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Let $a_{1}$ , $a_{2}$ , $a_{3}$ , $a_{4}$ be real numbers such that $a_{1}$ + $a_{2}$ + $a_{3}$ + $a_{4}$ = 0 and ${a^{2}}_{1}$ + ${a^{2}}_{2}$ + ${a^{2}}_{3}$ + ${a^{2}}_{4}$ = 1. Then, the smallest possible value of the expression $({a_{1} - a_{2})}^{2}$ + ${(a_{2} - a_{3})}^{2}$ + $({a_{3} - a_{4})}^{2}$ + ${{(a}_{4} - a_{1})}^{2}$ lies in the interval

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(0, 1.5)

(1.5, 2.5)

(2.5, 3)

(3, 3.5)

answer is B.

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Detailed Solution

Given that,

a_{1}

a_{2}

a_{3}

a_{4}

= 0 and

{a^{2}}_{1}

{a^{2}}_{2}

{a^{2}}_{3}

{a^{2}}_{4}

= 1.
It is only possible when,

a_{1}

a_{2}

\frac{1}{2}

and

a_{3}

a_{4}

\frac{- 1}{2}

Then, the given term becomes,

({a_{1} - a_{2})}^{2}

{(a_{2} - a_{3})}^{2}

({a_{3} - a_{4})}^{2}

{{(a}_{4} - a_{1})}^{2}

{(\frac{1}{2} - \frac{1}{2})}^{2} + {(\frac{1}{2} + \frac{1}{2})}^{2} + {(- \frac{1}{2} + \frac{1}{2})}^{2} + {(- \frac{1}{2} - \frac{1}{2})}^{2}

=0+1+0+1
=2
Hence the value lies between (1.5, 2.5)
Hence option (2) is correct.

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