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Q137 Measures of Location and Spread View

The grades of a teacher who participated in a selection process, in which the evaluation panel was composed of five members, are presented in the graph. It is known that each panel member assigned two grades to the teacher, one relating to specific knowledge in their area of work and another relating to pedagogical knowledge, and that the teacher's final grade was given by the arithmetic mean of all grades assigned by the evaluation panel.
Using a new criterion, this evaluation panel decided to discard the highest and lowest grades assigned to the teacher.
The new average, in relation to the previous average, is
(A) 0.25 point higher. (B) 1.00 point higher. (C) 1.00 point lower. (D) 1.25 points higher. (E) 2.00 points lower.

Q138 Permutations & Arrangements Forming Numbers with Digit Constraints View

A bank asked its customers to create a personal six-digit password, formed only by digits from 0 to 9, for access to their checking account via the internet.
However, an expert in electronic security systems recommended to the bank's management to re-register its users, requesting, for each one of them, the creation of a new six-digit password, now allowing the use of the 26 letters of the alphabet, in addition to digits from 0 to 9. In this new system, each uppercase letter was considered distinct from its lowercase version. Furthermore, the use of other types of characters was prohibited.
One way to evaluate a change in the password system is to verify the improvement coefficient, which is the ratio of the new number of password possibilities to the old one.
The improvement coefficient of the recommended change is
(A) $\frac{62^{6}}{10^{6}}$ (B) $\frac{62!}{10!}$ (C) $\frac{62! \cdot 4!}{10! \cdot 56!}$ (D) $62! - 10!$ (E) $62^{6} - 10^{6}$

Q145 Probability Definitions Finite Equally-Likely Probability Computation View

In a certain theater, the seats are divided into sectors. The figure presents the view of sector 3 of this theater, in which the dark chairs are reserved and the light ones have not been sold.
The ratio that represents the quantity of reserved chairs in sector 3 in relation to the total number of chairs in that same sector is
(A) $\frac{17}{70}$ (B) $\frac{17}{53}$ (C) $\frac{53}{70}$ (D) $\frac{53}{17}$ (E) $\frac{70}{17}$

Q146 Probability Definitions Finite Equally-Likely Probability Computation View

A store monitored the number of buyers of two products, A and B, during the months of January, February and March 2012. With this, it obtained this graph.
The store will draw a prize among the buyers of product A and another prize among the buyers of product B.
What is the probability that both winners made their purchases in February 2012?
(A) $\frac{1}{20}$ (B) $\frac{3}{242}$ (C) $\frac{5}{22}$ (D) $\frac{6}{25}$ (E) $\frac{7}{15}$

Q149 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View

The projections for rice production in the period 2012-2021, in a certain producing region, point to a perspective of constant growth in annual production. The table presents the quantity of rice, in tons, that will be produced in the first years of this period, according to this projection.

Year	Production projection (t)
2012	50.25
2013	51.50
2014	52.75
2015	54.00

The total amount of rice, in tonnes, that should be produced in the period from 2012 to 2021 will be
(A) 497.25. (B) 500.85. (C) 502.87. (D) 558.75. (E) 563.25.

Q150 Conditional Probability Direct Conditional Probability Computation from Definitions View

In a school with 1200 students, a survey was conducted on their knowledge of two foreign languages, English and Spanish.
In this survey, it was found that 600 students speak English, 500 speak Spanish, and 300 do not speak either of these languages.
If a student from this school is chosen at random and it is known that he does not speak English, what is the probability that this student speaks Spanish?
(A) $\frac{1}{2}$ (B) $\frac{5}{8}$ (C) $\frac{1}{4}$ (D) $\frac{5}{6}$ (E) $\frac{5}{14}$

Q151 Curve Sketching Identifying the Correct Graph of a Function View

During a Mathematics class, the teacher suggests to the students that a Cartesian coordinate system $(x, y)$ be established and represents on the board the description of five algebraic sets, I, II, III, IV and V, as follows:
I - is the circle with equation $x^{2} + y^{2} = 9$; II - is the parabola with equation $y = -x^{2} - 1$, with $x$ varying from $-1$ to $1$; III - is the square formed by the vertices $(-2,1)$, $(-1,1)$, $(-1,2)$ and $(-2,2)$; IV - is the square formed by the vertices $(1,1)$, $(2,1)$, $(2,2)$ and $(1,2)$; V - is the point $(0,0)$.
Next, the teacher correctly represents the five sets on the same grid, composed of squares with sides measuring one unit of length each, obtaining a figure.
Which of these figures was drawn by the teacher?
(A), (B), (C), (D), (E) [as shown in the figures]

