Q1812: A man goes to buy fruits from the market. The shopkeeper inf...

Class CBSE Class 12 Mathematics Matrices and Determinants Q #1812

COMPETENCY BASED

APPLY

5 Marks 2026 AISSCE(Board Exam) LA

A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60; 2 apples, 4 oranges and 6 bananas cost ₹ 90; whereas 6 apples, 2 oranges and 3 bananas cost ₹ 70. Using matrix method, find the cost of one fruit of each kind.

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AI Tutor Explanation

Detailed Solution

Step 1: Formulate the System of Equations

Let the cost of one apple, one orange, and one banana be $x$, $y$, and $z$ respectively. Based on the problem, we get: $$4x + 3y + 2z = 60$$ $$2x + 4y + 6z = 90$$ $$6x + 2y + 3z = 70$$

Step 2: Represent in Matrix Form

The system can be written as $AX = B$, where: $$A = \begin{pmatrix} 4 & 3 & 2 \\ 2 & 4 & 6 \\ 6 & 2 & 3 \end{pmatrix}, X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, B = \begin{pmatrix} 60 \\ 90 \\ 70 \end{pmatrix}$$

Step 3: Calculate Determinant and Inverse

First, find $|A| = 4(12-12) - 3(6-36) + 2(4-24) = 0 + 90 - 40 = 50$. Since $|A| \neq 0$, the inverse exists. Calculating the adjoint and multiplying by $1/|A|$, we find $X = A^{-1}B$.

Step 4: Solve for Variables

Solving the matrix equation yields: $$x = 5, y = 10, z = 5$$

Final Answer: Cost of one apple = ₹ 5, one orange = ₹ 10, one banana = ₹ 5

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because it requires the student to translate a real-world word problem into a mathematical model (system of linear equations) and then execute the matrix inversion algorithm.

Knowledge Dimension: PROCEDURAL

Justification: The student must follow a specific sequence of steps: defining variables, constructing the matrix, calculating the determinant, finding the adjoint, and performing matrix multiplication.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question aligns with the NEP 2020 focus on applying algebraic concepts to solve practical, everyday scenarios, moving beyond rote memorization.