InterviewSolution
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InterviewSolution
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If `a,b,c,d in R` and all the three roots of `az^3 + bz^2 + cZ + d=0` have negative real parts, thenA. `ab gt 0 `B. `bv gt 0`C. `ad gt 0`D. `bc-ad gt 0` |
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Answer» Correct Answer - A::B::C::D Let the roots of `az^(3) + bz^(2) + cz + d = 0 ` be `z_(1) = x_(1),z_(2) = x_(2) + iy_(2)` and `z_(3) = x_(2) -iy_(2)` `therefore z_(1) + z_(2) + z_(3) = -(b)/(a)` `rArr x_(1) +2x_(2) = - (b)/(a) lt 0` `rArr ab gt 0` `z_(1)z_(2) + z_(2)z_(3) + z_(1) +z_(3) = (c)/(a)` `x_(1)(x_(2) + iy_(2)) + (x_(2)^(2) + y_(2)^(2)) + x_(1)(x_(2) -iy_(2))` ` =2x_(1)x_(2) + x_(2)^(2) + y_(2)^(2) gt 0` `rArr (c)/(0) gt0` `rArr ac gt 0` `rArr a^(2) bc gt 0 ` `bc gt 0` Also, `z_(1)z_(2)z_(3)= x_(1)(x_(2)^(2) + y_(2)^(2)) = -(d)/(a)` `rArr ad gt 0` Futher,`-(bc)/(a^(2)) = (-(b)/(a))((c)/(a))` `= (x_(1)+ 2x_(2)) (2x_(1)x_(2) + x_(2)^(2) + y_(2)^(2))` `=x_(1)(x_(2)^(2) + y^(2)_(2)) + 2x_(1)^(2) x_(2) + 2x_(1)x_(2)^(2) + 2x_(2) (x_(2)^(2) + y_(2)^(2))` `lt x_(1)(x_(2)^(2) + y_(2)^(2))` `rArr -(bc)/(a^(2)) lt - (d)/(a)` `rArr bc gt ad` |
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