Q152 Completing the square and sketching Vertex and parameter conditions for a quadratic graph View

The interior of a cup was generated by the rotation of a parabola around an axis $z$, as shown in the figure.
The real function that expresses the parabola, in the Cartesian plane of the figure, is given by the law $f(x) = \frac{3}{2}x^{2} - 6x + C$, where $C$ is the measure of the height of the liquid contained in the cup, in centimetres. It is known that the point $V$, in the figure, represents the vertex of the parabola, located on the $x$ axis.
Under these conditions, the height of the liquid contained in the cup, in centimetres, is
(A) 1. (B) 2. (C) 4. (D) 5. (E) 6.

Q153 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

Many physiological and biochemical processes, such as heartbeats and respiration rate, present scales constructed from the relationship between surface area and mass (or volume) of the animal. One of these scales, for example, considers that ``the cube of the surface area $S$ of a mammal is proportional to the square of its mass $M$''.
This is equivalent to saying that, for a constant $k > 0$, the area $S$ can be written as a function of $M$ by means of the expression:
(A) $S = k \cdot M$ (B) $S = k \cdot M^{\frac{1}{3}}$ (C) $S = k^{\frac{1}{3}} \cdot M^{\frac{1}{3}}$ (D) $S = k^{\frac{1}{3}} \cdot M^{\frac{2}{3}}$ (E) $S = k^{\frac{1}{3}} \cdot M^{2}$

Q160 Measures of Location and Spread View

Five food companies are for sale. An entrepreneur, aiming to expand his investments, wishes to buy one of these companies. To choose which one he will buy, he analyses the profit (in millions of reais) of each one, as a function of their time (in years) of existence, deciding to buy the company that presents the highest average annual profit.
The table presents the profit (in millions of reais) accumulated over the time (in years) of existence of each company.

Company	Profit (in millions of reais)	Time (in years)
F	24	3.0
G	24	2.0
H	25	2.5
M	15	1.5
P	9	1.5

The entrepreneur decided to buy the company
(A) F. (B) G. (C) H. (D) M. (E) P.

Q162 Measures of Location and Spread View

A survey was conducted in the 200 hotels in a city, in which the values, in reais, of the daily rates for a standard double room and the number of hotels for each daily rate value were noted. The daily rate values were: $\mathrm{A} = \mathrm{R}\$ 200{,}00$; $\mathrm{B} = \mathrm{R}\$ 300{,}00$; $\mathrm{C} = \mathrm{R}\$ 400{,}00$ and $\mathrm{D} = \mathrm{R}\$ 600{,}00$. In the graph, the areas represent the numbers of hotels surveyed, in percentage, for each daily rate value.
The median value of the daily rate, in reais, for the standard double room in this city, is
(A) 300.00. (B) 345.00. (C) 350.00. (D) 375.00. (E) 400.00.

Q165 Permutations & Arrangements Circular Arrangement View

A jewellery artisan has at his disposal Brazilian stones of three colours: red, blue and green.
He intends to produce jewellery made of a metal alloy, based on a mould in the shape of a non-square rhombus with stones at its vertices, so that two consecutive vertices always have stones of different colours.
The figure illustrates a piece of jewellery, produced by this artisan, whose vertices $A$, $B$, $C$ and $D$ correspond to the positions occupied by the stones.
Based on the information provided, how many different pieces of jewellery, in this format, can the artisan obtain?
(A) 6 (B) 12 (C) 18 (D) 24 (E) 36

Q166 Laws of Logarithms Logarithmic Formula Application (Modeling) View

In September 1987, Goiânia was the site of the largest radioactive accident that occurred in Brazil, when a sample of caesium-137, removed from an abandoned radiotherapy device, was inadvertently handled by part of the population. The half-life of a radioactive material is the time required for the mass of that material to be reduced to half. The half-life of caesium-137 is 30 years and the amount of remaining mass of a radioactive material, after $t$ years, is calculated by the expression $M(t) = A \cdot (2.7)^{kt}$, where $A$ is the initial mass and $k$ is a negative constant.
Consider 0.3 as an approximation for $\log_{10} 2$.
What is the time required, in years, for an amount of caesium-137 mass to be reduced to 10\% of the initial amount?
(A) 27 (B) 36 (C) 50 (D) 54 (E) 100

