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Q.

Let f:(11,∞)→(0,∞) be given by f(x)=∏l=110 1(x−l)where for real number a1..........an∏l=1n al denotes the product a1×a2⋯×anStatement 1: ∫f(x)dx=∑l=110 (−1)llog⁡|x−l|(l−1)!(10−l)!Statement 2: For x∈[11,∞)f(x)=∑l=110 Alx−l where A1=∏j=110 jl−jl=1,2,10

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a

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

b

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

c

STATEMENT-1 is True, STATEMENT-2 is False

d

STATEMENT-1 is False, STATEMENT-2 is True

answer is C.

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Detailed Solution

f(x)=∏l=110 1x−l=A1x−1+A2x−2+⋯+A10x−10where Ai=1(i−1)⋯(i−(i+1))⋯(i−10)               =(−1)i(i−1)!(10−i)!∫f(x)dx=∑i=110 ∫Aix−idx=∑i=110 Ailog⁡|x−i|So statement 1 is true but not statement 2.
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