SATHEE: Theory of Equations 1 Question 47

Theory of Equations 1 Question 47

48. Find the set of all solutions of the equation

$2^{| y |} - | 2^{y - 1} - 1 | = 2^{y - 1} + 1$

(1997 C, 3M)

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Answer:

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

Solution:

Given, $2^{| y |} - | 2^{y - 1} - 1 | = 2^{y - 1} + 1$

Case I When $y \in (- \infty, 0]$

$\begin{array}{lc} ∴ & 2^{- y} + (2^{y - 1} - 1) = 2^{y - 1} + 1 \\ \Rightarrow & 2^{- y} = 2 \\ \Rightarrow & y = - 1 \in (- \infty, 0] \end{array}$

Case II When $y \in (0, 1]$

$\begin{aligned} ∴ 2^{y} + (2^{y - 1} - 1) = 2^{y - 1} + 1 \\ \Rightarrow 2^{y} = 2 \\ \Rightarrow y = 1 \in (0, 1] \end{aligned}$

Case III When $y \in (1, \infty)$

$\begin{aligned} ∴ & 2^{y} - 2^{y - 1} + 1 = 2^{y - 1} + 1 \\ \Rightarrow & 2^{y} - 2 \cdot 2^{y - 1} = 0 \\ \Rightarrow & 2^{y} - 2^{y} = 0 true for all y > 1 \end{aligned}$

From Eqs. (i), (ii) and (iii), we get

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

Theory of Equations 1 Question 47

48. Find the set of all solutions of the equation

Answer:

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Theory of Equations 1 Question 47

48. Find the set of all solutions of the equation

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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