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6. Which of the following statements must be true when 0 < a < 1 ?. sqrt a a >1II. 2a < 1III. a ^ 2 - a ^ 3 < 0A. I onlyB. III onlyC.I and III onlyD.II and III only

6 Which Of The Following Statements Must Be True When 0 Lt A Lt 1 Sqrt A A Gt1II 2a Lt 1III A 2 A 3 Lt 0A I OnlyB III OnlyCI And III OnlyDII And III Only class=

Answer :

Given:

The following inequalities:

[tex]\frac{\sqrt[]{a}}{a}>1\rightrightarrows\frac{1}{\sqrt[]{a}}[/tex]

When 0 < a < 1

So, the denominator will be always < 1

so, all the fraction will be greater than 1

So, the first inequality True

[tex]\begin{gathered} 2a<1 \\ a<\frac{1}{2} \end{gathered}[/tex]

when 0 < a < 1

The inequality will be true as a < 1/2

The inequality will be wrong when a >1/2

So, the second inequality is wrong

[tex]\begin{gathered} a^2-a^3<0 \\ a^2(1-a)<0 \end{gathered}[/tex]

The inequality true when a >1

so, for 0 < a < 1, the inequality is wrong

So, the answer will be option A. I only

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