Home
Class 12
MATHS
If a,b,c,d in R and all the three roots...

If `a,b,c,d in R` and all the three roots of `az^3 + bz^2 + cZ + d=0` have negative real parts, then

A

`ab gt 0`

B

`bc gt 0`

C

`ad gt 0`

D

`bc-ad gt 0`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`(a,b,c)` Let `z_(1)=x_(1)`, `z_(2)`, `z_(3)=x_(2)+-iy_(2)`
`implies z_(1)+z_(2)+z_(3)=-(b)/(a)`
`impliesx_(1)+2x_(2)=-(b)/(a) lt 0impliesab gt 0`
Also, `z_(1)z_(2)z_(3)=x_(1)[x_(2)^(2)+y_(2)^(2)]=-(d)/(a)lt0impliesad gt 0`
Also `z_(1)z_(2)+z_(2)z_(3)+z_(1)z_(3)=(c )/(a)`
`implies x_(1)(x_(2)+iy_(2))+x_(1)(x_(2)-iy_(2))+x_(2)^(2)+y_(2)^(2)=2x_(1)x_(2)+x_(2)^(2)+y_(2)^(2) gt 0`
`implies (c )/(a) gt 0`
`implies (b)/(a) (c )/(a) gt 0impliesbc gt 0`
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS

    CENGAGE|Exercise Matching Column|1 Videos

  • COMPLEX NUMBERS

    CENGAGE|Exercise Comprehension|11 Videos

  • COMPLEX NUMBERS

    CENGAGE|Exercise JEE Advanced Previous Year|14 Videos

  • CIRCLES

    CENGAGE|Exercise Question Bank|16 Videos

  • CONIC SECTIONS

    CENGAGE|Exercise Solved Examples And Exercises|87 Videos

Similar Questions

Explore conceptually related problems

Let a >0,b >0 and c >0 . Then, both the roots of the equation a x^2+b x+c=0 are a. real and negative b.have negative real parts c. have positive real parts d. None of the above

If a,b,c,d in R-{0} , such that a,b are the roots of equation x^(2)+cx+d=0 and c,d are the roots of equation x^(2)+ax+b=0, then |a|+|b|+|c|+|d| is equal to

If a ,b ,c ,d in R , then the equation (x^2+a x-3b)(x^2-c x+b)(x^2-dx+2b)=0 has a. 6 real roots b. at least 2 real roots c. 4 real roots d. none of these

If a,b,c in R and a+b+c=0, then the quadratic equation 3ax^2+2bx+c=0 has (a) at least one root in [0, 1] (b) at least one root in [1,2] (c) at least one root in [3/2, 2] (d) none of these

If b^2<2a c , then prove that a x^3+b x^2+c x+d=0 has exactly one real root.

If a ,b ,c in R^+a n d2b=a+c , then check the nature of roots of equation a x^2+2b x+c=0.

If the equation 4x^(3)+5x+k=0(k in R) has a negative real root then

If a ,b ,c in R , then find the number of real roots of the equation =|x c-b-c x a b-a x|=0

If a ,b ,c be the sides of A B C and equations a x^2+b x+c=0a n d5x^2+12x+13=0 have a common root, then find /_Cdot