(Originais Teachy 2022) - 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:

João is organizing a birthday party and wants to make a larger quantity of his favorite cake recipe. The original recipe yields a cake with 20 slices and uses 3 cups of flour, 2 cups of sugar, and 1 cup of oil. João wants to make a cake with 40 slices. a) How many cups of flour, sugar, and oil should he use to make a cake that yields 40 slices? b) If he wanted to make a cake with 60 slices, what would be the quantity of cups of each ingredient? c) João noticed that the number of cake slices is always half the number of cups of water used in the recipe. How many cups of water are needed for a cake of 80 slices?

Consider the linear system below: 2x + 3y = 10 5x - 4y = -7 Transform this linear system into a matrix form using a matrix A and a column vector X, representing the unknowns. Write down the resulting matrix equation and inform what is the dimension of matrix A.

In a civil engineering project, an architect needs to divide a rectangular plot of land into two equal parts for the construction of two semi-detached houses. The coordinates of the vertices of the land on the Cartesian plane are A(0, 0), B(30, 0), C(30, 20), and D(0, 20). Using concepts of analytic geometry, determine the coordinates of the point that represents the midpoint of side CD of the land, which will be the dividing point between the two new properties. Then, verify if the obtained midpoint point is indeed on segment CD, ensuring the accuracy of the project. Explain the reasoning used to determine the coordinates and to verify the positioning of the midpoint on the land.