Class 12 Determinants · Case Study4 Marksmatrix_method_applications
Class 12 Maths · Determinants · Case Study #34
A teacher distributes a fixed total amount equally among the class. If there were 8 fewer students, each would receive ₹10 more; if there were 16 more students, each would receive ₹10 less. Let x be the original student count and y the rupees originally given to each student.
Part (1) 1 Marks
Derive the linear equations.
Answer: 5x−4y=40; 5x−8y=−80.
- The fixed-total conditions are (x−8)(y+10)=xy and (x+16)(y−10)=xy. Expanding gives 10x−8y=80 and −10x+16y=160.
- Divide the first by 2 and the second by −2 to obtain 5x−4y=40 and 5x−8y=−80.
Part (2) 1 Marks
Express that system in matrix form.
Answer: [[5,−4],[5,−8]][x,y]ᵀ=[40,−80]ᵀ.
- Use the coefficients of x,y as matrix rows and the right sides as a column.
- The matrix equation is \(\begin{bmatrix}5 & -4\\5 & -8\end{bmatrix}\)[x,y]ᵀ=[40,−80]ᵀ.
Part (3) 2 Marks
Find the number of students and the total amount distributed.
Answer: 32 students; ₹960 total (₹30 each).
- The coefficient determinant is 5(−8)−(−4)5=−20≠0. Its inverse is (−1/20)[[−8,4],[−5,5]].
- Multiplying by [40,−80]ᵀ gives [x,y]ᵀ=[32,30]ᵀ. The fixed total is xy=32·30=960.
- There are 32 students, each receiving ₹30; the total distributed is ₹960.