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Differentiation of a Determinant

Determinant a...

Question

Determinant a. 1. 1 a. b. c. =(a-b)(b-c)(c-a). (a+b+c) a^3 b^3 c^3 Prove using properties LHS=RHS

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Q. Using properties of determinants, prove the following:

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

Q. Using the properties of determinants, show that:
(i)

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

∣ ∣ ∣ ∣ ∣ \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)

\frac{(a + b)^{3} + (b + c)^{3} + (c + a)^{3} - 3 (a + b) (b + c) (c + a)}{3 (a^{3} + b^{3} + c^{3} - 3 a b c)}

By using properties of determinants, show that:

(i)

(ii)

Q. The value of

\frac{{(a - b)}^{3} + {(b - c)}^{3} + {(c - a)}^{3}}{(a - b) (b - c) (c - a)}

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Differentiation of a Determinant

Standard XII Mathematics

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