Subtracting 'a' on both sides of (4), we get
$$a^2-a<0$$, Hence Option (iii) is also false.
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