Question

# Find the minimum value of a+b+c+d in the equation x5âˆ’ax4+bx3âˆ’cx2+dxâˆ’243=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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