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Consider the polynomial fucntion f(x) ...

Consider the polynomial fucntion `f(x) = |{:((1+x)^(a),,(1+2x)^(b),,1),(1,,(1+x)^(a),,(1+2x)^(b)),((1+2x)^(b),,1,,(1+x)^(a)):}|` a,b being positive integers. The constant term in f(x) is

A

2

B

1

C

-1

D

0

Text Solution

Verified by Experts

The correct Answer is:
D
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
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