Consider the polynomial fucntion
`f(x) = |{:((1+x)^(2),,(1+2x)^(b),,1),(1,,(1+x)^(a),,(1+2x)^(b)),((1+2x)^(b),,1,,(1+x)^(a)):}|`
a,b being positive integers.
Which of the following is true ?

Options

A. All the roots of the equation f(x)=0 are positive
B. All the roots of the equation f(x)=0 are negative
C. At least one of the equation f(x)=0 is repeating one .
D. None of these

Answer

C

Explanation

Let
`|{:((1+x)^(a),,(1+2x)^(b),,1),(1,,(1+x)^(a),,(1+2x)^(b)),((1+2x)^(b),,1,,(1+x)^(a)):}|=A +Bx+Cx^(2)+…….`
Putting x=0 we get
`A= |{:(1,,1,,1),(1,,1,,1),(1,,1,,1):}|=0`
now differenting both sides with respect to x and putting x=0 we get
`B= |{:(a,,2b,,0),(1,,1,,1),(1,,1,,1):}|+|{:(1,,1,,1),(0,,a,,2b),(1,,1,,1):}|+|{:(1,,1,,1),(1,,1,,1),(2b,,0,,a):}|=0`
Hence ,coefficient of x is 0 .Since f(x)=0 and f(0)=0 ,x=0 is a repreating root of the equation f(x)=0