JEE Main 2024 · 27 January · morning · Single correct · +4 / −1
Consider the matrix f(x)=[cos x, -sin x, 0; sin x, cos x, 0; 0, 0, 1].
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 :
- (a) Statement I is false but Statement II is true
- (b) Both Statement I and Statement II are false
- (c) Both Statement I and Statement II are true
- (d) Statement I is true but Statement II is false
Answer and solution
Answer (c) Both Statement I and Statement II are true
Solution
Maths sir · 4 steps
Prove Statement II first, then put y = -x in it and Statement I follows for free.
- Multiply the blocks. The third row and column of f(x) only carry a 1, so the product f(x)f(y) has the same 1 there. All the work is in the top-left 2 × 2 block.
(cos x, -sin x; sin x, cos x)(cos y, -sin y; sin y, cos y) = (cos(x+y), -sin(x+y); sin(x+y), cos(x+y))
- Statement II. The top-left entry is cos x cos y - sin x sin y = cos(x+y), and the bottom-left entry is sin x cos y + cos x sin y = sin(x+y). The other two entries match in the same way, so f(x)f(y) = f(x+y). True.
- Statement I. Put y = -x: f(x)f(-x) = f(0). Since cos 0 = 1 and sin 0 = 0, f(0) is the identity matrix I. Likewise f(-x)f(x) = f(0) = I, so f(-x) is the inverse of f(x). True.
- So. Both statements are true, which is option (c).
The idea to keep. A rotation matrix adds angles when you multiply, f(x)f(y) = f(x+y). So its inverse is the rotation by the negative angle, f(-x), and that is also its transpose.
Answer from the NTA final answer key. Solution by PrepWiser.