Consider the matrix \(\mathrm{f}(\mathrm{x})=\left[\begin{array}{ccc}\cos \mathrm{x} & -\sin \mathrm{x} & 0 \\ \sin \mathrm{x} & \cos \mathrm{x} & 0 \\ 0 & 0 & 1\end{array}\right]\). Given below are two statements : Statement I: \(\mathrm{f}(-\mathrm{x})\) is the inverse of the matrix \(\mathrm{f}(\mathrm{x})\). Statement II: \(\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y})=\mathrm{f}(\mathrm{x}+\mathrm{y})\). In the light of the above statements, choose the correct answer from the options given below

(D) Both Statement I and Statement II are true

(A) Statement I is false but Statement II is true

(B) Both Statement I and Statement II are false

(C) Statement I is true but Statement II is false

JEE Main · 2024 · 27 Jan 2024 · Shift 1 · Matrices

Consider the matrix

f (x) = cos x sin x 0 - sin x cos x 0 001

.

Given below are two statements :
Statement I:

f (- x)

is the inverse of the matrix

f (x)

.
Statement II:

f (x) f (y) = f (x + y)

.
In the light of the above statements, choose the correct answer from the options given below

Answer: (D)

f (x) f (- x)

multiplies a rotation by

x

with a rotation by

- x

, giving the identity matrix, so Statement I (

f (- x)

is the inverse of

f (x)

) is true.

f (x) f (y)

combines two rotations, and direct matrix multiplication using angle-addition identities gives exactly

f (x + y)

, so Statement II is also true. Both statements hold.

JEE Main · 2024 · 27 Jan 2024 · Shift 1 · Single correct option