Question 2

# 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

Solution

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.