1.

Find the value of y if the following equations has at least one solution. (i) 5sinx + 12cosx = y^(2)-4y+17 (ii) 4"cosec "^(2)pi(y+x)+y^(2)-4y=0

Answer»

Solution :(i) `-13 le 5sinx + 12 cosx le 13`…………(i)
Also, `y^(2)-4y+17=(y-2)^(2)+13 ge 13`……….(ii)
By (i) and (ii) we observe that SOLUTIONS is obtained at `5 sinx + 12 cosx = (y-2)^(2) + 13=13`
`RARR y=2`.
(ii) `4 " cosec "pi(y+x)=-y^(2)+4y-4+4=4-(y-2)^(2)`
As we KNOW that `4" cosec "^(2)pi(y+x) ge 4` and `4-(y-2)^(2) le 4`
Hence, solution is obtained at `4" cosec " ^(2)pi(y+x)=4-(y-2)^(2)=4 rArr y=2`


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