The number of orded 4-tuples `(x,y,z,w)` where `x,y,z,w in[0,10]` which satisfy the inequality
`2^(sin^(2)x)xx3^(cos^(2)y)xx4^(sin^(2)z)xx5^(cos^(2)w)gt120,` is

Options

A. `0`
B. `144`
C. `81`
D. infinite

Answer

B

Explanation

`:'x,y,z,w in [0,10]`
`2^(sin^(2)x.3^(cos^(2)y).4^(sin^(2)z).5^(cos^(2)w)ge2.3.4.5`
`:. 2^(cos^(2)x)3^(sin^(2)y).4cos^(2)z.5^(sin^(2)w)le1`
Then `cos^(2)xlog2+sin^(2)ylog3`
`+cos^(2)zlog4+sin^(2)wlog5 le 0`
It is possible when `cos^(2)x = 0`
`rArr x = mpi + (pi)/(2) : m in I`
`sin^(2)y = 0 rArr y = npi , n in I`
`cos^(2)z = 0 rArr z = rpi + (pi)/(2) , r in I`
`sin^(2)w = 0 rArr w = p pi, p in I`
`x,y,z,w in [0,10]`
`x = (pi)/(2), (3pi)/(2), (5pi)/(2),y=0,pi,2pi,3pi`
`z = (pi)/(2), (3pi)/(2),(5pi)/(2),w=0,pi,2pi,3pi`
`:. 3.4.4.3 = 144`