If a, b, c are distinct and |a a2 a3-1 b b2 b3-1 c c2 c3-1|=0 then abc equals

Question

If a, b, c are distinct and |a a2 a3-1 b b2 b3-1 c c2 c3-1|=0 then abc equals (A) 0 (B) 1 (C) -1 (D) -2

Tardigrade Admin · Accepted Answer

Write the determinant as =a b c Δ1-Δ2 where Δ1=|1 a a2 1 b b2 1 c c2|=(a-b)(b-c)(c-a) and Δ2=|a a2 1 b b2 1 c c2 1|=Δ1 ∴(a-b)(b-c)(c-a)(a b c-1)=0 Since, a, b, c are distinct, we get a b c-1=0 text or a b c=1

Q. If a,b,c are distinct and ∣∣​abc​a2b2c2​a3−1b3−1c3−1​∣∣​=0 then abc equals

0

1

-1

-2

Write the determinant as =abcΔ1​−Δ2​ where Δ1​=∣∣​111​abc​a2b2c2​∣∣​=(a−b)(b−c)(c−a) and Δ2​=∣∣​abc​a2b2c2​111​∣∣​=Δ1​ ∴(a−b)(b−c)(c−a)(abc−1)=0 Since, a,b,c are distinct, we get abc−1=0 or abc=1

Q. If $a, b, c$ are distinct and $∣ ∣ a b c a^{2} b^{2} c^{2} a^{3} - 1 b^{3} - 1 c^{3} - 1 ∣ ∣ = 0$ then $ab c$ equals