A) \[4,\,\,2\]
B) \[0,\,\,4\]
C) \[-1,\,\,3\]
D) \[5,\,\,1\]
Correct Answer: B
Solution :
Case I: When\[x<2\] \[\therefore \] \[{{(x-2)}^{2}}-(x-2)-6=0\] \[\Rightarrow \] \[{{x}^{2}}+4-4x-x-4=0\] \[\Rightarrow \] \[{{x}^{2}}-5x=0\] \[\Rightarrow \] \[x=0,\,\,x=5\] But we have\[x<2\] \[\therefore \]We take\[x=0\] Case II: When\[x\ge 2\] \[\therefore \] \[{{(x-2)}^{2}}+(x-2)-6=0\] \[\Rightarrow \] \[{{x}^{2}}+4-4x+x-8=0\] \[\Rightarrow \] \[{{x}^{2}}-3x-4=0\] \[\Rightarrow \] \[(x+1)(x-4)=0\] \[\Rightarrow \] \[x=-1,\,\,4\] \[\Rightarrow \] \[x=4\] \[(\because \,\,x\ge 2)\] \[\therefore \]Required roots are\[0,\,\,4\]. Alternative Solution: Let \[y=|x-2|\] \[{{y}^{2}}+y-6=0\] \[\Rightarrow \] \[{{y}^{2}}+3y-2y-6=0\] \[\Rightarrow \] \[(y+3)(y-2)=0\] \[\Rightarrow \] \[y=-3,\,\,2\]. It is clear from the figure that line \[y=2\] intersect the curve \[y=\,\,|x-2|\] at two distinct points whose \[x-\]coordinates are\[0,\,\,\,4\].You need to login to perform this action.
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