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Consider thepolynomial 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 beingpositiveintegers. Whichof the followingis true ? |
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Answer» All the rootsof theequationf(x)=0are positive `|{:((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` nowdifferentingboth sides withrespect 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 ,coefficientof x IS0 .Since f(x)=0 and f(0)=0 ,x=0 isa repreatingroot ofthe equation f(x)=0 |
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