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. the coefficient of x in f(x) isA. `2^(a)`B. `2^(a) -3xx 2^(b)+1`C. `0`D. none of these

Consider the polynomial fucntion `f(x) = |{:((1+x)^(2),,(1+2x)^(b),,1)

1.	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. the coefficient of x in f(x) isA. `2^(a)`B. `2^(a) -3xx 2^(b)+1`C. `0`D. none of these
Answer» Correct Answer - C 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