Mathematics

Step 1: Convert inequalities to equations and find points.
For :
If , . Point is (0, 6).
If , . Point is (4, 0).
The region is on the origin side of this line.
For :
If , . Point is (0, 4).
If , . Point is (4, 0).
The region is on the non-origin side of this line.
Step 2: Graph the lines and find the feasible region.
The feasible region is the area bounded by the lines that satisfies all constraints, including and . The corner points of the feasible region are:
A = (0, 4)
B = (4, 0)
C = (0, 6)
The intersection point of and is not needed as it lies outside the region defined by the combination of inequalities. The feasible region is the triangle formed by points A, B, and C.
Step 3: Evaluate the objective function at the corner points.
We evaluate at each vertex:
At A(0, 4):
At B(4, 0):
At C(0, 6):
Step 4: Determine the maximum value.
The maximum value of Z is 36, which occurs at the point (0, 6). So, the optimal solution is , , with a maximum value of .

Step 1: Define the variables.
Let be the weight of cement in kg.
Let be the weight of sand in kg.
Step 2: Formulate the objective function.
The objective is to minimize the cost of the brick. The cost function ( ) is: Step 3: Formulate the constraints.
The constraints are based on the conditions given:
Total weight constraint: The total weight must be at least 5 kg. Cement constraint: The brick must contain a minimum of 4 kg of cement. Sand constraint: The brick must contain not more than 2 kg of sand. Non-negativity constraints: The amount of cement and sand cannot be negative. The complete L.P.P. formulation is:
Minimize
Subject to the constraints:

When a bank discounts a bill of exchange, it calculates the interest on the face value of the bill for the unexpired period. This interest is called the Banker's Discount (B.D.). The bank deducts this amount from the face value and pays the remaining amount to the holder. This net amount received by the holder is known as the Cash Value (C.V.) or the proceeds of the bill.

An assignment problem is a special type of linear programming problem. Its primary objective is to assign a set of resources (e.g., persons, machines) to a set of tasks (e.g., jobs) on a one-to-one basis. The goal is to find the assignment that optimizes a given objective function, which is most commonly to minimize the total cost or time. While maximization problems (e.g., maximizing profit) also exist, the standard and most fundamental objective is minimization. Option (b) is the most complete and accurate description of the standard assignment problem.

A Del credere agent is a type of mercantile agent who, for an extra commission (called a del credere commission), guarantees the solvency of the third party and the performance of the contract. In the context of a sale, this means they guarantee that the buyer will pay for the goods. If the buyer fails to pay, the del credere agent is liable to pay the amount to the principal.

An annuity is a series of equal payments made at regular intervals. The key distinction between different types of annuities is the timing of the payments. In an ordinary annuity (or annuity immediate), the payments are made at the end of each payment period. In contrast, an annuity due has payments at the beginning of each period.