InterviewSolution
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InterviewSolution
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The number of real roots of the equation `5+|2^(x)-1|=2^(x)(2^(x)-2)is`A. 1B. 3C. 4D. 2 |
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Answer» Correct Answer - A Given equation `5+|2^(x)-1|=2^(x)(2^(x)-2)` Case I If `2^(x)-1 ge0impliesx ge0,then 5+2^(x)-1=2^(x)(2^(x)-2)` Put `2^(x)=t,`then `5+t-1=t^(2)-2timpliest^(2)-3t-4=0` `impliest^(2)-4t+t-4=0impliest(t-4)+1(t-4)=0` `impliest=4or-1impliest=4" "(becauset=2^(x)gt0)` `implies2^(x)=4impliesx=2gt0` `impliesx=2` is the solution. m Case II `If 2^(x)-1lt0impliesxlt0,` then `5+1-2^(x)=2^(x)(2^(x)-2)` Put `2^(x)=y, then 6-y=y^(2)-2y` `impliesy^(2)-y-6=0impliesy^(2)-3y+2y-6=0` `implies(y+2)(t-3)=0impliesy=3or -2` `impliesy=3(as y=2^(x)gt0)implies2^(x)=3` `impliesx=log_(2)3gt0` So, `x=log_(2)3` is not a solution. Therefore number of real roots is one. |
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