Class CBSE Class 12 Mathematics Matrices and Determinants Q #1810
COMPETENCY BASED
APPLY
5 Marks 2026 AISSCE(Board Exam) LA
On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.
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Detailed Solution

Step 1: Define Variables

Let $x$ be the number of ₹ 1,000 vouchers and $y$ be the number of ₹ 500 vouchers.

Step 2: Formulate Equations

Based on the problem statement: $$x + y = 60$$ $$x + 3y = 100$$

Step 3: Matrix Representation

The system can be written as $AX = B$, where: $$A = \begin{pmatrix} 1 & 1 \\ 1 & 3 \end{pmatrix}, X = \begin{pmatrix} x \\ y \end{pmatrix}, B = \begin{pmatrix} 60 \\ 100 \end{pmatrix}$$

Step 4: Solve using Inverse Matrix

Calculate determinant $|A| = (1)(3) - (1)(1) = 2$. Since $|A| \neq 0$, $A^{-1}$ exists. $$A^{-1} = \frac{1}{2} \begin{pmatrix} 3 & -1 \\ -1 & 1 \end{pmatrix}$$ $$X = A^{-1}B = \frac{1}{2} \begin{pmatrix} 3 & -1 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 60 \\ 100 \end{pmatrix}$$ $$X = \frac{1}{2} \begin{pmatrix} 180 - 100 \\ -60 + 100 \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 80 \\ 40 \end{pmatrix} = \begin{pmatrix} 40 \\ 20 \end{pmatrix}$$

Final Answer: Number of ₹ 1,000 vouchers = 40, Number of ₹ 500 vouchers = 20

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to translate a real-world scenario into a mathematical model and execute a specific algorithm (matrix inversion) to solve it.
Knowledge Dimension: PROCEDURAL
Justification: The question tests the student's ability to follow the step-by-step procedure of matrix algebra to solve a system of linear equations.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It aligns with the NEP 2020 focus on applying mathematical concepts to solve practical problems rather than rote memorization.

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