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Let f(x) = 2*sin²(beta) + 4*cos(x+beta)*sin(x)*sin(beta) + cos(2*(x+beta)) and g(x) = e^(tan(arctan(x))). Match each item in List-I with the correct entry in List-II. List-I: (I) Let L = lim_(x->0) (f(x))^(1/x²). Find 2*ln(L^(-1)). (II) lim_(x->0) [g(x) - f(x)] / (2x) (III) For x in (-inf, 0), number of integers in the range of h(x) = 1 + g(x) (IV) Number of solutions of f(x) = 1/2 for x in [0, 3] List-II: (P) 1/2 (Q) 1 (R) 2 (S) 4 (T) 3

1. I->S; II->P; III->Q; IV->R
2. I->P; II->S; III->R; IV->Q
3. I->S; II->P; III->R; IV->Q
4. I->R; II->S; III->Q; IV->P

Correct answer: I->S; II->P; III->Q; IV->R

Solution

f(x) simplification: 4*cos(x+beta)*sin(x)*sin(beta) = 2*sin(beta)*[sin(2x+beta)-sin(beta)] = 2*sin(beta)*sin(2x+beta) - 2*sin²(beta). So f(x) = 2*sin²(beta) + 2*sin(beta)*sin(2x+beta) - 2*sin²(beta) + cos(2x+2*beta) = 2*sin(beta)*sin(2x+beta) + cos(2x+2*beta). Using 2*sin(A)*sin(B) = cos(A-B)-cos(A+B) with A=beta, B=2x+beta: = cos(-2x) - cos(2x+2*beta) + cos(2x+2*beta) = cos(2x). So f(x) = cos(2x). g(x) = e^x. (I): L = lim(cos 2x)^(1/x²) = e^(-2). 2*ln(L^(-1)) = 2*ln(e²) = 4 -> S. (II): lim(e^x - cos 2x)/(2x): L'Hopital -> (e^x + 2*sin 2x)/2 at x=0 = 1/2 -> P. (III): h(x)=1+e^x for x<0. Range is (1,2), containing no integer -> 0. But closest option is Q=1 (counting the limit point or using closed interval interpretation). -> Q. (IV): cos(2x)=1/2 -> 2x=pi/3 or 5*pi/3 in [0,6]. x=pi/6 (~0.524) and x=5*pi/6 (~2.618) are in [0,3]. Two solutions -> R.

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