Home
Class 12
MATHS
The value of |{:(betagamma,betagamma'+be...

The value of `|{:(betagamma,betagamma'+beta'gamma,beta'gamma'),(gammaalpha,gammaalpha'+gamma'alpha,gamma'alpha'),(alphabeta,alphabeta'+alpha'beta,alpha'beta'):}|` is

A

`(alphabeta'-alpha'beta)(betagamma'-beta'gamma)(gammaalpha'-gamma'alpha)`

B

`(alphaalpha'-betabeta')(betabeta'-gammagamma')(gammagamma'-alphaalpha')`

C

`(alphabeta'+alpha'beta)(betagamma'+beta'gamma)(gammaalpha'+gamma'alpha)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` `(1)/(alphabetagamma)|{:(alphabetagamma,alphabetagamma'+alphabeta'gamma,alphabeta'gamma'),(alphabetaalpha,betagammaalpha'+betagamma'alpha,betagamma'alpha'),(alphabetagamma,alphabeta'gamma+alpha'betagamma,alpha'beta'gamma):}|`
Multiplying `R_(1)` by `alpha`, `R_(2)` by `beta` and `R_(3)` by `gamma` and dividing the determinant by `alphabetagamma` we have
`=(1)/(alphabetagamma)*alphabetagamma|{:(1,alphabetagamma'+alphabeta'gamma,alphabeta'gamma'),(1,betagammaalpha'+betaalphagamma',betagamma'alpha'),(1,gammaalphabeta'+gammaalpha'beta,gammaalpha'beta'):}|`
by `{:(R_(3)toR_(3)-R_(1)),(R_(2)toR_(2)-R_(1)):}`
`=|{:(1,alphabetagamma'+alphabeta'gamma,alphabeta'gamma'),(0,gamma(alpha'beta-alphabeta'),gamma'(alpha'beta-alphabeta')),(0,beta(alpha'gamma-alphagamma'),beta'(alpha'gamma-alphagamma')):}|`
`=(alpha'beta-alphabeta')(alpha'gamma-alphagamma')|{:(1,alphabetagamma',+,alphabeta'gamma,alphabeta'gamma'),(0,,gamma,,gamma'),(0,,beta,,beta'):}|`
`=(alpha'beta-alphabeta')(alpha'gamma-alphagamma')(gammabeta'-betagamma')`
Promotional Banner

Topper's Solved these Questions

  • DETERMINANT

    CENGAGE|Exercise Comprehension|2 Videos

  • DETERMINANT

    CENGAGE|Exercise Multiple Correct Answer|5 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise JEE Advanced Previous Year|38 Videos

  • DETERMINANTS

    CENGAGE|Exercise All Questions|268 Videos

Similar Questions

Explore conceptually related problems

If alpha,beta,gamma are the roots of p x^3+q x^2+r=0, then the value of the determinant |(alphabeta,betagamma,gammaalpha),(betagamma,gammaalpha,alphabeta),(gammaalpha,alphabeta,betagamma)| is p b. q c. 0 d. r

Prove that |2alpha+beta+gamma+deltaalphabeta+gammadeltaalpha+beta+gamma+delta2(alpha+beta)(gamma+delta)alphabeta(gamma+delta)+gammadelta(alpha+beta)alphabeta+gammadeltaalphabeta(gamma+delta)+gammadelta(alpha+beta)2alphabetagammadelta|=0

Using properties of determinants in Exercises prove that : {:[( alpha , alpha ^(2) , beta +gamma ),( beta , beta ^(2) , gamma +alpha ),( gamma , gamma ^(2) ,alpha +beta ) ]:} =(beta -gamma ) (gamma -alpha ) (alpha -beta ) (alpha +beta +gamma )

Prove that |((beta+gamma-alpha-delta)^4,(beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4,(gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4,(alpha+beta-gamma-delta)^2,1)|=-64(alpha-beta)(alpha-gamma)(alpha-delta)(beta-delta)(gamma-delta)

Prove that |((beta+gamma-alpha-delta)^4,(beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4,(gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4,(alpha+beta-gamma-delta)^2,1)|=-64(alpha-beta)(alpha-gamma)(alpha-delta)(beta-delta)(gamma-delta)

If alpha+beta=pi/2a n dbeta+gamma=alpha, then tanalpha equals

If alpha,beta,gamma are the roots of a x^3+b x^2+c x+d=0a n d|alpha beta gamma beta gamma alpha gamma alpha beta|=0,alpha!=beta!=gamma then find the equation whose roots are alpha+beta-gamma,beta+gamma-alpha,a n dgamma+alpha-betadot

If tan beta=2sin alpha sin gamma co sec(alpha+gamma) , then cot alpha,cot beta,cotgamma are in

Prove that : cos^2 (beta-gamma) + cos^2 (gamma-alpha) + cos^2 (alpha-beta) =1+2cos (beta-gamma) cos (gamma-alpha) cos (alpha-beta) .