Roots of the equation x m n 1 a x n 1 a b x 1 a b c 1 -0 are

The correct option is A independent of m and n x amp; m amp; n amp; 1 a amp; x amp; n amp; 1 a amp; b amp; x ...

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Standard XII

Mathematics

Polynomial Functions

Roots of the ...

Question

Roots of the equation $∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} x & m & n & 1 a & x & n & 1 a & b & x & 1 a & b & c & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ = 0$ are

independent of

m

and

n

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independent of

a, b

and

c

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depend on

m, n

and

a, b, c

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independent of

m, n

and

a, b, c

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Solution

The correct option is A independent of $m$ and $n$
$∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} x & m & n & 1 a & x & n & 1 a & b & x & 1 a & b & c & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ = 0$
Applying $R_{4} \to R_{4} - R_{3}$ and expanding
$∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} x & m & n & 1 a & x & n & 1 a & b & x & 1 0 & 0 & c - x & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ = (c - x) ∣ ∣ ∣ ∣ \begin{matrix} x & m & 1 a & x & 1 a & b & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0$
Applying $R_{3} \to R_{3} - R_{2}$ and expanding
$(c - x) ∣ ∣ ∣ ∣ \begin{matrix} x & m & 1 a & x & 1 a & b & 1 \end{matrix} ∣ ∣ ∣ ∣ = (c - x) ∣ ∣ ∣ ∣ \begin{matrix} x & m & 1 a & x & 1 0 & b - x & 0 \end{matrix} ∣ ∣ ∣ ∣ \Rightarrow (c - x) (b - x) (x - a) = 0$
Hence, roots of the quation are independent of m and n.

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Roots of the equation ∣∣ ∣ ∣ ∣∣xmn1axn1abx1abc1∣∣ ∣ ∣ ∣∣=0 are

Roots of the equation $∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} x & m & n & 1 a & x & n & 1 a & b & x & 1 a & b & c & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ = 0$ are