Theory of Equations 1 Question 47
48. Find the set of all solutions of the equation
$ 2^{|y|}-\left|2^{y-1}-1\right|=2^{y-1}+1 $
(1997 C, 3M)
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Answer:
Correct Answer: 48. [$y \in(-1) \cup (1, \infty)$]
Solution:
- Given, $2^{|y|}-\left|2^{y-1}-1\right|=2^{y-1}+1$
Case I When $y \in(-\infty, 0]$
$ \begin{array}{lc} \therefore & 2^{-y}+\left(2^{y-1}-1\right)=2^{y-1}+1 \\ \Rightarrow & 2^{-y}=2 \\ \Rightarrow & y=-1 \in(-\infty, 0] \end{array} $
Case II When $y \in(0,1]$
$ \begin{aligned} & \therefore \quad 2^{y}+\left(2^{y-1}-1\right)=2^{y-1}+1 \\ & \Rightarrow \quad 2^{y}=2 \\ & \Rightarrow \quad y=1 \in(0,1] \end{aligned} $
Case III When $y \in(1, \infty)$
$ \begin{array}{rlrl} \therefore & 2^{y}-2^{y-1}+1 =2^{y-1}+1 \\ \Rightarrow & 2^{y}-2 \cdot 2^{y-1} =0 \\ \Rightarrow & 2^{y}-2^{y} =0 \text { true for all } y>1 \end{array} $
From Eqs. (i), (ii) and (iii), we get
[$y \in(-1) \cup (1, \infty)$]
