Advertisements
Advertisements
प्रश्न
A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society?
Advertisements
उत्तर
Let the monthly fees paid by poor and rich children be Rs x and Rs y, respectively.
For batch I:
20x + 5y = 9000 .....(1)
For batch II:
5x + 25y = 26000 .....(2)
The system of equations can be written as
\[AX = B\]
\[\begin{matrix}20 & 5 \\ 5 & 25\end{matrix}\binom{x}{y} = \binom{9000}{26000}\]
\[\text { Here }, A = \begin{matrix}20 & 5 \\ 5 & 25\end{matrix}, X = \binom{x}{y} \text { and } B = \binom{9000}{26000}\]
\[\left| A \right| = \begin{vmatrix}20 & 5 \\ 5 & 25\end{vmatrix} = 500 - 25 = 475 \neq 0\]
\[C_{11} = \left( - 1 \right)^{1 + 1} \left( 25 \right) = 25, C_{12} = \left( - 1 \right)^{1 + 2} \left( 5 \right) = - 5\]
\[ C_{21} = \left( - 1 \right)^{2 + 1} \left( 5 \right) = - 5, C_{22} = \left( - 1 \right)^{2 + 2} \left( 20 \right) = 20\]
\[\text { Adj }A = \begin{bmatrix}25 & - 5 \\ - 5 & 20\end{bmatrix}^T = \begin{bmatrix}25 & - 5 \\ - 5 & 20\end{bmatrix}\]
\[ \therefore A^{- 1} = \frac{AdjA}{\left| A \right|} = \frac{1}{475}\begin{bmatrix}25 & - 5 \\ - 5 & 20\end{bmatrix}\]
So, the given system has a unique solution given by X = A−1B.
\[\therefore X = A^{- 1} B\]
\[ \Rightarrow \binom{x}{y} = \frac{1}{475}\begin{bmatrix}25 & - 5 \\ - 5 & 20\end{bmatrix}\binom{9000}{26000}\]
\[ \Rightarrow \binom{x}{y} = \frac{1}{475}\binom{95000}{475000}\]
\[ \Rightarrow \binom{x}{y} = \binom{200}{1000}\]
\[ \Rightarrow x = 200, y = 1000\]
Hence, the monthly fees paid by each poor child is Rs 200 and the monthly fees paid by each rich child is Rs 1000.
By offering discount to the poor children, the coaching institute offers an unbiased chance for the development and enhancement of the weaker section of our society.
APPEARS IN
संबंधित प्रश्न
if A = [(1,1,1),(1,1,1),(1,1,1)], Prove that A" = `[(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))]` `n in N`
If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N
If A = `[(α, β),(γ, -α)]` is such that A2 = I, then ______.
Use product `[(1,-1,2),(0,2,-3),(3,-2,4)][(-2,0,1),(9,2,-3),(6,1,-2)]` to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3
If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − AT is a skew-symmetric matrix.
Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:
`[(6, 0),(0, 6)]`
Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:
`[(3, 0, 0),(0, 5, 0),(0, 0, 1/3)]`
Identify the following matrix is singular or non-singular?
`[(3, 5, 7),(-2, 1, 4),(3, 2, 5)]`
Find k if the following matrix is singular:
`[(4, 3, 1),(7, "k", 1),(10, 9, 1)]`
If A = `[(3, 1),(-1, 2)]`, prove that A2 – 5A + 7I = 0, where I is unit matrix of order 2
Answer the following question:
If A = diag [2 –3 –5], B = diag [4 –6 –3] and C = diag [–3 4 1] then find B + C – A
Choose the correct alternative:
If B = `[(6, 3),(-2, "k")]` is singular matrix, then the value of k is ______
State whether the following statement is True or False:
If A and B are two square matrices such that AB = BA, then (A – B)2 = A2 – 2AB + B2
If A is a square matrix of order 2 such that A(adj A) = `[(7, 0),(0, 7)]`, then |A| = ______
If A = `[(3, 1),(-1, 2)]`, then prove that A2 – 5A + 7I = O, where I is unit matrix of order 2
If A = `[(1, 3, 3),(3, 1, 3),(3, 3, 1)]`, then show that A2 – 5A is a scalar matrix
If A = `[(3, -4),(1, 1),(2, 0)]` and B = `[(2, 1, 2),(1, 2, 4)]`, then verify (BA)2 ≠ B2A2
If the matrix A `= [(5,2,"x"),("y",2,-3),(4, "t",-7)]` is a symmetric matrix, then find the value of x, y and t respectively.
If `[(1,2),(3,4)],` then A2 - 5A is equal to ____________.
A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an
The number of all possible matrices of order 3/3, with each entry 0 or 1 is
Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| ≠ 0. Consider the following two statements:
(P) If A1I2, then |A| = –1
(Q) If |A| = 1, then tr(A) = 2,
where I2 denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then ______.
Let A = `[(0, -2),(2, 0)]`. If M and N are two matrices given by M = `sum_(k = 1)^10 A^(2k)` and N = `sum_(k = 1)^10 A^(2k - 1)` then MN2 is ______.
If A = `[(0, -tan θ/2),(tan θ/2, 0)]` and (I2 + A) (I2 – A)–1 = `[(a, -b),(b, a)]` then 13(a2 + b2) is equal to ______.
A matrix which is both symmetric and skew symmetric matrix is a ______.
Matrices are classified into different types based on which two features?
A matrix with only one row is called a:
What is the order of a row matrix with \[n\] columns?
How is an identity matrix denoted when its order is \[n\]?
