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JEE MainMathematicsDifferentiation

Let A(x) = pmatrix f(x) & g(x) -g(x) & f(x) pmatrix be a matrix defined for all real x , where f(x) and g(x) are differentiable functions. If A(x+y) = A(x)A(y) for all x, y R , and f'(0) = 2 , g'(0) = 3 , then the value of (f'(x₀))^2 + (g'(x₀))^2 at x₀ = 1 2 (3) is equal to _____.

Correct answer

117

Step-by-step solution

Given A(x+y) = A(x)A(y) . Differentiating partially with respect to y and setting y = 0 : A'(x) = A(x)A'(0) We are given f'(0) = 2 and g'(0) = 3 , so: A'(0) = pmatrix 2 & 3 -3 & 2 pmatrix Thus, pmatrix f'(x) & g'(x) -g'(x) & f'(x) pmatrix = pmatrix f(x) & g(x) -g(x) & f(x) pmatrix pmatrix 2 & 3 -3 & 2 pmatrix Equating the elements of the first row, we get: f'(x) = 2f(x) - 3g(x) g'(x) = 3f(x) + 2g(x) We need to find H(x) = (f'(x))^2 + (g'(x))^2 . H(x) = (2f(x) - 3g(x))^2 + (3f(x) + 2g(x))^2 H(x) = 4f(x)^2 - 12f(x)g(

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