Verified Solution • 2025–26 Session moderate

QJEEM21Ma89b2: JEE Main 2021 Mathematics - Class 11 JEEM21Ma89b2 Solution: Let [x] denote the greatest integer less than or equal to x.…

Let [x] denote the greatest integer less than or equal to x. Then, the values of x ∈\in R satisfying the equation [ex]2+[ex+1]−3=0{[{e^x}]^2} + [{e^x} + 1] - 3 = 0 lie in the interval :

Multiple Choice Options

  • A [0,1e)\left[ {0,{1 \over e}} \right)
  • B [loge2, loge3)
  • C [1, e)
  • D [0, loge2)

Step-by-Step Verified Solution

[ex]2+[ex+1]−3=0{[{e^x}]^2} + [{e^x} + 1] - 3 = 0

⇒[ex]2+[ex]+1−3=0\Rightarrow {[{e^x}]^2} + [{e^x}] + 1 - 3 = 0

Let [ex]=t[{e^x}] = t

⇒t2+t−2=0\Rightarrow {t^2} + t - 2 = 0

⇒t=−2,1\Rightarrow t = - 2,1

[ex]=−2[{e^x}] = - 2 (Not possible)

or [ex]=1[{e^x}] = 1 ∴\therefore 1 \le {e^x} < 2

\Rightarrow \ln (1) \le x < \ln (2)

\Rightarrow 0 \le x < \ln (2)

⇒x∈[0,ln⁡2)\Rightarrow x \in [0,\ln 2)

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