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AP EAMCET201923 Apr 2019Morning ShiftMathematicsLimitsActual

Let ([x] ) denote the greatest integer not exceeding (x ). If (l₁= _ x 2⁺ (x^2+[x] ) ), (l₂= _ x 3⁻ (2 x-[x]) ) and (l₃= _ x 2 ( x x- 2 ) ), then

Options

  1. A(I₂ < I₃ < I₁ )
  2. B(I₁ < I₃ < I₂ )
  3. C(I₁ < I₂ < I₃ )
  4. D(I₃ < I₂ < I₁ )

Correct answer

D. (I₃ < I₂ < I₁ )

Step-by-step solution

Given, ( aligned l₁ & = _ x 2⁺ (x^2+[x] ) & = _ x 2⁺ (x^2+2 ) [ as x 2⁺,[x]=2 ] & =4+2 l₁ & =6 l₂ & = _ x 3⁻ (2 x-[x]) [ as x 3⁻,[x]=2 ] & = _ x 3⁻ (2 x-2) & =2(3)-2 l₂ & =4 l₃ & = _ x 2 ( x x- 2 ) aligned ) Put (x- 2 =y ) and as (x 2 ), then (y 0 ) ( aligned x & = 2 +y & = _ y 0 ( 2 +y ) y & = _ y 0 - y y l₃ & =-1 & l₃ < l₂ < l₁ aligned ) ( ) Hence, answer is (d).

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