Let Δ=∣∣∣∣∣1+x1y11+x2y11+x3y11+x1y21+x2y21+x3y21+x1y31+x2y31+x3y3∣∣∣∣∣ then value of Δ is

[d] We can write Δ=Δ1+y1Δ2, where Δ1=∣∣∣∣∣1111+x1y21+x2y21+x3y21+x1y31+x2y31+x3y3∣∣∣∣∣ and Δ2=∣∣∣∣∣x1x2x31+x1y21+x2y21+x3y21+x1y31+x2y31+x3y3∣∣∣∣∣ In Δ1, use C2→C2−C1 and C3→C3−C1 so that, Δ1=∣∣∣∣∣111x1y2x2y2x3y2x1y3x2y3x3y3∣∣∣∣∣=0 [∵C2 and C3 are proportional] In  Δ2, us C2→C2−y2C1 and C3→C3−y3C1 to get Δ2=∣∣∣∣x1x2x3111111∣∣∣∣=0 [∵C2 and C3 are identical] ∴Δ=0