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Question

Statement1: If the function f(x)=ax22bx[x]+c[x]2, where a,b,cN is periodic with period 1, then a,b,c are in A.P., G.P. and H.P.

Statement2: Three non-zero numbers are in A.P., G.P. and H.P. if and only if they are equal.
( Here, [.] denotes the greatest integer function)


A
Statement1 is true, Statement2 is true and Statement2 is correct explanation for Statement1.
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B
Statement1 is true, Statement2 is true and Statement2 is NOT the correct explanation for Statement1.
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C
Statement1 is true, Statement2 is false.
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D
Statement1 is false, Statement2 is true.
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Solution

The correct option is D Statement1 is true, Statement2 is true and Statement2 is correct explanation for Statement1.
f(x)=ax22bx[x]+c[x]2
Put [x]=x{x}
f(x)=ax22bx(x{x})+c(x{x})2f(x)=(a2b+c)x2+(2b2c)x{x}+c{x}2
We know that {x} is periodic with period 1, so f(x) is periodic if 
(a2b+c)=0, (2b2c)=0b=c=a
Hence, three non-zero equal numbers are in A.P., G.P. and H.P.

Mathematics

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