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Q136 Exponential Functions Applied/Contextual Exponential Modeling View

A loan was made at a monthly rate of $i\%$, using compound interest, in eight fixed and equal installments of $P$.
The debtor has the possibility of paying off the debt early at any time, paying for this the present value of the remaining installments. After paying the $5^{\text{th}}$ installment, he decides to pay off the debt when paying the $6^{\text{th}}$ installment.
The expression that corresponds to the total amount paid for the loan settlement is
(A) $P \left[ 1 + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } \right]$
(B) $P \left[ 1 + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { 2i } { 100 } \right) } \right]$
(C) $P \left[ 1 + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } \right]$
(D) $P \left[ \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { 2i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { 3i } { 100 } \right) } \right]$
(E) $P \left[ \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 3 } } \right]$

Q137 Laws of Logarithms Logarithmic Formula Application (Modeling) View

To take the trip of her dreams, a person needed to take out a loan in the amount of $\mathrm{R}\$ 5000.00$. To pay the installments, she has at most $\mathrm{R}\$ 400.00$ monthly. For this loan amount, the installment value ($P$) is calculated as a function of the number of installments ($n$) according to the formula
$$P = \frac { 5000 \times 1.013 ^ { n } \times 0.013 } { \left( 1.013 ^ { n } - 1 \right) }$$
If necessary, use 0.005 as an approximation for $\log 1.013$; 2.602 as an approximation for $\log 400$; 2.525 as an approximation for $\log 335$.
According to the given formula, the smallest number of installments whose values do not compromise the limit defined by the person is
(A) 12.
(B) 14.
(C) 15.
(D) 16.
(E) 17.

Q138 Trig Graphs & Exact Values View

Rays of sunlight are hitting the surface of a lake forming an angle $x$ with its surface, as shown in the figure.
Under certain conditions, one can assume that the light intensity of these rays, on the lake surface, is given approximately by $I(x) = K \cdot \sin(x)$, where $k$ is a constant, and assuming that $x$ is between $0^{\circ}$ and $90^{\circ}$.
When $x = 30^{\circ}$, the light intensity is reduced to what percentage of its maximum value?
(A) $33\%$
(B) $50\%$
(C) $57\%$
(D) $70\%$
(E) $86\%$

Q139 Travel graphs View

Traffic congestion constitutes a problem that afflicts thousands of Brazilian drivers every day. The graph illustrates the situation, representing, over a defined time interval, the variation in the speed of a vehicle during a traffic jam.
How many minutes did the vehicle remain stationary throughout the total time interval analyzed?
(A) 4
(B) 3
(C) 2
(D) 1
(E) 0

Q140 Circles Area and Geometric Measurement Involving Circles View

A waiter needs to choose a tray with a rectangular base to serve four glasses of sparkling wine that need to be arranged in a single row, parallel to the longer side of the tray, and with their bases completely supported on the tray. The base and upper edge of the glasses are circles with radius 4 cm and 5 cm, respectively.
The tray to be chosen should have a minimum area, in square centimeters, equal to
(A) 192.
(B) 300.
(C) 304.
(D) 320.
(E) 400.

Q141 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View

In a cafeteria, the success of summer sales are juices prepared based on fruit pulp. One of the best-selling juices is strawberry with acerola, which is prepared with $\frac{2}{3}$ of strawberry pulp and $\frac{1}{3}$ of acerola pulp.
For the merchant, the pulps are sold in packages of equal volume. Currently, the strawberry pulp package costs $\mathrm{R}\$ 18.00$ and the acerola one costs $\mathrm{R}\$ 14.70$. However, a price increase is expected for the acerola pulp package next month, rising to $\mathrm{R}\$ 15.30$.
To not increase the price of the juice, the merchant negotiated with the supplier a reduction in the price of the strawberry pulp package.
The reduction, in reais, in the price of the strawberry pulp package should be
(A) 1.20.
(B) 0.90.
(C) 0.60.
(D) 0.40.
(E) 0.30.

Q142 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View

A couple is moving to a new home and needs to place a cubic object, with 80 cm edges, in a cardboard box, which cannot be disassembled. They have five boxes available, with different dimensions, as described:

Box 1: $86 \mathrm{~cm} \times 86 \mathrm{~cm} \times 86 \mathrm{~cm}$
Box 2: $75 \mathrm{~cm} \times 82 \mathrm{~cm} \times 90 \mathrm{~cm}$
Box 3: $85 \mathrm{~cm} \times 82 \mathrm{~cm} \times 90 \mathrm{~cm}$
Box 4: $82 \mathrm{~cm} \times 95 \mathrm{~cm} \times 82 \mathrm{~cm}$
Box 5: $80 \mathrm{~cm} \times 95 \mathrm{~cm} \times 85 \mathrm{~cm}$

The couple needs to choose a box in which the object fits, so that the least free space remains in its interior.
The box chosen by the couple should be number
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(E) 5.

