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)

Show Answer

Answer:

Correct Answer: 48. [$y \in(-1) \cup (1, \infty)$]

Solution:

  1. 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

[$y \in(-1) \cup (1, \infty)$]



NCERT Chapter Video Solution

Dual Pane