Home
Class 12
MATHS
If A, G and H are the arithmetic mean, g...

If A, G and H are the arithmetic mean, geometric mean and harmonic mean between unequal positive integers. Then, the equation `Ax^2 -IGlx- H=0` has (a) both roots are fractions (b) atleast one root which is negatuve fraction (c) exactly one positive root (d) atleast one root which is an integer

A

both roots are fractioins

B

atleast one root which is negative fraction

C

exactly one positive root

D

atleast one root which is an integer

Text Solution

Verified by Experts

The correct Answer is:
B, C
Given equation is
`Ax^(2)-|G|x-H=0`………i
`:.` Discriminant `=(-|G|)^(20)-4A(-H)`
`=G^(@)+4AH`
`=G^(2)+4G^(2)[:'G^(2)=AH]`
`=5G^(2)gt0`
`:.` Roots of Eq. (i) are real and distinct.
`:'A=(a+b)/2gt0, G=sqrt(ab)gt0,H=(2ab)/(a+b)gt0`
[ `:'` a and b are two unequal positive integers]
Let `alpha` and `beta` be the roots of Eq. (i). Then
`alpha+ beta=(|G|)/Agt0`
and `alpha beta =-H/Alt0`
and `alpha-beta=(sqrt(D))/A=(Gsqrt(5))/Agt0`
`:.alpha=(|G|+Gsqrt(5))/(2A)gt0`
and `beta=(|G|-Gsqrt(5))/(2A)lt0`
Exactly one positive root and atleat one root which is negative fraction.
Promotional Banner

Topper's Solved these Questions

  • THEORY OF EQUATIONS

    ARIHANT MATHS|Exercise Exercise (Passage Based Questions)|21 Videos

  • THEORY OF EQUATIONS

    ARIHANT MATHS|Exercise Exercise (Single Integer Answer Type Questions)|10 Videos

  • THEORY OF EQUATIONS

    ARIHANT MATHS|Exercise SCQ_TYPE|1 Videos

  • THE STRAIGHT LINES

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|18 Videos

  • THREE DIMENSIONAL COORDINATE SYSTEM

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|44 Videos

Similar Questions

Explore conceptually related problems

Let A,G and H are the AM,GM and HM respectively of two unequa positive integers. Then the equation Ax^(2)-1Gx-H=0 has (a) both roots as fractions (b) at least one root which is a negative fraction ( c ) exactly one positive root (d) at least one root which is an integer roots

If A, G and H are the arithmetic, geometric and harmonic means between a and b respectively, then which one of the following relations is correct?

Let A;G;H be the arithmetic; geometric and harmonic means of two positive no.a and b then A>=G>=H

If A and G are respectively arithmetic and geometric mean between positive no. a and b ; then the quadratic equation having a,b as its roots is x^2-2Ax+G^2=0.

The quadratic equation whose roots are the arithmetic mean and the harmonic mean of the roots of the equation x^(2)+7x-1=0 is

Let A;G;H be the arithmetic; geometric and harmonic means between three given no.a;b;c then the equation having a;b;c as its root is x^(3)-3Ax^(2)+3(G^(3))/(H)x-G^(3)=0

If 3 arithmetic means, 3 geometric means and 3 harmonic means are inserted between 1 and 5, then the cubic equation whose roots are first A.M., second G.M. and third H.M. between 1 and 5, is

If a,b,c in R and abc<0, then equation bcx^(2)+(2b+c-a)x+a=0 has (a).both positive roots (b) both negative roots (c).real roots (d) one positive and one negative root

If positive numbers a,b,c are in H.P.then equation x^(2)-kx+2b^(101)-a^(101)-c^(101)=0(k in R) has (a)both roots positive (b)both roots negative (c)one positive and one negative root (d)both roots imaginary