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If f(x)=|((1+x)^a,(1+2x)^b,1),(1,(1+x)^a...

If `f(x)=|((1+x)^a,(1+2x)^b,1),(1,(1+x)^a,(1+2x)^b), ((1+2x)^b,1,(1+x)^a)|` then find constant term and coefficient of x

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

Text Solution

Verified by Experts

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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