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:
Express the given information algebraically using matrices.
Show Hint
Solution and Explanation
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: \[ \begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 180 \end{bmatrix}. \]
Check whether the system of matrix equations so obtained is consistent or not.
Show Hint
Solution and Explanation
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.
Find the number of scholarships of each kind given by the school using matrices.
Show Hint
Solution and Explanation
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).
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?
Show Hint
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: \[ {Expenditure} = 30(3000) + 20(4000) = 90000 + 80000 = 170000. \]
Top Questions on Matrices
- If \( A \) and \( B \) are two skew-symmetric matrices, then \( AB + BA \) is:
- If \(A = [a_{ij}] = \begin{bmatrix} 2 & -1 & 5 \\ 1 & 3 & 2 \\ 5 & 0 & 4 \end{bmatrix}\) and \( c_{ij} \) is the cofactor of element \( a_{ij} \), then the value of \( a_{21}c_{11} + a_{22}c_{12} + a_{23}c_{13} \) is:
- If \(A = \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix}\) and \( A^2 - kA - 5I = 0 \), then the value of \( k \) is:
- If \(\begin{bmatrix} a & c & 0 \\ b & d & 0 \\ 0 & 0 & 5 \end{bmatrix}\)is a scalar matrix, then the value of \( a + 2b + 3c + 4d \) is:
- If \(A = \begin{bmatrix} 2 & 1 \\ -4 & -2 \end{bmatrix}\), then the value of \( I - A + A^2 - A^3 + \ldots \) is:
Questions Asked in CBSE CLASS XII exam
A bacteria sample of a certain number of bacteria is observed to grow exponentially in a given amount of time. Using the exponential growth model, the rate of growth of this sample of bacteria is calculated. The differential equation representing the growth is:
\[ \frac{dP}{dt} = kP, \] where \( P \) is the bacterial population.
Based on this, answer the following:
- CBSE CLASS XII - 2024
- Differentiability
Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves. A student may spend 1 hour to 6 hours in a day upskilling self. The probability distribution of the number of hours spent by a student is given below:
\[ P(X = x) = \begin{cases} kx^2 & {for } x = 1, 2, 3, \\ 2kx & {for } x = 4, 5, 6, \\ 0 & {otherwise}. \end{cases} \]
Based on the above information, answer the following:
- CBSE CLASS XII - 2024
- Probability Distribution
- Using integration, find the area of the region enclosed between the curve \( y = \sqrt{4 - x^2} \), and the lines \( x = -1 \), \( x = 1 \), and the \( x \)-axis.
- CBSE CLASS XII - 2024
- Integration
- The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that the volume of the cylinder so formed is maximum.
- CBSE CLASS XII - 2024
- Absolute maxima and Absolute minima
- It is given that the function \( f(x) = x^4 - 62x^2 + ax + 9 \) attains a local maximum value at \( x = 1 \). Find the value of \( a \), and hence obtain all other points where the given function \( f(x) \) attains local maximum or local minimum values.
- CBSE CLASS XII - 2024
- Absolute maxima and Absolute minima