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16 maths questions

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) $86\%$
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.
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}}$
P10.5176442
P20.478428
P30.357624
P40.244121
P50.1648

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
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:
EvaluationAverage 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.
DisciplinesGrades\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}}$
X555106
Y49395
Z55856

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
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 pressure78
Maximum pressure120
Number of heartbeats per minute90

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 Completing the square and 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.
OptionFormat
ILDDDDD
IIDDDDDD
IIILLDDDD
IVDDDDD
VLLLDD

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