Class CBSE Class 12 Mathematics Matrices and Determinants Q #1474
KNOWLEDGE BASED
REMEMBER
3 Marks 2025 AISSCE(Board Exam) SA
Let $2x+5y-1=0$ and $3x+2y-7=0$ represent the equations of two lines on which the ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.
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Detailed Solution

Step 1: Write the system of equations in matrix form.

The given equations are: $2x + 5y = 1$ $3x + 2y = 7$ We can write this system in matrix form as $AX = B$, where $A = \begin{bmatrix} 2 & 5 \\ 3 & 2 \end{bmatrix}$, $X = \begin{bmatrix} x \\ y \end{bmatrix}$, and $B = \begin{bmatrix} 1 \\ 7 \end{bmatrix}$.

Step 2: Find the inverse of matrix A.

First, find the determinant of $A$: $|A| = (2)(2) - (5)(3) = 4 - 15 = -11$ Since $|A| \neq 0$, the inverse exists. The inverse of $A$ is given by: $A^{-1} = \frac{1}{|A|} \begin{bmatrix} 2 & -5 \\ -3 & 2 \end{bmatrix} = \frac{1}{-11} \begin{bmatrix} 2 & -5 \\ -3 & 2 \end{bmatrix} = \begin{bmatrix} -2/11 & 5/11 \\ 3/11 & -2/11 \end{bmatrix}$

Step 3: Solve for X using the inverse.

We have $AX = B$, so $X = A^{-1}B$. $X = \begin{bmatrix} -2/11 & 5/11 \\ 3/11 & -2/11 \end{bmatrix} \begin{bmatrix} 1 \\ 7 \end{bmatrix} = \begin{bmatrix} (-2/11)(1) + (5/11)(7) \\ (3/11)(1) + (-2/11)(7) \end{bmatrix} = \begin{bmatrix} -2/11 + 35/11 \\ 3/11 - 14/11 \end{bmatrix} = \begin{bmatrix} 33/11 \\ -11/11 \end{bmatrix} = \begin{bmatrix} 3 \\ -1 \end{bmatrix}$ Therefore, $x = 3$ and $y = -1$.

Step 4: State the common point.

The point common to the paths of the ants is $(3, -1)$.

Final Answer: (3, -1)

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because the student needs to recall the matrix method for solving a system of linear equations.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a specific algorithm (matrix method) to solve a system of linear equations. This involves knowing the steps to find the inverse of a matrix and using it to solve for the variables.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of solving linear equations using matrices, a standard topic in the syllabus.

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