The maximum value of \( f(x, y, z) = 10x + 6y - 8z \) subject to the constraints \[ 5x - 2y + 6z \leq 20, \quad 10x + 4y - 6z \leq 30, \quad x, y, z \geq 0, \] is equal to …………. (round off to TWO decimal places).
The maximum value of \( f(x, y, z) = 10x + 6y - 8z \) subject to the constraints \[ 5x - 2y + 6z \leq 20, \quad 10x + 4y - 6z \leq 30, \quad x, y, z \geq 0, \] is equal to …………. (round off to TWO decimal places).
Show Hint
Solution and Explanation
Step 1: Linear programming formulation. The problem involves maximizing \( f(x, y, z) = 10x + 6y - 8z \) subject to the given constraints.
Step 2: Solving using the simplex method. By applying the simplex method or computational tools, the optimal point is determined to be \( (x, y, z) = (3.33, 5, 0) \).
Step 3: Calculating \( f(x, y, z) \). Substituting into \( f(x, y, z) \), the maximum value is \( 56.66 \).
Step 4: Conclusion. The maximum value is \( {56.66} \).
Top Questions on Product of Matrices
- \(\text{ If } P = \begin{bmatrix} -1 \\ 2 \\ 1 \end{bmatrix} \text{ and } Q = \begin{bmatrix} 2 & -4 & 1 \end{bmatrix} \text{ are two matrices, then } (PQ)^T \text{ will be:}\)
- CUET (UG) - 2024
- Mathematics
- Product of Matrices
Consider the following Linear Programming Problem $ P $: Minimize $ x_1 + 2x_2 $, subject to
$ 2x_1 + x_2 \leq 2 $,
$ x_1 + x_2 = 1 $,
$ x_1, x_2 \geq 0 $.
The optimal value of the problem $ P $ is equal to:- GATE MA - 2024
- Mathematics
- Product of Matrices
Let $D = \{(x, y) \in \mathbb{R}^2 : x > 0 \text{ and } y > 0\}$. If the following second-order linear partial differential equation
$y^2 \frac{\partial^2 u}{\partial x^2} - x^2 \frac{\partial^2 u}{\partial y^2} + y \frac{\partial u}{\partial y} = 0$ on $D$
is transformed to
$\left( \frac{\partial^2 u}{\partial \eta^2} - \frac{\partial^2 u}{\partial \xi^2} \right) + \left( \frac{\partial u}{\partial \eta} + \frac{\partial u}{\partial \xi} \right) \frac{1}{2\eta} + \left( \frac{\partial u}{\partial \eta} - \frac{\partial u}{\partial \xi} \right) \frac{1}{2\xi} = 0$ on $D$,
for some $a, b \in \mathbb{R}$, via the coordinate transform $\eta = \frac{x^2}{2}$ and $\xi = \frac{y^2}{2}$, then which one of the following is correct?- GATE MA - 2024
- Mathematics
- Product of Matrices
- Let \( T, S : {R}^4 \to {R}^4 \) be two non-zero, non-identity \( {R} \)-linear transformations. Assume \( T^2 = T \). Which of the following is/are true?
- GATE MA - 2024
- Mathematics
- Product of Matrices
Let \( p_1<p_2 \) be the two fixed points of the function \( g(x) = e^x - 2 \), where \( x \in {R} \). For \( x_0 \in {R} \), let the sequence \( (x_n)_{n \geq 1} \) be generated by the fixed-point iteration \[ x_n = g(x_{n-1}), \quad n \geq 1. \] Which one of the following is/are correct?
- GATE MA - 2024
- Mathematics
- Product of Matrices
Questions Asked in GATE MA exam
- Let \( \alpha, \beta \) be distinct non-zero real numbers, and let \( Q(z) \) be a polynomial of degree less than 5. If the function \[ f(z) = \frac{\alpha^6 \sin \beta z - \beta^6 (e^{2az} - Q(z))}{z^6} \] satisfies Morera's theorem in \( \mathbb{C} \setminus \{0\} \), then the value of \( \frac{\alpha}{4\beta} \) is equal to (in integer).
- GATE MA - 2025
- Function
- A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?
- GATE MA - 2025
- GATE AG - 2025
- Combinations
A square paper, shown in figure (I), is folded along the dotted lines as shown in figures (II) and (III). Then a few cuts are made as shown in figure (IV). Which one of the following patterns will be obtained when the paper is unfolded?
- GATE MA - 2025
- GATE AG - 2025
- Paper Folding and Cutting
- A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled thrice. What is the probability of rolling ‘6’ exactly once?
- GATE MA - 2025
- GATE AG - 2025
- Probability
In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and the line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative