SATHEE: Matrices and Determinants 2 Question 26

Matrices and Determinants 2 Question 26

29. Given that $x = - 9$ is a root of $| \begin{matrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{matrix} | = 0$ , the other two roots are… and… .

(1983, 2M)

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Answer:

Correct Answer: 29. (2 and 7)

Solution:

Given, $| \begin{matrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{matrix} | = 0$

Applying $R_{1} \to R_{1} + R_{2} + R_{3}$

$\Rightarrow | \begin{matrix} x + 9 & x + 9 & x + 9 \\ 2 & x & 2 \\ 7 & 6 & x \end{matrix} | \Rightarrow 0 \Rightarrow (x + 9) | \begin{matrix} 1 & 1 & 1 \\ 2 & x & 2 \\ 7 & 6 & x \end{matrix} | = 0$

Applying $C_{2} \to C_{2} - C_{1}$ and $C_{3} \to C_{3} - C_{1}$

$\begin{aligned} \Rightarrow (x + 9) | \begin{array}{c} 1 & 0 & 0 \\ 2 & x - 2 & 0 \\ 7 & - 1 & x - 7 \end{array} | \Rightarrow 0 \Rightarrow (x + 9) (x - 2) (x - 7) = 0 \\ \Rightarrow x = - 9, 2, 7 are the roots. \end{aligned}$

$∴$ Other two roots are 2 and 7 .

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Matrices and Determinants 2 Question 26

29. Given that x=−9 is a root of |x372x276x|=0, the other two roots are… and… .

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29. Given that $x = - 9$ is a root of $| \begin{matrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{matrix} | = 0$ , the other two roots are… and… .