Class CBSE Class 12 Mathematics Matrices and Determinants Q #852
KNOWLEDGE BASED
APPLY
1 Marks 2023 MCQ SINGLE
5. The value $|\begin{matrix}x+y & y+z & z+x\\ z & x & y\\ 1 & 1 & 1\end{matrix}|$ is :
(A) 0
(B) 1
(C) $x+y+z$
(D) $2(x+y+z)$
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Step-by-Step Solution

  1. Apply the property of determinants: $C_1 \rightarrow C_1 + C_2$ $|\begin{matrix}x+y & y+z & z+x\\ z & x & y\\ 1 & 1 & 1\end{matrix}| = |\begin{matrix}x+y+y+z & y+z & z+x\\ z+x & x & y\\ 1+1 & 1 & 1\end{matrix}| = |\begin{matrix}x+2y+z & y+z & z+x\\ z+x & x & y\\ 2 & 1 & 1\end{matrix}|$
  2. Apply the property of determinants: $R_1 \rightarrow R_1 - (x+y+z)R_3$ $|\begin{matrix}x+y & y+z & z+x\\ z & x & y\\ 1 & 1 & 1\end{matrix}| = |\begin{matrix}x+y-(x+y+z) & y+z-(x+y+z) & z+x-(x+y+z)\\ z & x & y\\ 1 & 1 & 1\end{matrix}| = |\begin{matrix}-z & -x & -y\\ z & x & y\\ 1 & 1 & 1\end{matrix}|$
  3. Take -1 common from R1 $|\begin{matrix}-z & -x & -y\\ z & x & y\\ 1 & 1 & 1\end{matrix}| = -|\begin{matrix}z & x & y\\ z & x & y\\ 1 & 1 & 1\end{matrix}|$
  4. Since R1 and R2 are identical, the value of the determinant is 0.

Correct Answer: 0

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AI Suggestion: Option A

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the properties of determinants to simplify and evaluate the given determinant.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure (applying determinant properties) to arrive at the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of determinants and their properties as covered in the textbook.

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