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" If " f(alpha , beta) = |{:(cos alpha ...

`" If " f(alpha , beta) = |{:(cos alpha ,,-sin alpha,,1),(sin alpha,,cos alpha,,1),(cos (alpha+beta),,-sin (alpha+beta),,1):}|` then

A

independent of `alpha`

B

independent of `beta`

C

independent of `alpha and beta`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
We have, `|(cos alpha,- sin alpha,1),(sin alpha,cos alpha,1),(cos (alpha + beta),- sin (alpha + beta),1)|`
`= |(cos alpha,- sin alpha,1),(sin alpha,cos alpha,1),(0,0,1 + sin beta - cos beta)| " " ["Applying "R_(3) rarr R_(3) - R_(1) (cos beta) + R_(2) (sin beta)]`
`= (1 + sin beta - cos beta) (cos^(2) alpha + sin^(2) alpha)`
`= 1 + sin beta - cos beta`, which is independent of `alpha`
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