Home
Class 12
MATHS
If a1, a2, a3 ......an (n>= 2) are real...

If `a_1, a_2, a_3 ......a_n (n>= 2) ` are real and `(n-1) a_1^2 -2na_2 < 0` then prove that at least two roots of the equation ` x^n+a_1 x^(n-1) +a_2 x^(n-2) +......+a_n = 0 `are imaginary.

Text Solution

AI Generated Solution

To prove that at least two roots of the equation \[ x^n + a_1 x^{n-1} + a_2 x^{n-2} + \ldots + a_n = 0 \] are imaginary given that ...
Promotional Banner

Topper's Solved these Questions

  • THEORY OF EQUATIONS

    ARIHANT MATHS ENGLISH|Exercise Exercise For Session 1|11 Videos

  • THEORY OF EQUATIONS

    ARIHANT MATHS ENGLISH|Exercise Exercise For Session 2|10 Videos

  • THE STRAIGHT LINES

    ARIHANT MATHS ENGLISH|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|17 Videos

  • THREE DIMENSIONAL COORDINATE SYSTEM

    ARIHANT MATHS ENGLISH|Exercise Three Dimensional Coordinate System Exercise 12 : Question Asked in Previous Years Exam|2 Videos

Similar Questions

Explore conceptually related problems

If a_1, a_2, a_3,.....a_n are in H.P. and a_1 a_2+a_2 a_3+a_3 a_4+.......a_(n-1) a_n=ka_1 a_n , then k is equal to

Let a_1, a_2, a_3, ...a_(n) be an AP. then: 1 / (a_1 a_n) + 1 / (a_2 a_(n-1)) + 1 /(a_3a_(n-2))+......+ 1 /(a_(n) a_1) =

If the sequence a_1, a_2, a_3,....... a_n ,dot forms an A.P., then prove that a_1^2-a_2^2+a_3^2-a_4^2+.......+ a_(2n-1)^2 - a_(2n)^2=n/(2n-1)(a_1^2-a_(2n)^2)

If a_1, a_2,...... ,a_n >0, then prove that (a_1)/(a_2)+(a_2)/(a_3)+(a_3)/(a_4)+.....+(a_(n-1))/(a_n)+(a_n)/(a_1)> n

If a_1, a_2, a_3, ,a_(2n+1) are in A.P., then (a_(2n+1)-a_1)/(a_(2n+1)+a_1)+(a_(2n)-a_2)/(a_(2n)+a_2)++(a_(n+2)-a_n)/(a_(n+2)+a_n) is equal to a. (n(n+1))/2xx(a_2-a_1)/(a_(n+1)) b. (n(n+1))/2 c. (n+1)(a_2-a_1) d. none of these

A sequence of positive integers a_1 , a_2, a_3, …..a_n is given by the rule a_(n + 1) = 2a_(n) + 1 . The only even number in the sequence is 38. What is the value of a_2 ?

If a_1, a_2, a_3, a_n are in H.P. and f(k)=(sum_(r=1)^n a_r)-a_k ,t h e n (a_1)/(f(1)),(a_2)/(f(2)),(a_3)/(f(3)), ,(a_n)/(f(n)), are in a. A.P b. G.P. c. H.P. d. none of these

Let a_1, a_2, a_3, ,a_(11) be real numbers satisfying a_1=15 , 27-2a_2>0 a n da_k=2a_(k-1)-a_(k-2) for k=3,4, , 11. If (a1 2+a2 2+...+a 11 2)/(11)=90 , then the value of (a1+a2++a 11)/(11) is equals to _______.

cIf a_1,a_2,a_3,..,a_n in R then (x-a_1)^2+(x-a_2)^2+....+(x-a_n)^2 assumes its least value at x=

If x,a_1,a_2,a_3,…..a_n epsilon R and (x-a_1+a_2)^2+(x-a_2+a_3)^2+…….+(x-a_(n-1)+a_n)^2le0 , then a_1,a_2,a_3………a_n are in (A) AP (B) GP (C) HP (D) none of these

ARIHANT MATHS ENGLISH-THEORY OF EQUATIONS-Exercise (Questions Asked In Previous 13 Years Exam)
  1. If a1, a2, a3 ......an (n>= 2) are real and (n-1) a1^2 -2na2 < 0 then...

    Text Solution

    |

  2. In the quadratic equation ax^2 + bx + c = 0. if delta = b^2-4ac and al...

    Text Solution

    |

  3. Let S denote the set of all polynomials P(x) of degree lt=2 such th...

    Text Solution

    |

  4. If the roots of x^2-b x+c=0 are two consecutive integers, then b^2-4c ...

    Text Solution

    |

  5. If the equation a(n)x^(n)+a(n-1)x^(n-1)+..+a(1)x=0, a(1)!=0, n ge2, ha...

    Text Solution

    |

  6. If both the roots of the quadratic equation x^(2)-2kx+k^(2)+k-5=0 are ...

    Text Solution

    |

  7. Let aa n db be the roots of the equation x^2-10 c x-11 d=0 and those o...

    Text Solution

    |

  8. Let a,b,c be the sides of a triangle. No two of them are equal and lam...

    Text Solution

    |

  9. All the values of m for whilch both the roots of the equation x^2-2m x...

    Text Solution

    |

  10. If the roots of the quadratic equation x^2+p x+q=0 are tan30^0a n dtan...

    Text Solution

    |

  11. Let alpha,beta be the roots of the equation x^2-p x+r=0a n dalpha//2,2...

    Text Solution

    |

  12. If the difference between the roots of the equation x^2+a x+1=0 is les...

    Text Solution

    |

  13. Let a ,b , c ,p ,q be real numbers. Suppose alpha,beta are the roots o...

    Text Solution

    |

  14. The quadratic equations x^2""-6x""+""a""=""0""a n d ""x^2""-c x""+""...

    Text Solution

    |

  15. How many real solutions does the equation x^7+14 x^5+16 x^3+30 x-560=0...

    Text Solution

    |

  16. Suppose the cubic x^(3)-px+q has three distinct real roots, where pgt0...

    Text Solution

    |

  17. The smallest value of k, for which both the roots of the equation, x^2...

    Text Solution

    |

  18. If the roots of the equation b x^2+""c x""+""a""=""0 be imaginary, t...

    Text Solution

    |

  19. Let p and q real number such that p!= 0,p^3!=q and p^3!=-q. if alpha a...

    Text Solution

    |

  20. solve 0=1+2x+3x^2

    Text Solution

    |

  21. Find the roots of x^2-6x-2=0

    Text Solution

    |