Let α be a root of the equation a-cx2+b-ax+c-b=0 where a, b, c are distinct real numbers such that the matrix α2α1111abc is singular. Then the value of (a−c)2(b−a)(c−b)+(b−a)2(a−c)(c−b)+(c−b)2(a−c)(b−a) is

(a−c)α2+(b−a)α+(c−b)=0⇒α=1and (a−c)2(b−a)(c−b)+(b−a)2(a−c)(c−b)+(c−b)2(a−c)(b−a)=(a−c)3+(b−a)3+(c−b)3(b−a)(c−b)(a−c)=3 ∵x+y+z=0⇒x3+y3+z3=3xyz

Let α be a root of the equation a-cx2+b-ax+c-b=0 where a, b, c are distinct real numbers such that the matrix α2α1111abc is singular. Then the value of (a−c)2(b−a)(c−b)+(b−a)2(a−c)(c−b)+(c−b)2(a−c)(b−a) is

(a−c)α2+(b−a)α+(c−b)=0⇒α=1and (a−c)2(b−a)(c−b)+(b−a)2(a−c)(c−b)+(c−b)2(a−c)(b−a)=(a−c)3+(b−a)3+(c−b)3(b−a)(c−b)(a−c)=3 ∵x+y+z=0⇒x3+y3+z3=3xyz

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Let $α$ be a root of the equation $(a - c) x^{2} + (b - a) x + (c - b) = 0$ where a, b, c are distinct real numbers such that the matrix $[\begin{array}{ccc} α^{2} & α & 1 \\ 1 & 1 & 1 \\ a & b & c \end{array}]$ is singular. Then the value of $\frac{(a - c)^{2}}{(b - a) (c - b)} + \frac{(b - a)^{2}}{(a - c) (c - b)} + \frac{(c - b)^{2}}{(a - c) (b - a)}$ is

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Detailed Solution

$\begin{array}{l} (a - c) α^{2} + (b - a) α + (c - b) = 0 \Rightarrow α = 1 \\ a n d \frac{(a - c)^{2}}{(b - a) (c - b)} + \frac{(b - a)^{2}}{(a - c) (c - b)} + \frac{(c - b)^{2}}{(a - c) (b - a)} \\ = \frac{(a - c)^{3} + (b - a)^{3} + (c - b)^{3}}{(b - a) (c - b) (a - c)} \\ = 3 (\begin{array}{ll} ∵ & x + y + z = 0 \\ \Rightarrow & x^{3} + y^{3} + z^{3} = 3 xyz \end{array}) \end{array}$