Find the sum of the roots (real or complex) of the equation `x^2001 + (1/2-x)^2001=0` .

Correct Answer - C `(c )` ` (a^(2)+b^(2))x^(2)-2b(a+c)x+b^(2)+c^(2)=0` `D=4b^(2)(a+c)^(2)-4(a^(2)+b^(2))(b^(2)+c^(2))` `=-4(b^(4)-2b^(2)ac+a^(2)c^(2))` `=-(n^(2)-ac)^(2)` For real roots, `D ge 0` `implies -4(b^(2)-ac)^(2) ge 0` `implies b^(2)-ac=0` `implies` Roots are real and equal. `:.` Roots...

Correct Answer - C `(c )` `"Sum=Product"` `(4)/(a)=a implies a^(2)=4` Also `D lt 0 implies (4)^(2)-4a^(3) lt 0` `4(4-a^(3)) lt 0` `a=2`

Correct Answer - D `(d)` `alpha+beta=-(b)/(a)`, `alpha beta=(c )/(a)`, Sum of roots of the equation `a^(2)x^(2)+(b^(2)-2ac)x+b^(2)-4ac=0` ,brgt `x_(1)+x_(2)=(2ac-b^(2))/(a^(2))` `=(2c)/(a)-((b)/(a))^(2)` `=2alpha beta-(alpha+beta)^(2)` `=-(alpha^(2)+beta^(2))`

Correct Answer - C::D `(c,d)` Let `P=2alpha+2` `implies alpha=(P-2)/(3)` Since `9alpha^(3)-7alpha+6=0` `implies (9(P-2)^(3))/(27)-(7)/(3)(P-2)+6=0` `implies (1)/(3)(P^(3)-8-6P^(2)+12P)-(7)/(3)P+(14)/(3)+6=0` `implies P^(3)-6P^(2)+12P-8-7P+14+18=0` `impliesP^(3)-6P^(2)+5P+24=0` so, the equation `x^(3)-6x^(2)+5x+24=0` has roots `3alpha+2`, `3beta+2`, `3gamma+2`.

Correct Answer - B `(b)` Let us consider `alpha` as the real and `ibeta` as the imaginergy root. Then `alpha+ibeta=-a` `implies alpha-ibeta=-bara` `implies2alpha=-(a+bara)` and `2ibeta=-(a-bara)` `implies 4ialphabeta=a^(2)-bara^(2)` `implies a^(2)-bara^(2)=4b`

Correct Answer - B::C `(b,c)` `alpha,beta,gamma,delta` are in `H.P.` `implies(1)/(alpha)`, `(1)/(beta)`, `(1)/(gamma)`, `(1)/(delta)` are in `A.P.` Let `d` be the common difference of the `A.P.` Since `alpha`, `gamma` are roots of...

Correct Answer - `(x-2)`

Correct Answer - `0005`

correct answer is D