Let `Delta = |{:(-bc,,b^(2)+bc,,c^(2)+bc),(a^(2)+ac,,-ac,,c^(2)+ac),(a^(2)+ab,,b^(2)+ab,,-ab):}|` and the equation `px^(3) +qx^(2) +rx+s=0` has roots a,b,c where `a,b,c in R^(+)` `" if " Delta =27 " and " a^(2) +b^(2)+c^(2) =3`thenA. `3p+2q=0`B. `4p+3q=0`C. `3p+q=0`D. none of these

Let `Delta = |{:(-bc,,b^(2)+bc,,c^(2)+bc),(a^(2)+ac,,-ac,,c^(2)+ac),(a

1.	Let `Delta = \|{:(-bc,,b^(2)+bc,,c^(2)+bc),(a^(2)+ac,,-ac,,c^(2)+ac),(a^(2)+ab,,b^(2)+ab,,-ab):}\|` and the equation `px^(3) +qx^(2) +rx+s=0` has roots a,b,c where `a,b,c in R^(+)` `" if " Delta =27 " and " a^(2) +b^(2)+c^(2) =3`thenA. `3p+2q=0`B. `4p+3q=0`C. `3p+q=0`D. none of these
Answer» Correct Answer - C Multiplying `R_(1) ,R_(2),R_(3) " by " a,b,c ` respectively ,and then taking a,b,c common from` C_(1),C_(2)" and " C_(3)` we get `Delta = \|{:(-bc,,ab+ac,,ac+ab),(ab+bc,,-ac,,bc+ab),(ac+bc,,bc+ac,,-ab):}\|` Now using `C_(2) to C_(2)-C_(1)" and " C_(3) to C_(3) -C_(1) ` and then taking (ab+bc+ca) common from `C_(2) " and " C_(3)` we get `Delta =\|{:(-bc,,1,,1),(ab+bc,,-1,,0),(ac+bc,,0,,-1):}\|xx (ab +bc +ca)^(2)` Now applying `R_(2) to R_(2) +R_(1)` we get `Delta = \|{:(-bc,,1,,1),(ab,,0,,1),(ac+bc,,0,,-1):}\| (ab+bc+ca)^(2)` Expanding along `c_(2)` we get `Delta =(ab+bc+ca)^(2)[ac+bc+ca)^(2)` `=(ab+bc+ca)^(2)` `=(r//p)^(3) =r^(3)//p^(3)` Now given a,b,c are all positive then `A.M ge G.M.` `rArr (ab+bc+ac)/(3) ge (abxx bcxx ac)^(1//3)` `" or " (ab+bc+ac)^(3) ge 27a^(2)b^(2)c^(2)` `" or " (ab+bc+ca)^(3) ge 27(s^(2)//p^(2))` if `Delta =27` then ab+bc+ca =3 and given that `a^(2) +b^(2)+c^(2)=3` From `(a+b+c)^(2) =a^(2) +b^(2)+c^(2) +2 (ab+bc+ca)` we have `a+b+c = ne 3` `rArr a+b+c =3` `rArr 3p+q=0`