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Question

# If 2x−12x3+3x2+x is positive, then x lies in the interval-

A
(1.5,0.5)
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B
(2.5,)
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C
(0.5,2.5)
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D
(,1.5)
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Solution

## The correct options are B (−1.5,0.5) C (0.5,2.5) D (−∞,−1.5)Given 2x−12x3+3x2+x is positive.⇒2x−12x3+3x2+x>0⇒2x−1x(2x2+3x+1)>0⇒2x−1x(2x+1)(x+1)>0Now let us compute the signs of each term.2x−1=0⇒x=122x−1<0⇒x<122x−1>0⇒x>12x=0⇒x=0,x<0⇒x<0,x>0⇒x>0x+1=0⇒x=−1,x+1<0⇒x<−1,x+1>0⇒x>−12x+1=0⇒x=−122x+1<0⇒x<−122x+1>0⇒x>−12Now summarize the above in a table: (−∞,−1) −1 (−1,−1/2) −1/2 (−1/2,0) 0(0,1/2) 1/2 (1/2,∞)2x−1 − − − − − − −0 +x − − − − − 0 + + + x+1 − 0 + + ++ + + + 2x+1 − − − 0 ++ + ++ 2x−1x(x+1)(2x+1) + ND− ND+ ND −0 +Now consider the above table.Check the column (−∞,−1), we get 2x−1x(x+1)(2x+1)=−(−)(−)(−)>0 .Now check column (−1/2,0), we get 2x−1x(x+1)(2x+1)=−(−)(+)(+)>0 .and check column (1/2,∞), we get 2x−1x(x+1)(2x+1)=+(+)(+)(+)>0 .Thus 2x−12x3+3x2+x is positive at (−∞.−1),(−1/2,0),(1/2,∞).Thus the correct answers are (−1.5,0.5),(0.5,2.5)and(−∞,−1.5)

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