Repeated roots : If equation f(x) = 0, where f(x) is a polyno- mial function, has roots `alpha,alpha,beta,… or alpha` root is repreated root, then f(x) = 0 is equivalent to `(x-alpha)^(2)(x-beta)…=0,` from which we can conclude that `f(x)=0 or 2(x-alpha)[(x-beta)...]+(x-alpha)^(2)[(x-beta)...]'=0 or (x-alpha) [2 {(x-beta)...}+(x-alpha){(x-beta)...}']=0` has root `alpha`. Thus, if `alpha` root occurs twice in the, equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if `alpha` root occurs thrice in equation, then it is common in the equations f(x)=0, f'(x)=0, and f'''(x)=0.
If `a_(1)x^(3)+b_(1)x^(2)+c_(1)x+d_(1)=0 and a_(2)x^(3)+b_(2)x^(2)+c_(2)x+d_(2)=0` have a pair of repeated roots common, then
`|{:(3a_(1),2b_(1),c_(1)),(3a_(2),2b_(2),c_(2)),(a_(2)b_(1)-a_(1)b_(2),c_(1)a_(2)-c_(2)a_(1),d_(1)a_(2)-d_(2)a_(1)):}|=`

Options

A. 0
B. 1
C. -1
D. 2

Explanation

`"Let "f(x)=a_(1)x^(3)+b_(1)x^(2)+c_(1)x+d_(1)=0" has roots "alpha, alpha,beta.`
`"Then "g(x)=a_(2)x^(3)+b_(2)x^(2)+c_(2)x+d_(2)=0" must have roots "alpha,alpha, gamma.` Then
`a_(1)alpha^(3)+b_(1)alpha^(2)+c_(1)alpha+d_(1)=0" (1)"`
`"and "a_(2)alpha^(3)+b_(2)alpha^(2)+c_(2)alpha+d_(2)=0" (2)"`
`alpha" is alos a root of equations "f'(x)=3a_(1)x^(2)+2b_(1)x+c_(1)=0 and `
`g'(x)=3a_(2)x^(2)+2b_(2)x+c_(2)=0.` Then
`3a_(1)alpha^(2)+2b_(1)alpha+c_(1)=0" (3)"`
`"and "3a_(2)alpha^(2)+2b_(2)alpha+c_(2)=0" (4)"`
`"Also, from "a_(2)(1)-a_(1)(2),"we have"`
`(a_(2)b_(1)-a_(1)b_(2))alpha^(2)+(c_(1)a_(2)-c_(2)a_(1))alpha+d_(1)a_(2)-d_(2)a_(1)=0" (5)"`
Eliminating `alpha" from "(3),(4), and (5)` we have
`|{:(3a_(1),2b_(1),c_(1)),(3a_(2),2b_(2),c_(2)),(a_(2)b_(1)-a_(1)b_(2),c_(1)a_(2)-c_(2)a_(1),d_(1)a_(2)-d_(2)a_(1)):}|=0.`