Home
Class 12
MATHS
If alpha,beta,gamma are such that alpha+...

If `alpha,beta,gamma` are such that `alpha+beta+gamma=2,""alpha^2+beta^2+gamma^2=6,""alpha^3+beta^3+gamma^3=8,t h e n""alpha^4+beta^4+gamma^4` is a. 18 b. 10 c. 15 d. 36

Text Solution

Verified by Experts

We have,
`(alpha+beta+gamma)^(2)=alpha^(2)+beta^(2)+gamma^(2)+2(betagamma+gammaalpha+alphabeta)`
`rArr 4 = 6 + 2 (betagamma+gammaalpha+alphabeta)`
`rArr betagamma+gammaalpha+alphabeta = -1`
Also, `alpha^(3) + beta^(3) + gamma^(3) - 3alphabeta gamma`
`=(alpha+beta+gamma)(alpha^(2)+beta^(2)+gamma^(2))(betagamma+gammaalpha+alphabeta)`
or " " 8 - 3 `alpha+beta+gamma = 2(6+1)`
or `3alpha+beta+gamma = 8 - 14 = -6`
or `alpha+beta+gamma = - 2`
Now, `(alpha^(2)+beta^(2)+gamma^(2))^(2)=Sigmaalpha^(4)+2Sigmaalpha^(2)beta^(2)`
`=Sigmaalpha^(4)+2[(Sigmaalphabeta)^(2)-2alphabetagamma(Sigmaalpha)]`
or `=Sigmaalpha^(4)= 36 - 2 [(-1)^(2)-2(-2)(2)] = 18`
Promotional Banner

Topper's Solved these Questions

  • THEORY OF EQUATIONS

    CENGAGE ENGLISH|Exercise SOLVED EXAMPLES|14 Videos

  • THEORY OF EQUATIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 2.1|3 Videos

  • THEORY OF EQUATIONS

    CENGAGE ENGLISH|Exercise Multiple Correct Answer|6 Videos

  • STRAIGHT LINES

    CENGAGE ENGLISH|Exercise ARCHIVES (NUMERICAL VALUE TYPE)|1 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE ENGLISH|Exercise All Questions|294 Videos

Similar Questions

Explore conceptually related problems

If alpha, beta, gamma are the cube roots of 8 , then |(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta)|=

If sin alpha+ sin beta+ sin gamma=3, then sin^3alpha+sin^3beta+sin^3 gamma=

If 2x^(3) + 3x^(2) + 5x +6=0 has roots alpha, beta, gamma then find alpha + beta + gamma, alphabeta + betagamma + gammaalpha and alpha beta gamma

If alpha+beta+gamma=2pi , then show that tan.alpha/2 + tan.beta/2 + tan.gamma/2 = tan.alpha/2 tan.beta/2 tan.gamma/2 .

Determine the values of alpha ,beta , gamma when [{:( 0, 2beta, gamma),( alpha ,beta, -gamma),( alpha ,-beta, gamma):}] is orthogonal.

If alpha+beta, gamma epsilon R^(+) such that alpha gamma =1/(beta) then maximum value of alpha^(beta+gamma), .beta^(beta+gamma).gamma^(alpha+beta) is_____

If alpha,beta,gamma in R, alpha+beta+gamma=4 " and " alpha^(2)+beta^(2)+gamma^(2)=6 , the number of integers lie in the exhaustive range of alpha is ……… .

if alpha, beta, gamma are the roots of the equation x^(3) + 3x + 2=0 " then " (alpha^(3) +beta^(3)+gamma^(3))/(alpha^(2) +beta^(2)+gamma^(2))

If alpha, beta, gamma are the roots of x^(3) + ax^(2) + b = 0 , then the value of |(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta)| , is

If alpha,beta,gamma are roots of the equation x^(3)+px+q=0 then the value of |{:(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta):}| is