Comprehension
A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs.
Every year a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2022–23, the school offered monthly scholarships of ₹3,000 each to some girl students and ₹4,000 each to meritorious achievers in academics as well as sports.
In all, 50 students were given the scholarships, and the monthly expenditure incurred by the school on scholarships was ₹1,80,000.
Based on the above information, answer the following questions:
Question: 1
Express the given information algebraically using matrices.
Express the given information algebraically using matrices.
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To express a system of linear equations in matrix form, write the coefficients as the matrix \(A\), variables as column matrix \(X\), and constants as column matrix \(B\). The system is represented as \(AX = B\).
Updated On: Feb 6, 2025
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Solution and Explanation
Step 1: Define the variables
Let: \[ x = \text{number of girl child scholarships}, \] \[ y = \text{number of meritorious achievers}. \] Step 2: Formulate the equations
We are given: \[ x + y = 50 \quad \text{(1)}, \] \[ 3000x + 4000y = 180000 \quad \text{(2)}. \] Divide equation (2) by 1000: \[ 3x + 4y = 180 \quad \text{(3)}. \] Step 3: Represent the equations in matrix form
The equations can be expressed in matrix form as: \[ \begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 180 \end{bmatrix}. \]
Let: \[ x = \text{number of girl child scholarships}, \] \[ y = \text{number of meritorious achievers}. \] Step 2: Formulate the equations
We are given: \[ x + y = 50 \quad \text{(1)}, \] \[ 3000x + 4000y = 180000 \quad \text{(2)}. \] Divide equation (2) by 1000: \[ 3x + 4y = 180 \quad \text{(3)}. \] Step 3: Represent the equations in matrix form
The equations can be expressed in matrix form as: \[ \begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 180 \end{bmatrix}. \]
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Question: 2
Check whether the system of matrix equations so obtained is consistent or not.
Check whether the system of matrix equations so obtained is consistent or not.
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To check the consistency of a matrix equation, calculate the determinant of the coefficient matrix \(A\). If \(\det(A) \neq 0\), the system is consistent and has a unique solution.
Updated On: Feb 6, 2025
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Solution and Explanation
Step 1: Write the coefficient matrix
The coefficient matrix is: \[ A = \begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix}. \] Step 2: Compute the determinant of \(A\)
The determinant is: \[ \det(A) = (1)(4) - (1)(3) = 4 - 3 = 1. \] Step 3: Check consistency
Since \( \det(A) \neq 0 \), the system of equations is consistent, meaning it has a unique solution.
The coefficient matrix is: \[ A = \begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix}. \] Step 2: Compute the determinant of \(A\)
The determinant is: \[ \det(A) = (1)(4) - (1)(3) = 4 - 3 = 1. \] Step 3: Check consistency
Since \( \det(A) \neq 0 \), the system of equations is consistent, meaning it has a unique solution.
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Question: 3
Find the number of scholarships of each kind given by the school using matrices.
Find the number of scholarships of each kind given by the school using matrices.
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To solve a matrix equation \(AX = B\), compute the inverse \(A^{-1}\) and multiply it by \(B\) to find \(X\).
Updated On: Feb 6, 2025
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Solution and Explanation
Step 1: Write the matrix equation
The matrix equation is: \[ \begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 180 \end{bmatrix}. \] Step 2: Find the inverse of \(A\)
The inverse of \(A\) is: \[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} 4 & -1 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} 4 & -1 \\ -3 & 1 \end{bmatrix}. \] Step 3: Solve for \(X\)
Using \(X = A^{-1}B\): \[ X = \begin{bmatrix} 4 & -1 \\ -3 & 1 \end{bmatrix} \begin{bmatrix} 50 \\ 180 \end{bmatrix}. \] Multiply the matrices: \[ X = \begin{bmatrix} 4(50) - 1(180) \\ -3(50) + 1(180) \end{bmatrix} = \begin{bmatrix} 200 - 180 \\ -150 + 180 \end{bmatrix} = \begin{bmatrix} 20 \\ 30 \end{bmatrix}. \] Thus, \( x = 20 \) (girl students) and \( y = 30 \) (meritorious achievers).
The matrix equation is: \[ \begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 180 \end{bmatrix}. \] Step 2: Find the inverse of \(A\)
The inverse of \(A\) is: \[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} 4 & -1 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} 4 & -1 \\ -3 & 1 \end{bmatrix}. \] Step 3: Solve for \(X\)
Using \(X = A^{-1}B\): \[ X = \begin{bmatrix} 4 & -1 \\ -3 & 1 \end{bmatrix} \begin{bmatrix} 50 \\ 180 \end{bmatrix}. \] Multiply the matrices: \[ X = \begin{bmatrix} 4(50) - 1(180) \\ -3(50) + 1(180) \end{bmatrix} = \begin{bmatrix} 200 - 180 \\ -150 + 180 \end{bmatrix} = \begin{bmatrix} 20 \\ 30 \end{bmatrix}. \] Thus, \( x = 20 \) (girl students) and \( y = 30 \) (meritorious achievers).
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Question: 4
Had the amount of scholarship given to each girl child and meritorious student been interchanged, what would be the monthly expenditure incurred by the school?
Had the amount of scholarship given to each girl child and meritorious student been interchanged, what would be the monthly expenditure incurred by the school?
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When the values of variables are known, substitute them into the equation to compute the required quantity, such as expenditure.
Updated On: Feb 6, 2025
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Solution and Explanation
Recalculate the expenditure:
If the scholarship amounts are interchanged, girl students receive ₹4,000, and meritorious achievers receive ₹3,000.
The expenditure becomes: \[ {Expenditure} = 30(3000) + 20(4000) = 90000 + 80000 = 170000. \]
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