(Originais Teachy 2023) - Question Easy of Mathematics

A store decides to reward its customers. Each customer will receive one of the six possible gifts available, according to their arrival order at the store. The gifts to be distributed are: a ball, a keychain, a pen, a soda, an ice cream, and a CD, in that order. The first customer at the store receives a ball, the second receives a keychain, the third receives a pen, the fourth receives a soda, the fifth receives an ice cream, and the sixth receives a CD. The seventh receives a ball, the eighth receives a keychain, and so on, according to the order of the gifts. The one-thousandth customer will receive as a gift a:

In a large engineering system, a building is being designed and energy efficiency is a major concern. To ensure the efficiency of heating and cooling, the air conditioning system must be sized according to the specific needs of the building. The engineer in charge of the project must solve a system of linear equations to determine the ideal operating temperatures. Let the following be the system of equations that models the thermal behavior of the building: 2x + y + 3z = 10, 4x - 6y + 8z = 5, 3x + 7y + 5z = 18. The engineer must rewrite this system of equations in the matrix form Ax = b, where A is the matrix of coefficients, x is the vector of unknowns, and b is the vector of constant terms.

In a science project, students need to measure different volumes of liquids for their experiments. The scale used to measure the volume of water can only measure in multiples of 100 milliliters. When measuring, the scale shows 450.725 grams of water. We know that the density of water is 1 g/ml. (1) Round the volume measured by the scale to the nearest milliliter. (2) Explain why the scale provided a number in grams instead of milliliters and how you used the density of water to convert the measurement back to volume in milliliters.

Consider a Cartesian coordinate system where a point P has coordinates (a, b) and a point Q has coordinates (c, d). The absolute value of a number is defined as its distance from the origin of this system. If point P is in the second quadrant and point Q is in the fourth quadrant, with a, b, c, and d being integers, and it is known that the absolute difference between the y coordinates of P and Q is 15, determine the value of a - c, knowing that a < 0 and c > 0.