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Question

For what values of m, the equation(1+m)x22(1+3m)x+(1+8m)=0 has (mR) at least one negative root.

A
m(,0)(3,)
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B
m(,1)(3,)
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C
m(1,18)
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D
m(1,18)
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Solution

The correct option is D m(1,18)
(1+m)x22(1+3m)x+(1+8m)=0
We need to find the values of m for which the above equation has at-least one negative root.
Lets consider that the above equation has both non-negative roots, then the condition for both roots are non-negative is "Products of roots 0" and "Sum of roots 0"
(i) Product of roots 0
1+8m1+m0
m(,1)[18,)
(ii) Sum of roots 0
2(1+3m)(1+m)0
m(,1)[13,)

Taking intersection of (i) and (ii), we get
m(,1)[18,)

Now for the final values of m for which the equation have at-least one negative root, is
mRintersection of (i) and (ii)
mR(,1)[18,)
m[1,18)

Checking at the corner values, we get that m=1 does not satisfies the equation conditions.
m(1,18)

Hence, option D.

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