Advertisements
Advertisements
प्रश्न
| To raise money for an orphanage, students of three schools A, B and C organised an exhibition in their residential colony, where they sold paper bags, scrap books and pastel sheets made by using recycled paper. Student of school A sold 30 paper bags, 20 scrap books and 10 pastel sheets and raised ₹ 410. Student of school B sold 20 paper bags, 10 scrap books and 20 pastel sheets and raised ₹ 290. Student of school C sold 20 paper bags, 20 scrap books and 20 pastel sheets and raised ₹ 440. |
Answer the following question:
- Translate the problem into a system of equations.
- Solve the system of equation by using matrix method.
- Hence, find the cost of one paper bag, one scrap book and one pastel sheet.
उत्तर
i. Let the cost of one paper bag, one scrap book and one pastel sheet be x, y and z respectively.
30x + 20y + 10z = 410
`\implies` 3x + 2y + z = 41
20x + 10y + 20z = 290
`\implies` 2x + y + 2z = 29
20x + 20y + 20z = 440
`\implies` x + y + z = 22
ii. Given system of equations is equivalent to AX = B
Where `A = [(3, 2, 1),(2, 1, 2),(1, 1, 1)], X = [(x),(y),(z)], B = [(41),(29),(22)]`
|A| = –2 ≠ 0 `\implies` A–1 exists.
adj A = `[(-1, -1, 3),(0, 2, -4),(1, -1, -1)]`
Thus A–1 = `1/|A| adj A`
= `-1/2[(-1, -1, 3),(0, 2, -4),(1, -1, -1)]`
AX = B
`\implies` X = A–1B
= `-1/2[(-1, -1, 3),(0, 2, -4),(1, -1, -1)][(41),(29),(22)]`
= `[(2),(15),(5)]`
∴ x = 2, y = 15, z = 5
iii. The cost of one paper bag, one scrap book and one pastel sheet be Rs. 2, Rs. 15 and Rs. 5 respectively.
APPEARS IN
संबंधित प्रश्न
Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. School A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively with a total award money of Rs 1,600. School B wants to spend Rs 2,300 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for an award.
Verify A (adj A) = (adj A) A = |A|I.
`[(1,-1,2),(3,0,-2),(1,0,3)]`
If A = `[(3,1),(-1,2)]` show that A2 – 5A + 7I = O. Hence, find A–1.
Let A = `[(1,-2,1),(-2,3,1),(1,1,5)]` verify that
- [adj A]–1 = adj (A–1)
- (A–1)–1 = A
Find the adjoint of the following matrix:
\[\begin{bmatrix}\cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}\]
Find the inverse of the following matrix:
Find the inverse of the following matrix:
Show that the matrix, \[A = \begin{bmatrix}1 & 0 & - 2 \\ - 2 & - 1 & 2 \\ 3 & 4 & 1\end{bmatrix}\] satisfies the equation, \[A^3 - A^2 - 3A - I_3 = O\] . Hence, find A−1.
prove that \[A^{- 1} = A^3\]
If \[A = \begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}\] , show that \[A^{- 1} = A^3\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & - 1 & 4 \\ 4 & 0 & 7 \\ 3 & - 2 & 7\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}3 & 0 & - 1 \\ 2 & 3 & 0 \\ 0 & 4 & 1\end{bmatrix}\]
If A is a singular matrix, then adj A is ______.
If A satisfies the equation \[x^3 - 5 x^2 + 4x + \lambda = 0\] then A-1 exists if _____________ .
The matrix \[\begin{bmatrix}5 & 10 & 3 \\ - 2 & - 4 & 6 \\ - 1 & - 2 & b\end{bmatrix}\] is a singular matrix, if the value of b is _____________ .
If \[A^2 - A + I = 0\], then the inverse of A is __________ .
If A and B are invertible matrices, which of the following statement is not correct.
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] be such that \[A^{- 1} = kA\], then k equals ___________ .
If A is an invertible matrix, then det (A−1) is equal to ____________ .
Find A−1, if \[A = \begin{bmatrix}1 & 2 & 5 \\ 1 & - 1 & - 1 \\ 2 & 3 & - 1\end{bmatrix}\] . Hence solve the following system of linear equations:x + 2y + 5z = 10, x − y − z = −2, 2x + 3y − z = −11
An amount of Rs 10,000 is put into three investments at the rate of 10, 12 and 15% per annum. The combined income is Rs 1310 and the combined income of first and second investment is Rs 190 short of the income from the third. Find the investment in each using matrix method.
If A and B are invertible matrices, then which of the following is not correct?
|A–1| ≠ |A|–1, where A is non-singular matrix.
A square matrix A is invertible if det A is equal to ____________.
If A is a square matrix of order 3 and |A| = 5, then |adj A| = ______.
Given that A is a square matrix of order 3 and |A| = –2, then |adj(2A)| is equal to ______.
