A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
B) Statement-1 is true, Statement-2 is true; Statement-2 is NOT a correct explanation for Statement-1
C) Statement-1 is true, Statement-2 is false
D) Statement-1 is false, Statement-2 is true.
Correct Answer: A
Solution :
\[f(x)={{(x-1)}^{2}}+1,x\ge 1\] \[f:[1,\infty )\to [1,\infty )\]is a bijective function \[\Rightarrow \]\[y={{(x-1)}^{2}}+1\Rightarrow {{(x-1)}^{2}}=y-1\] \[\Rightarrow \]\[x=1\pm \sqrt{y-1}\Rightarrow {{f}^{-1}}(y)=1\pm \sqrt{y-1}\] \[\Rightarrow \]\[{{f}^{-1}}(x)=1+\sqrt{x-1}\{\therefore x\ge 1\}\] so statement-2 is correct Now \[f(x)={{f}^{-1}}(x)\Rightarrow f(x)=x\Rightarrow {{(x-1)}^{2}}+1=x\] \[\Rightarrow \]\[{{x}^{2}}-3x+2=0\Rightarrow x=1,2\] so statement-1 is correct
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