If `f{x}=[{:(cos x,-sinx,0),(sin x,cos x,0),(" "0," "0,1):}]`, then which of the following are correct ?
1. `f(theta) xx f(phi)=f(theta + phi)`
2. The value of the determinant of the matrix `f(theta) xx f(phi)` is 1.
3. The determinant of f(x) is an even function.

Options

A. 1 and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3

Answer

D

Explanation

`f(theta)=[(cos theta,-sin theta,0),(sin theta, cos theta,0),(0,0,1)]&f(phi)[(cosphi,-sin phi,0),(sinphi, cos phi,0),(0,0,1)]`
`f(theta)xxf(phi)=[(costheta cosphi-sin thetasin phi,-cos thetasin phi-sin theta cos phi,0),(sin theta cos phi+costheta sin phi,-sintheta sin phi+cos thetacos phi,0),(0,0,1)]`
`" (using Trigonometric Identities)"`
`rArr" "f(theta)xxf(phi)=f(theta+phi)`
Also. `det,[f(theta)xxf(phi)]=1[cos^(2)(theta+phi)-(-sin^(2)(theta+phi))]`
`=cos^(2)(theta+phi)+sin^(2)(theta+phi)=1.`
`&det.(f(x))=(cos^(2)x-(-sin^(2)x))=cos^(2)x+sin^(2)x=1.`
For `x=-x`
`det.(f(x))=cos^(2)(-x)+sin^(2)(-x)=1`
Hence, `det.(f(-x))=det.(f(x))`
Hence, `det.(f(x))` is even function.