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Question

Find the minimum value of a+b+c+d in the equation x5ax4+bx3cx2+dx243=0, if it is given that the roots are positive real numbers ?


A

780

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B

660

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C

30

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D

720

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Solution

The correct option is D

780


There are 5 roots in this equation.

Here the product of the roots is given as 243. Hence the sum will be minimum when the roots are equal. In fact a,b,c and d will be minimum when the roots are equal.

243 = 35

Hence, the roots will be 3,3,3,3 and 3

a= sum of the roots = 15

b=Sum of product of roots taken 2 at a time = (3 × 3)+(3 × 3)...10 times or = 5C2×9 = 90

c= Sum of product of roots taken 3 at a time = (3 × 3× 3)+(3× 3× 3)...5 times or = 5C3×27 = 270

d=Sum of product of roots taken 4 at a time = (3× 3× 3× 3)....5 times= 405

Hence, a+b+c+d = 780


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