A) and (ii) are true while (iii) is false.
B) and (iii) are true while (ii) is false.
C) (ii) and (iii) are true while (i) is false.
D) Neither (i) and (ii) nor (iii) is true.
Correct Answer: A
Solution :
(i) \[\left( 1-\frac{1}{2} \right)\,\left( 1-\frac{1}{3} \right)\,\left( 1-\frac{1}{4} \right)\,.........\left( 1-\frac{1}{x} \right)\] \[=\frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \,.........\times \frac{x-1}{x}=\frac{1}{x}\] (ii) \[2=x+\frac{1}{1+\frac{1}{\frac{12+1}{4}}}\,\,\,\,\,\,\,\,\,=x+\frac{1}{1+\frac{1}{\frac{13}{4}}}\] \[=x+\frac{1}{\frac{13+4}{13}}=x+\frac{13}{17}\] \[\therefore \,\,\,\,\,\,\,\,\,\,x=12-\frac{13}{17}=\frac{21}{17}\] (iii) \[(999\times 6)\,+\left( \frac{1}{7}+\frac{2}{7}+\frac{3}{7}+\frac{4}{7}+\frac{5}{7}+\frac{6}{7} \right)\] \[=5994+\frac{21}{7}\,=5994+3=5997\]
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