Question

All the pair (x,y) that satisfy the inequality `2^(sqrt(sinx-2sinx+5)).(1)/(4^(sin^(2)y))le1` also satisfy the equation
A. `2|sinx|=3siny`
B. `sinx=siny`
C. `sin x=2siny`
D. `2 sin x=siny`

Answer 1

Correct Answer - B
Given, inequality is
`2^(sqrt(sin^(2)n-2sinx+5)),(1)/(4^(sin^(2)y))le1`
`implies2^(sqrt((sinx-1)^(2)+4)).2^(2sin^(2)y)le1`
`implies2^(sqrt((sinx-1)^(2)+4))le2^(sin^(2)y)`
`impliessqrt((sinx-1)^(2)+4)le2sin^(2)y`
`" "[if agt1and a^(m)lea^(n)impliesmlen]`
`because` Range of `sqrt((sinx-1)^(2)+4)is[2,2sqrt2]`
and range of `2 sin ^(2)y is[0,2].`
`therefore` the above inequality holds, iff `sqrt((sinx-1)^(2)+4)=2=2sin^(2)y`
`impliessinx=1and sin^(2)y=1`
`impliessinx=|siny|" "["from the options"]`