find the number of integral values of a in the interval [0,100] so that the range of the function f(x)= (x+a)/(x²-1) contains the interval [0,1]???

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{x}+\mathrm{a}}{{\mathrm{x}}^2-1}=\frac{\mathrm{x}+\mathrm{a}}{\left(\mathrm{x}-1\right)\left(\mathrm{x}+1\right)} \\ \mathrm{When} \mathrm{a}=1 \\ \mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{x}+1}{\left(\mathrm{x}-1\right)\left(\mathrm{x}+1\right)} \\ \mathrm{f}\left(\mathrm{x}\right) \mathrm{cannot} \mathrm{be} 0 \mathrm{in} \mathrm{this} \mathrm{case} \mathrm{as} \mathrm{numerator} \mathrm{becomes} 0 \mathrm{at} \mathrm{x}=-1 \mathrm{but} \mathrm{x}=-1 \\ \mathrm{make} \mathrm{denominator} 0 \mathrm{too} \mathrm{which} \mathrm{makes} \mathrm{function} \mathrm{undefined}. \\ \mathrm{Hence} \mathrm{a}\ne 1 \mathrm{if} \mathrm{we} \mathrm{want} \mathrm{range} \mathrm{to} \mathrm{have} \mathrm{the} \mathrm{interval} \left[0,1\right] \\ \mathrm{For} \mathrm{all} \mathrm{other} \mathrm{values} \mathrm{of} \mathrm{a}, \mathrm{you} \mathrm{will} 0 \mathrm{in} \mathrm{range} \mathrm{at} \mathrm{x}=-\mathrm{a} \\ \mathrm{e}.\mathrm{g}. \\ \mathrm{for} \mathrm{a}=2 \\ \mathrm{f}\left(\mathrm{x}\right) \mathrm{will} \mathrm{be} 0 \mathrm{at} \mathrm{x}=-1 \\ \mathrm{Set} \mathrm{f}\left(\mathrm{x}\right)=1 \\ \Rightarrow \frac{\mathrm{x}+\mathrm{a}}{{\mathrm{x}}^2-1}=1 \\ \Rightarrow \mathrm{x}+\mathrm{a}={\mathrm{x}}^2-1 \\ \Rightarrow \mathrm{a}={\mathrm{x}}^2-\mathrm{x}-1 \\ \Rightarrow {\mathrm{x}}^2-\mathrm{x}-1-\mathrm{a}=0 \\ \mathrm{Now} \mathrm{a} \mathrm{is} \mathrm{integer} \mathrm{in} \mathrm{the} \mathrm{interval} \left[0,100\right] \\ \mathrm{a}=1 \mathrm{is} \mathrm{already} \mathrm{discarded} \\ \mathrm{for} \mathrm{a}=0,2,3,4....,100 \\ -1-\mathrm{a} \mathrm{will} \mathrm{always} \mathrm{be} \mathrm{negative}. \\ \mathrm{Roots} \mathrm{of} {\mathrm{Ax}}^2+\mathrm{Bx}+\mathrm{C} \mathrm{is} \mathrm{always} \mathrm{real} \mathrm{when} \mathrm{AC}<0 \\ \mathrm{Here} \mathrm{A}=1 \mathrm{and} \mathrm{C}=-1-\mathrm{a} \\ \Rightarrow \mathrm{AC}<0 \\ \mathrm{Hence} \mathrm{roots} \mathrm{of} {\mathrm{x}}^2-\mathrm{x}-1-\mathrm{a}=0 \mathrm{will} \mathrm{be} \mathrm{real}. \\ \mathrm{Hence} 1 \mathrm{will} \mathrm{also} \mathrm{be} \mathrm{part} \mathrm{of} \mathrm{range} \\ \Rightarrow \left[0,1\right] \mathrm{will} \mathrm{be} \mathrm{for} \mathrm{all} \mathrm{a}\in \left[0,100\right] \mathrm{except} \mathrm{a}=1 \\ \mathbf{Hence}\mathbf{ }\mathbf{100}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{answer} \end{gathered}$$

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