Is |a| > a?

1. a^{2} < a

2. (a/2) > (2/a)

The magnitude of ‘a’ will be greater than a only if ‘a’ is a negative number. For positive numbers the magnitude of ‘a’ will be equal to ‘a’.

So, what we have to determine using the two statements is whether a is negative.

Statement 1: a^{2} < a

This inequality holds good only when 0 < a < 1. i.e., we can conclude that a is positive.

Hence, we can answer the question – whether a is negative with a definite NO.

Therefore, statement 1 is SUFFICIENT.

Statement 2: (a/2) > (2/a)

a is a variable and can therefore, take both positive and negative values.

We can derive two options from the information given in statement 2

Option 1: a^{2} > 4 only if a > 0

i.e., a^{2} – 4 > 0 and a > 0

or (a + 2) (a – 2) > 0 and a > 0

or a > 2 or a < -2 and a > 0

Given that option 1 holds good only if a > 0, a > 2 and a cannot be less than -2.

Option 1 therefore, points to the result that a is positive.

Option 2: a^{2} < 4 if a < 0

i.e., a^{2} – 4 < 0 and a < 0

or (a + 2) (a – 2) < 0 and a < 0

or -2 < a < 2 and a < 0

Given that option 2 holds good only if a < 0, the range of values that a can take narrows down to -2 < a < 0.

Option 2 therefore, points to the result that a is negative.

Statement 2 leaves us with both the possibilities – ‘a’ could be positive or ‘a’ could be negative.

Therefore, statement 2 is NOT SUFFICIENT.

Choice A is the correct answer.

## Queries, answers, comments welcome