1.

The number of ordered pairs (x, y) satisfying `4("log"_(2) x^(2))^(2) + 1 = 2 "log"_(2)y " and log"_(2)x^(2) ge "log"_(2) y`, isA. 1B. 2C. more than 2 but finiteD. infinite

Answer» Correct Answer - D
We have,
`4("log"_(2) x)^(2) + 1 = 2"log"_(2) y " and log"_(2) x^(2) ge "log"_(2)y`
`rArr 4("log"_(2)x)^(2) + "log"_(2)2 le 2 "log"_(2)x^(2)`
`rArr 4("log"_(2) x)^(2) - 4"log"_(2) x + 1 le 0`
`rArr (2"log"_(2)x-1)^(2) le 0 rArr "log"_(2) = (1)/(2) rArr x = sqrt(2)`
Now,
`"log"_(2) x^(2) ge "log"_(2) y`
`rArr x^(2) ge y " and " y gt 0 rArr 2 ge y gt 0 rArr 0 lt y le 2`
Hence, the ordered pairs are `(sqrt(2), y)`, where `0 lt y lt 2`


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