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Question

If a1 < a2< a3 < a4 < a5 < a6, then the equation (xa1)(xa3)(xa5)+2(xa2)(xa4)(xa6) = 0 has



A

Three real roots 

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B

One real root

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C

One real root in each interval ( ,), ( , ) and ( , )

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D

None of these

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Solution

The correct option is D

One real root in each interval ( ,), ( , ) and ( , )


f(x)=(xa1)(xa3)(xa5)+2(xa2)(xa4)(xa6) = 0 

a1 < a2< a3 < a4 < a5 < a6

f( a1 ) = 2 (a1a2)(a1a4)(a1a6)<0

f( a2 ) = 2 (a2a1)(a2a3)(a2a5)>0

  at least one real root lies in (a1 ,a2 )

Similarly, at least one real root lies in each interval (a3 , a4 )  and (a5 , a6 )

But f(x) is cubic, therefore there are only three roots.

Hence, the equation f(x) = 0 has one real root in each interval (a1  ,a2 ), (a3  , a4 ) and (a5  , a6 )


Mathematics

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