Q168 Probability Definitions Conditional Probability and Bayes' Theorem View

A screw factory has two machines, I and II, for the production of a certain type of screw.
In September, machine I produced $\frac{54}{100}$ of the total screws produced by the factory. Of the screws produced by this machine, $\frac{25}{1000}$ were defective. In turn, $\frac{38}{1000}$ of the screws produced in the same month by machine II were defective.
The combined performance of the two machines is classified according to the table, in which $P$ indicates the probability of a randomly chosen screw being defective.
$$\begin{aligned} 0 \leq P < \frac{2}{100} & \quad \text{Excellent} \\ \frac{2}{100} \leq P < \frac{4}{100} & \quad \text{Good} \\ \frac{4}{100} \leq P < \frac{6}{100} & \quad \text{Fair} \\ \frac{6}{100} \leq P < \frac{8}{100} & \quad \text{Poor} \\ \frac{8}{100} \leq P \leq 1 & \quad \text{Very Poor} \end{aligned}$$
The combined performance of these machines in September can be classified as
(A) excellent. (B) good. (C) fair. (D) poor. (E) very poor.

Q169 Combinations & Selection Combinatorial Probability View

Consider the following betting game:
On a ticket with 60 available numbers, a bettor chooses from 6 to 10 numbers. Among the available numbers, only 6 will be drawn. The bettor will be awarded if the 6 drawn numbers are among the numbers chosen by him on the same ticket.
The table presents the price of each ticket, according to the quantity of numbers chosen.

\begin{tabular}{ c } Quantity of numbers

chosen on a ticket

& Ticket price (R\$) \hline 6 & 2.00 \hline 7 & 12.00 \hline 8 & 40.00 \hline 9 & 125.00 \hline 10 & 250.00 \hline \end{tabular}
Five bettors, each with R\$ 500.00 to bet, made the following choices:
Arthur: 250 tickets with 6 numbers chosen; Bruno: 41 tickets with 7 numbers chosen and 4 tickets with 6 numbers chosen; Caio: 12 tickets with 8 numbers chosen and 10 tickets with 6 numbers chosen; Douglas: 4 tickets with 9 numbers chosen; Eduardo: 2 tickets with 10 numbers chosen.
The two bettors with the highest probabilities of being awarded are
(A) Caio and Eduardo. (B) Arthur and Eduardo. (C) Bruno and Caio. (D) Arthur and Bruno. (E) Douglas and Eduardo.

Q171 Simultaneous equations View

In the calibration of a new traffic light, the times are adjusted so that, in each complete cycle (green-yellow-red), the yellow light remains on for 5 seconds, and the time in which the green light remains on is equal to $\frac{2}{3}$ of the time in which the red light stays on. The green light is on, in each cycle, for $X$ seconds and each cycle lasts $Y$ seconds.
Which expression represents the relationship between $X$ and $Y$?
(A) $5X - 3Y + 15 = 0$ (B) $5X - 2Y + 10 = 0$ (C) $3X - 3Y + 15 = 0$ (D) $3X - 2Y + 15 = 0$ (E) $3X - 2Y + 10 = 0$

Q172 Solving quadratics and applications Geometric or real-world application leading to a quadratic equation View

The temperature $T$ of an oven (in degrees Celsius) is reduced by a system from the moment it is turned off ($t = 0$) and varies according to the expression $T(t) = -\frac{t^{2}}{4} + 400$, with $t$ in minutes. For safety reasons, the oven lock is only released for opening when the oven reaches a temperature of $39^{\circ}C$.
What is the minimum waiting time, in minutes, after turning off the oven, for the door to be opened?
(A) 19.0 (B) 19.8 (C) 20.0 (D) 38.0 (E) 39.0

Q173 Arithmetic Sequences and Series Find Specific Term from Given Conditions View

The Sun's magnetic activity cycle has a period of 11 years. The beginning of the first recorded cycle occurred at the beginning of 1755 and extended until the end of 1765. Since then, all cycles of the Sun's magnetic activity have been recorded.
In the year 2101, the Sun will be in the magnetic activity cycle number
(A) 32. (B) 34. (C) 33. (D) 35. (E) 31.

Q175 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View

In recent years, television has undergone a true revolution in terms of image quality, sound and interactivity with viewers. This transformation is due to the conversion of the analog signal to the digital signal. However, many cities still do not have this new technology. Seeking to bring these benefits to three cities, a television station intends to build a new transmission tower that sends signal to antennas A, B and C, already existing in these cities. The locations of the antennas are represented in the Cartesian plane.
The tower must be located in a place equidistant from the three antennas.
The appropriate location for the construction of this tower corresponds to the point with coordinates
(A) $(65; 35)$. (B) $(53; 30)$. (C) $(45; 35)$. (D) $(50; 20)$. (E) $(50; 30)$.

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