Q55 / 100 · BITSAT 2024 (Memory Based Paper 3) · Mathematics

Number of real solution of $\sqrt{5-\log _2|x|}$ $=3-\log _2|x|$ is equal to

  • A
    1
  • Correct answer
    B
    2
  • C
    3
  • D
    4

Solution

$\begin{aligned} \text { } & \text { Let } & \log _2|x| & =t \ldots(i)\\ & \therefore & \sqrt{5-t} & =3-t \\ \Rightarrow & & 5-t & =(3-t)^2 \\ \Rightarrow & & 5-t & =9+t^2-6 t \end{aligned}$ $\begin{aligned} & \Rightarrow \quad t^2-5 t+4=0 \\ & \Rightarrow \quad(t-4)(t-1)=0 \\ & \Rightarrow \quad t=1 \text { or } 4 \text{[rejected]}\\ & {[\because \sqrt{5-4}=3-4,1 \neq-1]} \\ & \Rightarrow \quad \log _2|x|=1 \\ & \Rightarrow \quad|x|=2 \\ & \Rightarrow \quad x= \pm 2 \end{aligned}$ $\therefore$ Total 2 real solutions.

Hence, the correct answer is option: 'B'.

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