Consider a2 x-bx-ca-ba-c- b2 x-cx-ab-cb-a- c2 x-ax-bc-ac-b- x2- then which of the following is-are true

A^2 (x b)(x c)(a b)(a c)+ nbsp;b^2 (x c)(x a)(b c)(b a)+ nbsp;c^2 (x a)(x b)(c a)(c b)= x^2 By observation, x = a, x = b, x = c satisfy the given equation. ...

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Byju's Answer

Standard XIII

Mathematics

Quadratic Identity

Consider a2 x...

Question

Consider $\frac{a^{2} (x - b) (x - c)}{(a - b) (a - c)} + \frac{b^{2} (x - c) (x - a)}{(b - c) (b - a)} + \frac{c^{2} (x - a) (x - b)}{(c - a) (c - b)} = x^{2},$ then which of the following is/are true

Given equation is an identity

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a, b, c

are the roots of the given equation

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Given equation is not an identity

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Roots of the given equation are equal

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Solution

The correct option is B $a, b, c$ are the roots of the given equation
$\frac{a^{2} (x - b) (x - c)}{(a - b) (a - c)} + \frac{b^{2} (x - c) (x - a)}{(b - c) (b - a)} + \frac{c^{2} (x - a) (x - b)}{(c - a) (c - b)} = x^{2}$
By observation,
$x = a, x = b, x = c$ satisfy the given equation.
$\Rightarrow a, b, c$ are the roots
and there are $3$ roots which is greater than the degree of the given equation.
$\Rightarrow$ It is a identity.

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Similar questions

Q. Consider

\frac{a^{2} (x - b) (x - c)}{(a - b) (a - c)} + \frac{b^{2} (x - c) (x - a)}{(b - c) (b - a)} + \frac{c^{2} (x - a) (x - b)}{(c - a) (c - b)} = x^{2},

then which of the following is/are true

Q. Consider

\frac{a^{2} (x - b) (x - c)}{(a - b) (a - c)} + \frac{b^{2} (x - c) (x - a)}{(b - c) (b - a)} + \frac{c^{2} (x - a) (x - b)}{(c - a) (c - b)} = x^{2},

then which of the following is/are true

Q. Prove the identities:
(1)

\frac{a^{2} (x - b) (x - c)}{(a - b) (a - c)} + \frac{b^{2} (x - c) (x - a)}{(b - c) (b - a)} + \frac{c^{2} (x - a) (x - b)}{(c - a) (c - b)} = x^{2}

Q. If

∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a + b + c)

, then the determinant

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 (x - a)^{2} & (x - b)^{2} & (x - c)^{2} (x - b) (x - c) & (x - c) (x - a) & (x - a) (x - b) \end{matrix} ∣ ∣ ∣ ∣ ∣

vanishes when

Q. If

∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{2} & b^{2} & c^{2} \end{matrix} ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a + b + c)

,
where

a, b, c

are all different,then the determinant

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 (x - a)^{2} & (x - b)^{2} & (x - c)^{2} (x - b) (x - c) & (x - c) (x - a) & (x - a) (x - b) \end{matrix} ∣ ∣ ∣ ∣ ∣

vanishes when

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Consider a2(x−b)(x−c)(a−b)(a−c)+b2(x−c)(x−a)(b−c)(b−a)+c2(x−a)(x−b)(c−a)(c−b)=x2, then which of the following is/are true

Consider $\frac{a^{2} (x - b) (x - c)}{(a - b) (a - c)} + \frac{b^{2} (x - c) (x - a)}{(b - c) (b - a)} + \frac{c^{2} (x - a) (x - b)}{(c - a) (c - b)} = x^{2},$ then which of the following is/are true