If `Delta(x) = |{:(x ,, 1+ x^(2) ,, x^(3)), ("log" (1+ x ^(2)) ,, e^(x) ,, sin x), ( cos x ,, tan x , , sin^(2)x):}|` , then-

Options

A. `Delta(x)` is divisible by x
B. `Delta(x) = 0 `
C. `Delta'(x) = 0`
D. None of these

Answer

A

Explanation

`Delta(x) = |{:(x , 1+ x^(2) , x^(3)), ("log"(1+ x^(2)) , e^(x) , sinx ), (cos x , tan x , sin^(2)x):}| = a_(0) + a_(1) x + a_(2)x^(2) + a_(3)x^(3)+"......"`
put x = 0 `" " a_(0) = Delta (0) = 0`
`Delta'(x) = a_(1) + 2a_(2) x + 3a_(3)x^(2)+ "......."`
`a_(1) = 0 + 1 + 0 = 1 `
`Delta(x) = x + a_(2)x^(2) + a_(3)x^(3) + "......."`
`Delta(x)` is divisible by x