Question

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 a monthly scholarship 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:
(i) Express the given information algebraically using matrices.
(ii) Check whether the system of matrix equations so obtained is consistent or not.
(iii)(a) Find the number of scholarships of each kind given by the school using matrices.
(iii)(b) 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?

Hint:

Represent a system of linear equations in matrix form as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix.

Answer 1

(i) Express the given information algebraically using matrices:

Let:

\[ x = \text{Number of girl students}, \quad y = \text{Number of meritorious students}. \]

The given conditions are:

\[ x + y = 50 \quad (\text{Total students}). \] \[ 3000x + 4000y = 180000 \quad (\text{Total expenditure}). \]

Write this system in matrix form:

\[ \begin{bmatrix} 1 & 1 \\ 3000 & 4000 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 180000 \end{bmatrix}. \]