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Correct answer: (S) 97
(A) In (5^(1/6)+2^(1/8))¹⁰⁰: general term = C(100,r)*(5^(1/6))^(100-r)*(2^(1/8))^r = C(100,r)*5^((100-r)/6)*2^(r/8). For rational terms: (100-r)/6 must be integer AND r/8 must be integer. LCM(6,8)=24. r must be multiple of 8: r=0,8,16,...,96 => r/8 integer. Also (100-r)/6 must be integer: 100-r divisible by 6 => r=100-6k => r must give 100-r div by 6. r=4,10,16,22,...,100 are values where 100-r is div by 6. Intersection: r divisible by 8 AND 100-r divisible by 6. r=16 (100-16=84=6*14 YES, 16/8=2 YES), r=40 (100-40=60=6*10 YES, 40/8=5 YES), r=64 (100-64=36=6*6 YES, 64/8=8 YES), r=88 (100-88=12=6*2 YES, 88/8=11 YES). Also r=100: 100/8=12.5 NO. r=0: 100/6 not integer. So rational terms: r=16,40,64,88 => 4 rational terms. Total terms = 101. Irrational terms = 101-4=97. lambda=97. Lambda is greater than 82 (R) and less than 97? Actually lambda=97, which is greater than 96, so greater than P(29), Q(58), R(82). Not greater than S(97) since 97 is not greater than 97. Not greater than T(98). So A matches R (lambda>82 is true, and lambda=97>82). (D) Sum of coefficients at x=1: (1/3-15+15)²⁰¹³ * (17-17+3)²⁰¹⁷ = (1/3)²⁰¹³ * 3²⁰¹⁷ = 3⁴=81. n=81. Dissimilar terms in (1+x)⁸¹ = 82. Number of dissimilar terms (82) is less than T(98) and S(97). So D matches S (82<97) or T(82<98). Matching: A->R, D->T or D->S. Given the options only go up to (S) 97, and (T) 98 is the fifth option not listed as an answer choice, D->S (82<97). The answer option (S) 97 represents one of the Column-II entries that is a valid match.