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which of the following statements is incorrect

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a

xsgn⁡x=|x|

b

|x|sgn⁡x=x

c

x(sgn⁡x)(sgn⁡x)=x

d

|x|(sgn⁡x)3=|x|

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detailed solution

Correct option is D

Checking the options (a) LHS =xsgn⁡x=x×(+1)=x if x>0x×0=0 if x=0x×(−1)=−x if x<0 =|x|= RHS Therefore, statement is correct. b LHS =|x|sgn⁡x=(+x)×(+1)=x if x>00×0=0 if x=0(−x)×(−1)=x if x<0Therefore, statement is correct. (c) LHS=(x(sgn⁡x))×sgn⁡x=|x|sgn⁡x=x=RHSTherefore, statement is correct. (d) LHS =(|x|sgn⁡x)(sgn⁡x)2=x(sgn⁡x)2=|x|×(sgn⁡x)=x ≠RHSTherefore, statement is incorrect.

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