  Question

Statement−1: If the function f(x)=ax2–2bx[x]+c[x]2, where a,b,c∈N is periodic with period 1, then a,b,c are in A.P., G.P. and H.P. Statement−2: 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.  B
Statement1 is true, Statement2 is true and Statement2 is NOT the correct explanation for Statement1.  C
Statement1 is true, Statement2 is false.  D
Statement1 is false, Statement2 is true.  Solution

The correct option is D Statement−1 is true, Statement−2 is true and Statement−2 is correct explanation for Statement−1.f(x)=ax2–2bx[x]+c[x]2 Put [x]=x−{x} ⇒f(x)=ax2−2bx(x−{x})+c(x−{x})2⇒f(x)=(a−2b+c)x2+(2b−2c)x{x}+c{x}2 We know that {x} is periodic with period 1, so f(x) is periodic if  (a−2b+c)=0, (2b−2c)=0⇒b=c=a Hence, three non-zero equal numbers are in A.P., G.P. and H.P.Mathematics

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