A manufacturer produces three products \(x,y,z\) which he sells in two markets. Annual sales are indicated below:Market Products I 10000 2000 18000 II 6000 20000 8000
(a)If unit sale prices of \(x,y\) and \(z\) are Rs 2.50,Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.
| Market | Products | ||
| I | 10000 | 2000 | 18000 |
| II | 6000 | 20000 | 8000 |
(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.
Solution and Explanation
(a) The unit sale prices of \(x, y\), and \(z\) are respectively given as Rs 2.50, Rs 1.50, and Rs 1.00. Consequently, the total revenue in market I can be represented in the form of a matrix as:
\(\begin{bmatrix}10000& 2000& 18000\end{bmatrix}\begin{bmatrix}2.50\\ 1.50\\ 1.00\end{bmatrix}\)
\(=10000\times2.50+2000\times1.50+18000\times1.00]\)
\(=25000+3000+18000\)
\(=46000\)
The total revenue in market II can be represented in the form of a matrix as:
\(\begin{bmatrix}6000& 20000& 8000\end{bmatrix}\begin{bmatrix}2.50\\ 1.50\\ 1.00\end{bmatrix}\)
\(=6000\times2.50+20000\times1.50+8000\times1.00\)
\(=15000+30000+8000\)
\(=53000\)
Therefore, the total revenue in market I is Rs 46000 and the same in market II is Rs 53000.
(b) The unit cost prices of x, y, and z are respectively given as Rs 2.00, Rs 1.00, and 50 paise.
Consequently, the total cost prices of all the products in market I can be represented in the form of a matrix as:
\(\begin{bmatrix}10000& 2000& 18000\end{bmatrix}\begin{bmatrix}2.00\\ 1.00\\ 0.50\end{bmatrix}\)
\(=10000\times2.00+2000\times1.00+18000\times0.50\)
\(=20000+2000+9000\)
\(=31000\)
Since the total revenue in market I is Rs 46000, the gross profit in this market is (Rs 46000−Rs 31000) Rs 15000.
The total cost prices of all the products in market II can be represented in the form of a matrix as:
\(\begin{bmatrix}6000& 20000& 8000\end{bmatrix}\begin{bmatrix}2.00\\ 1.00\\ 0.50\end{bmatrix}\)
\(=6000\times2.00+20000\times1.00+8000\times0.50\)
\(=12000+20000+4000\)
\(=Rs 36000\)
Since the total revenue in market II isRs 53000, the gross profit in this market is (Rs53000 − Rs 36000) Rs 17000.
Top Questions on Matrices
Let
\( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix} \)
and \(|2A|^3 = 2^{21}\) where \(\alpha, \beta \in \mathbb{Z}\). Then a value of \(\alpha\) is:
- If \(A = \begin{bmatrix} 3 & 2 \\ -1 & 1 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -1 & 0 \\ 2 & 5 \\ 3 & 4 \end{bmatrix},\)then \((BA)^T\) is equal to:
- If \( A = \begin{bmatrix} K & 4 \\ 4 & K \end{bmatrix} \) and \( |A^3| = 729 \), then the value of \( K^8 \) is:
- Let $A$ be a matrix such that $A^2 = I$, where $I$ is an identity matrix. Then $(I + A)^4 - 8A$ is equal to:
- Let $A = I_2 - MM^T$, where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^T M = I_1$ holds. If $\lambda$ is a real number such that the relation $AX = \lambda X$ holds for some non-zero real matrix $X$ of order $2 \times 1$, then the sum of squares of all possible values of $\lambda$ is equal to:
Questions Asked in CBSE CLASS XII exam
What is the Planning Process?
- CBSE CLASS XII - 2023
- Planning process steps
Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)
- CBSE CLASS XII - 2021
- Determinants
- Find the vector and the cartesian equations of the lines that pass through the origin and(5,-2,3).
- CBSE CLASS XII - 2021
- Three Dimensional Geometry
- Find the integrals of the function: \(cos\, 2x\, cos\, 4x\, cos\, 6x\)
- CBSE CLASS XII
- integral
Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2 }. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
- CBSE CLASS XII
- Relations and Functions