Q1438: A furniture workshop produces three types of furniture chair...

Class CBSE Class 12 Mathematics Matrices and Determinants Q #1438

COMPETENCY BASED

UNDERSTAND

5 Marks 2025 AISSCE(Board Exam) LA

A furniture workshop produces three types of furniture chairs, tables and beds each day. On a particular day the total number of furniture pieces produced is 45. It was also found that production of beds exceeds that of chairs by 8, while the total production of beds and chairs together is twice the production of tables. Determine the units produced of each type of furniture, using matrix method.

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AI Tutor Explanation

Detailed Solution

Step 1: Define Variables

Let $x$ be the number of chairs produced, $y$ be the number of tables produced, and $z$ be the number of beds produced.

Step 2: Formulate Equations

We are given the following information:

The total number of furniture pieces produced is 45: $x + y + z = 45$
Production of beds exceeds that of chairs by 8: $z = x + 8$, which can be written as $-x + z = 8$
The total production of beds and chairs together is twice the production of tables: $z + x = 2y$, which can be written as $x - 2y + z = 0$

So, we have the following system of linear equations:

$$x + y + z = 45$$$$-x + z = 8$$$$x - 2y + z = 0$$

Step 3: Express in Matrix Form

We can write the system of equations in matrix form as $AX = B$, where:

$$A = \begin{bmatrix} 1 & 1 & 1 \\ -1 & 0 & 1 \\ 1 & -2 & 1 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 45 \\ 8 \\ 0 \end{bmatrix}$$

Step 4: Calculate the Determinant of A

$$|A| = \begin{vmatrix} 1 & 1 & 1 \\ -1 & 0 & 1 \\ 1 & -2 & 1 \end{vmatrix} = 1(0 - (-2)) - 1(-1 - 1) + 1(2 - 0) = 2 + 2 + 2 = 6$$Since $|A| \neq 0$, the system has a unique solution.

Step 5: Find the Adjoint of A

The matrix of cofactors is:

$$C = \begin{bmatrix} 2 & 2 & 2 \\ -3 & 0 & 3 \\ 1 & -2 & 1 \end{bmatrix}$$The adjoint of $A$ is the transpose of the cofactor matrix:

$$adj(A) = C^T = \begin{bmatrix} 2 & -3 & 1 \\ 2 & 0 & -2 \\ 2 & 3 & 1 \end{bmatrix}$$

Step 6: Calculate the Inverse of A

The inverse of $A$ is given by $A^{-1} = \frac{1}{|A|} adj(A)$:

$$A^{-1} = \frac{1}{6} \begin{bmatrix} 2 & -3 & 1 \\ 2 & 0 & -2 \\ 2 & 3 & 1 \end{bmatrix}$$

Step 7: Solve for X

We have $X = A^{-1}B$:

$$X = \frac{1}{6} \begin{bmatrix} 2 & -3 & 1 \\ 2 & 0 & -2 \\ 2 & 3 & 1 \end{bmatrix} \begin{bmatrix} 45 \\ 8 \\ 0 \end{bmatrix} = \frac{1}{6} \begin{bmatrix} 2(45) - 3(8) + 1(0) \\ 2(45) + 0(8) - 2(0) \\ 2(45) + 3(8) + 1(0) \end{bmatrix} = \frac{1}{6} \begin{bmatrix} 90 - 24 \\ 90 \\ 90 + 24 \end{bmatrix} = \frac{1}{6} \begin{bmatrix} 66 \\ 90 \\ 114 \end{bmatrix} = \begin{bmatrix} 11 \\ 15 \\ 19 \end{bmatrix}$$Therefore, $x = 11$, $y = 15$, and $z = 19$.

Step 8: State the Solution

The number of chairs produced is 11, the number of tables produced is 15, and the number of beds produced is 19.

Final Answer: Chairs: 11, Tables: 15, Beds: 19

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the problem, formulate equations, and apply the matrix method to solve the system of equations.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply a specific procedure (matrix method) to solve a system of linear equations.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question requires the student to apply the concept of matrices to solve a real-world problem, demonstrating competency in applying mathematical tools.