Q143 Combinations & Selection Counting Integer Solutions to Equations View

A children's toy truck-carrier is formed by a trailer and ten small cars transported on it. In the production sector of the company that manufactures this toy, all the small cars are painted so that the toy looks more attractive. The colors used are yellow, white, orange and green, and each small car is painted with only one color. The truck-carrier has a fixed color. The company determined that in every truck-carrier there must be at least one small car of each of the four available colors. Change of position of the small cars on the truck-carrier does not generate a new model of the toy.
Based on this information, how many distinct models of the truck-carrier toy can this company produce?
(A) $C_{6,4}$
(B) $C_{9,3}$
(C) $C_{10,4}$
(D) $6^{4}$
(E) $4^{6}$

Q145 Confidence intervals Find a specific bound or margin of error from the CI formula View

An electoral research institute receives an order in which the margin of error should be at most 2 percentage points (0.02).
The institute has 5 recent surveys, P1 to P5, on the subject of the order and will use the one with an error smaller than requested.
The data on the surveys are as follows:

Survey	$\boldsymbol{\sigma}$	$\boldsymbol{N}$	$\sqrt{\boldsymbol{N}}$
P1	0.5	1764	42
P2	0.4	784	28
P3	0.3	576	24
P4	0.2	441	21
P5	0.1	64	8

The error $e$ can be expressed by
$$|e| < 1.96 \frac{\sigma}{\sqrt{N}}$$
where $\sigma$ is a parameter and $N$ is the number of people interviewed by the survey. Which survey should be used?
(A) P1
(B) P2
(C) P3
(D) P4
(E) P5

Q149 Probability Definitions Finite Equally-Likely Probability Computation View

The figure illustrates a game of Minesweeper, the game present in practically every personal computer. Four squares on a $16 \times 16$ board were opened, and the numbers on their faces indicate how many of their 8 neighbors contain mines (to be avoided). The number 40 in the lower right corner is the total number of mines on the board, whose positions were chosen at random, uniformly, before opening any square.
In his next move, the player must choose among the squares marked with the letters $P, Q, R, S$ and $T$ one to open, and should choose the one with the lowest probability of containing a mine.
The player should open the square marked with the letter
(A) $P$.
(B) $Q$.
(C) $R$.
(D) $S$.
(E) $T$.

Q151 Measures of Location and Spread View

The evaluation of student performance in a university course is based on the weighted average of grades obtained in the disciplines by their respective number of credits, as shown in the table:

Evaluation	Average grade (M)
Excellent	$9 < M \leq 10$
Good	$7 \leq M \leq 9$
Regular	$5 \leq M < 7$
Poor	$3 \leq M < 5$
Very Poor	$M < 3$

A certain student knows that if he obtains a ``Good'' or ``Excellent'' evaluation, he will be able to enroll in the disciplines he desires. He has already taken the exams for 4 of the 5 disciplines in which he is enrolled, but has not yet taken the exam for discipline I, as shown in the table.

Disciplines	Grades	\begin{tabular}{ c } Number
of credits

\hline I & & 12 \hline II & 8.00 & 4 \hline III & 6.00 & 8 \hline IV & 5.00 & 8 \hline V & 7.50 & 10 \hline \end{tabular}
In order to achieve his objective, the minimum grade he must obtain in discipline I is
(A) 7.00.
(B) 7.38.
(C) 7.50.
(D) 8.25.
(E) 9.00.

Q155 Measures of Location and Spread View

Three students, X, Y, and Z, are enrolled in an English course. To evaluate these students, the teacher chose to give five tests. In order to pass this course, the student must have an arithmetic mean of the grades from the five tests greater than or equal to 6. The table shows the grades each student received on each test.

Student	$\mathbf{1}^{\mathbf{st}}$	$\mathbf{2}^{\mathbf{nd}}$	$\mathbf{3}^{\mathbf{rd}}$	$\mathbf{4}^{\mathbf{th}}$	$\mathbf{5}^{\mathbf{th}}$
X	5	5	5	10	6
Y	4	9	3	9	5
Z	5	5	8	5	6

Based on the data in the table and the information given, will be/will fail
(A) only student Y.
(B) only student Z.
(C) only students X and Y.
(D) only students X and Z.
(E) students X, Y, and Z.

Q161 Stationary points and optimisation Geometric or applied optimisation problem View

Lobster hatcheries are built, by local fishing cooperatives, in the shape of right-rectangular prisms, fixed to the ground and with flexible nets of the same height, capable of withstanding marine corrosion. For each hatchery to be built, the cooperative uses entirely 100 linear meters of this net, which is used only on the sides.
What should be the values of $X$ and $Y$, in meters, so that the area of the base of the hatchery is maximum?
(A) 1 and 49
(B) 1 and 99
(C) 10 and 10
(D) 25 and 25
(E) 50 and 50

Q162 Curve Sketching Asymptote Determination View

The English physiologist Archibald Vivian Hill proposed, in his studies, that the velocity $v$ of contraction of a muscle when subjected to a weight $p$ is given by the equation $( p + a )( v + b ) = K$, with $a$, $b$, and $K$ constants.
A physiotherapist, with the intention of maximizing the beneficial effect of the exercises he would recommend to one of his patients, wanted to study this equation and classified it as follows:

Type of curve

Oblique half-line

Horizontal half-line

Branch of parabola

Arc of circle

Branch of hyperbola

The physiotherapist analyzed the dependence between $v$ and $p$ in Hill's equation and classified it according to its geometric representation in the Cartesian plane, using the coordinate pair ($p$; $v$). Assume that $K > 0$.
The graph of the equation that the physiotherapist used to maximize the effect of the exercises is of the type
(A) oblique half-line.
(B) horizontal half-line.
(C) branch of parabola.
(D) arc of circle.
(E) branch of hyperbola.

