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Question Number 204869 by BaliramKumar last updated on 29/Feb/24
How many distinct positive integer valued solution                           exist the equation (x^2  − 7x + 11)^((x^2  −13x + 42))  = 1               (a) 2              (b) 4                (c) 6                 (d) 8
$$\mathrm{How}\:\mathrm{many}\:\mathrm{distinct}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{valued}\:\mathrm{solution}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{exist}\:\mathrm{the}\:\mathrm{equation}\:\left({x}^{\mathrm{2}} \:−\:\mathrm{7}{x}\:+\:\mathrm{11}\right)^{\left({x}^{\mathrm{2}} \:−\mathrm{13}{x}\:+\:\mathrm{42}\right)} \:=\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\mathrm{8} \\ $$
Answered by A5T last updated on 29/Feb/24
x^2 −7x+11=1⇒x=5 or 2  x^2 −7x+11=−1⇒x=4 or 3  x^2 −13x+42=0⇒x=7 or 6                                ⇒(c)
$${x}^{\mathrm{2}} −\mathrm{7}{x}+\mathrm{11}=\mathrm{1}\Rightarrow{x}=\mathrm{5}\:{or}\:\mathrm{2} \\ $$$${x}^{\mathrm{2}} −\mathrm{7}{x}+\mathrm{11}=−\mathrm{1}\Rightarrow{x}=\mathrm{4}\:{or}\:\mathrm{3} \\ $$$${x}^{\mathrm{2}} −\mathrm{13}{x}+\mathrm{42}=\mathrm{0}\Rightarrow{x}=\mathrm{7}\:{or}\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\left({c}\right) \\ $$
Commented by BaliramKumar last updated on 29/Feb/24
thanks
$$\mathrm{thanks} \\ $$

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