Home
Class 12
MATHS
If roots of the equation f(x)=x^6-12 x^5...

If roots of the equation `f(x)=x^6-12 x^5+bx^4+cx^3+dx^2+ex+64=0`are positive, then
Which has the greatest absolute value ?

A

b

B

c

C

d

D

e

Text Solution

Verified by Experts

The correct Answer is:
C
Let roots of equation `x^(6) - 12x^(5) + bx^(4) + cx^(3) + dx^(2) + ex + 64 = 0`
be `x_(i), I = 1,2…..6` Now,
`x_(1) + x_(2) + x_(3) + x_(4) + x_(5) + x_(6) = 12`
and `x_(1) x_(2) x_(4) x_(5) x_(6) = 64`
Thus,
`(x_(1) + x_(2) …. + x_(6))/(6) = 2` and `(x_(1) x_(2) x_(3) x_(5) x_(6))^(1//6) = 2`
`implies a.M = G.M`
`implies x_(1) = x_(2) = x_(3) - x_(4) = x_(5) = x_(6) = 2`
Hence, the given equation is equivalent to
`(x - 2)^(6) = 0`
or `x^(6) - 12 x^(5) + 60x^(4) - 160 x^(3) + 240x^(2) - 192 x - 64 = 0`
`:. f(1) = 1 - 12 + 60 - 160 + 240 - 192 + 64 = 1`
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise (Numerical) & JEE Previous Year|11 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise (Multiple)|5 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE|Exercise Solved Examples And Exercises|222 Videos

Similar Questions

Explore conceptually related problems

If roots of the equation f(x)=x^6-12 x^5+bx^4+cx^3+dx^2+ex+64=0 are positive, then remainder when f(x) is divided by x-1 is

If the roots of the equation x^3 + bx^2 + cx - 1=0 form an increasing G.P, then

One root of the equation (12x -1)(6x -1)(4x-1)(3x-1)=5 is

The roots of the equation x^(2)-5x+6=0 are .......

How many roots of the equation 3x^4+6x^3+x^2+6x+3=0 are real ?

If he roots of the equation 12 x^2-m x+5=0 are in the ratio 2:3 then find the value of mdot

Find the real roots of the equation . x^2+5|x|+6=0

If aa n db are positive numbers and eah of the equations x^2+a x+2b=0a n dx^2+2b x+a=0 has real roots, then the smallest possible value of (a+b) is_________.

Consider the graph of y = f(x) as shown in the following figure. (i) Find the sum of the roots of the equation f (x) = 0. (ii) Find the product of the roots of the equation f(x) = 4. (iii) Find the absolute value of the difference of the roots of the equation f(x) = x+2 .

The common root of the equation x^(2) - bx + c = 0 and x^(2) + b x - a= 0 is ………..