Q166 Harmonic Form View

A scientist, in his studies to model a person's blood pressure, uses a function of the type $P(t) = A + B\cos(kt)$ where $A$, $B$, and $K$ are positive real constants and $t$ represents the time variable, measured in seconds. Consider that a heartbeat represents the time interval between two successive maximum pressures.
When analyzing a specific case, the scientist obtained the data:

Minimum pressure	78
Maximum pressure	120
Number of heartbeats per minute	90

The function $P(t)$ obtained by this scientist when analyzing the specific case was
(A) $P(t) = 99 + 21\cos(3\pi t)$
(B) $P(t) = 78 + 42\cos(3\pi t)$
(C) $P(t) = 99 + 21\cos(2\pi t)$
(D) $P(t) = 99 + 21\cos(t)$
(E) $P(t) = 78 + 42\cos(t)$

Q168 Curve Sketching Number of Solutions / Roots via Curve Analysis View

The Church of Saint Francis of Assisi, a modernist architectural work by Oscar Niemeyer, located at Pampulha Lake, in Belo Horizonte, has parabolic vaults. Figure 2 provides a front view of one of the vaults, with hypothetical measurements to simplify the calculations.
What is the measure of the height H, in meters, indicated in Figure 2?
(A) $\frac{16}{3}$
(B) $\frac{31}{5}$
(C) $\frac{25}{4}$
(D) $\frac{25}{3}$
(E) $\frac{75}{2}$

Q174 Measures of Location and Spread View

The graph presents the unemployment rate (in \%) for the period from March 2008 to April 2009, obtained based on data observed in the metropolitan regions of Recife, Salvador, Belo Horizonte, Rio de Janeiro, São Paulo, and Porto Alegre.
The median of this unemployment rate, in the period from March 2008 to April 2009, was
(A) $8.1\%$
(B) $8.0\%$
(C) $7.9\%$
(D) $7.7\%$
(E) $7.6\%$

Q175 Binomial Distribution MCQ Selecting a Binomial Probability Expression or Value View

On an avenue there are 10 traffic lights. Due to a system failure, the traffic lights were without control for one hour, and fixed their lights only in green or red. The traffic lights operate independently; the probability of showing green is $\frac{2}{3}$ and of showing red is $\frac{1}{3}$. A person walked the entire avenue during the period of the failure, observing the color of the light of each of these traffic lights. What is the probability that this person observed exactly one signal in green?
(A) $\frac{10 \times 2}{3^{10}}$
(B) $\frac{10 \times 2^{9}}{3^{10}}$
(C) $\frac{2^{10}}{3^{100}}$
(D) $\frac{2^{90}}{3^{100}}$
(E) $\frac{2}{3^{10}}$

Q177 Permutations & Arrangements Forming Numbers with Digit Constraints View

A company will build its page on the internet and expects to attract an audience of approximately one million customers. To access this page, a password with a format to be defined by the company will be required. There are five format options offered by the programmer, described in the table, where ``L'' and ``D'' represent, respectively, uppercase letter and digit.

Option	Format
I	LDDDDD
II	DDDDDD
III	LLDDDD
IV	DDDDD
V	LLLDD

The letters of the alphabet, among the 26 possible ones, as well as the digits, among the 10 possible ones, can be repeated in any of the options.
The company wants to choose a format option whose number of possible distinct passwords is greater than the expected number of customers, but such that this number is not greater than twice the expected number of customers.
The option that best suits the company's conditions is
(A) I.
(B) II.
(C) III.
(D) IV.
(E) V.

Q178 Combinations & Selection Basic Combination Computation View

Not being fans of practicing sports, a group of friends decided to hold a soccer tournament using a video game. They decided that each player plays only once against each of the other players. The champion will be the one who gets the highest number of points. They observed that the number of matches played depends on the number of players, as shown in the table:

\begin{tabular}{ c } Number of

players

& 2 & 3 & 4 & 5 & 6 & 7 \hline

Number of

matches

& 1 & 3 & 6 & 10 & 15 & 21 \hline \end{tabular}
If the number of players is 8, how many matches will be played?
(A) 64
(B) 56
(C) 49
(D) 36
(E) 28

Q179 Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View

A resident of a metropolitan region has a 50\% probability of being late for work when it rains in the region; if it does not rain, his probability of being late is 25\%. For a given day, the meteorological service estimates a 30\% probability of rain occurring in that region.
What is the probability that this resident will be late for work on the day for which the rain estimate was given?
(A) 0.075
(B) 0.150
(C) 0.325
(D) 0.600
(E) 0.800

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