If `A = |(sin (theta + alpha),cos (theta + alpha),1),(sin (theta + beta),cos (theta + beta),1),(sin (theta + gamma),cos (theta + gamma),1)|`, then

Options

A. `A = 0 " for all " theta`
B. A is an odd function of `theta`
C. A = 0 for `theta = alpha + beta + gamma`
D. A is independent of `theta`

Answer

D

Explanation

We have,
`A =|(sin (theta + alpha),cos (theta + alpha),1),(sin (theta + beta),cos (theta + beta),1),(sin (theta + gamma),cos (theta + gamma),1)|`
`rArr A =|(sin (theta + alpha),cos (theta + alpha),1),(sin (theta + beta) - sin (theta + alpha),cos (theta + beta) - cos (theta + alpha),0),(sin (theta + gamma) - sin (theta + alpha) ,cos (theta + gamma) - cos (theta + alpha),0)| "Applying " R_(2) rarr R_(2) - R_(1), R_(3) rarr R_(3) - R_(1)`
`rArr A = {sin (theta + beta) - sin (theta + alpha)} {cos (theta + gamma) - cos (theta + alpha)}`
`- {sin (theta + gamma) - sin (theta + alpha)} {cos (theta + beta) - cos (theta + alpha)}`
`A = 4 sin ((beta - alpha)/(2)) cos ((2 theta + alpha + beta)/(2)) sin ((2 theta + alpha + gamma)/(2))`
`sin ((alpha - gamma)/(2)) - 4 sin (2 theta + alpha + gamma)/(2)) sin ((2 theta + alpha + gamma)/(2)) sin ((alpha - beta)/(2))`
`rArr A = 4 sin ((alpha - beta)/(2)) sin ((gamma - alpha)/(2)) sin ((gamma - beta)/(2))`
Hence, A is independent of `theta`