Let `a, b, c , d` he real numbers such that `a + b+c+d = 10`, then the minimum value of `a^2 cot 9^@-b^2 cot 27^@+c^2 cot 63^@+d^2 cot 81^@` is `sqrtn ,(n in N)` find n.

Options

A. even
B. odd
C. prime
D. divisible by `'5'`

Answer

B, D

Explanation

f `(a_(1), a_(2)……..,a_(n)), (b_(1)b_(2)…..,b_(n))` are `2n` real numbers
then `(a_(1)b_(1) + a_(2)b_(2) + ……. + a_(n)b_(n) )^(2) le (a_(1)^(2) +a_(2)^(2) + …. + a_(n)^(2))(b_(1)^(2) + b_(2)^(2) + …….+ b_(n)^(2))…..(1)`
let `a_(1) = asqrt(cot)g^(@) , a_(2) = bsqrt(cot27)^(@) , a_(3) = c sqrt(cot)63^(@)`
`dsqrt(cot)81^(@) = a_(4)`
`b_(1) = sqrt(tan)9^(@), b_(2) = sqrt(tan)27^(@)`
`b_(3) = sqrt(tan)63^(@) , b_(4) = sqrt(tan)81^(@)`
`(a + b + c + d)^(2) le (a^(2)cot9^(@) + b^(2)cot27^(@) + c^(2)cot63^(@) + d^(2)cot81^(@))(tan9^(@) + tan27^(@) + tan81^(@)).....(2)`
put `a + b + c + d = 10`
`tan9^(@) + tan81^(@) + tan27^(@) + tan63^(@)`
`(1)/(sin9^(@)cos9^(@)) + (1)/(sin27^(@)cos27^(@))`
`= (2)/(sin18^(@)) + (2)/(sin54^(@)) = (2)/((sqrt(5) -1)/(8)) + (2)/((sqrt(5) + 1)/(8)) = 4sqrt(5)`
required `sqrt(n) = sqrt(125)`
`n = 125`