SATHEE: Matrices and Determinants 4 Question 22

Matrices and Determinants 4 Question 22

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23. Consider the system of equations $x - 2 y + 3 z = - 1$ , $x - 3 y + 4 z = 1$ and $- x + y - 2 z = k$

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Assertion and Reason

For the following questions, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows.

(a) Statement I is true, Statement II is also true; Statement II is the correct explanation of Statement I

(b) Statement I is true, Statement II is also true; Statement II is not the correct explanation of Statement I

(d) Statement I is false; Statement II is true.

####23. Consider the system of equations $x - 2 y + 3 z = - 1$ , $x - 3 y + 4 z = 1$ and $- x + y - 2 z = k$

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Statement I The system of equations has no solution for $k \neq 3$ and

Statement II The determinant $| \begin{array}{rrr} 1 & 3 & - 1 \\ - 1 & - 2 & k \\ 1 & 4 & 1 \end{array} | \neq 0$ , for

$k \neq 0$ .

(2008, 3M)

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Answer:

Correct Answer: 23. (a)

Solution:

The given system of equations can be expressed as

$[\begin{array}{rrr} 1 & - 2 & 3 \\ 1 & - 3 & 4 \\ - 1 & 1 & - 2 \end{array}] [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{array}{r} - 1 \\ 1 \\ k \end{array}]$

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

$\sim [\begin{array}{lll} 1 & - 2 & 3 \\ 0 & - 1 & 1 \\ 0 & - 1 & 1 \end{array}] [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{array}{r} - 1 \\ 2 \\ k - 1 \end{array}]$

Applying $R_{3} \to R_{3} - R_{2}$

$\sim [\begin{array}{ccc} 1 & - 2 & 3 \\ 0 & - 1 & 1 \\ 0 & 0 & 0 \end{array}] [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{array}{r} - 1 \\ 2 \\ k - 3 \end{array}]$