Tüm alıştırma soruları

Consider the following system of linear equations in variables

x

and

y

, where

k

is a real constant:

\begin{aligned} kx + 4y &= 8 \\ x + ky &= k + 2 \end{aligned}

Which of the following statements are true? Select all that apply.

Geçerli olan tümünü seçin

If

k = 2

, the system has infinitely many solutions.

If

k = -2

, the system has no solutions.

If

k = 0

, the unique solution to the system is

(x, y) = (2, 2)

.

If

k = 3

, the system has no solutions.

If

k = 1

, the solution to the system satisfies

x + y = 4

.

Cevabı ve açıklamayı göster

Cevap: If $k = 2$ , the system has infinitely many solutions.; If $k = -2$ , the system has no solutions.; If $k = 0$ , the unique solution to the system is $(x, y) = (2, 2)$ .

Cevap

The correct statements are that setting

k = 2

results in infinitely many solutions, setting

k = -2

results in no solutions, and setting

k = 0

yields the unique solution

(2, 2)

.

The system has a coefficient matrix determinant of

k^2 - 4

. Setting

k = 2

produces identical equations (

x + 2y = 4

), giving infinitely many solutions. Setting

k = -2

produces parallel equations with different constants (

-x + 2y = 4

vs

-x + 2y = 0

), giving no solutions. Setting

k = 0

reduces the system directly to

y = 2

and

x = 2

, confirming the unique point

(2, 2)

.

Adım Adım Çözüm

1

Analyze the determinant of the coefficient matrix to identify conditions for unique vs. non-unique solutions.

The coefficient matrix determinant is

\Delta = k\cdot k - 4\cdot 1 = k^2 - 4 = (k - 2)(k + 2)

.

If

\Delta \neq 0

(i.e.,

k \neq \pm 2

), the system has a unique solution. If

\Delta = 0

(i.e.,

k = 2

or

k = -2

), the lines are either identical or parallel.

2

Test

k = 2

in the system.

Equation 1 becomes

2x + 4y = 8 \implies x + 2y = 4

. Equation 2 becomes

x + 2y = 4

.

Since the equations are identical, the lines coincide, resulting in infinitely many solutions.

3

Test

k = -2

in the system.

Equation 1 becomes

-2x + 4y = 8 \implies -x + 2y = 4

. Equation 2 becomes

x - 2y = 0 \implies -x + 2y = 0

.

The slopes are equal (

1/2

) but the y-intercepts differ (

2

vs

0

), meaning the lines are parallel and distinct, yielding zero solutions.

4

Test

k = 0

and

k = 1

to evaluate remaining choices.

For

k = 0

,

4y = 8 \implies y = 2

and

x + 0 = 2 \implies x = 2

, giving

(2, 2)

. For

k = 1

, Equation 2 is directly

x + y = 3

.

This verifies that the statement for

k = 0

is correct, while the statement for

k = 1

claiming

x + y = 4

is false.

Anahtar Kavram

Parametric Linear Systems and Conditions for Solvability

Soru 162Soru →

A survey of $120$ university students evaluated course enrollment in Computer Science and Mathematics. Of the students surveyed, $70$ are enrolled in Computer Science, $65$ are enrolled in Mathematics, and $25$ are enrolled in neither course. How many students are enrolled in both Computer Science and Mathematics?

15

25

40

50

95

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

40 students are enrolled in both Computer Science and Mathematics.

The correct answer is 40. From the total population of 120 students, 25 take neither subject, meaning

120 - 25 = 95

students take at least one subject. According to the Principle of Inclusion-Exclusion,

|C \cup M| = |C| + |M| - |C \cap M|

. Substituting the known values yields

95 = 70 + 65 - |C \cap M|

, which simplifies to

95 = 135 - |C \cap M|

, giving

|C \cap M| = 40

.

Adım Adım Çözüm

1

Calculate the total number of students enrolled in at least one of the two courses.

Total in at least one course =

120 - 25 = 95

.

Subtracting the students enrolled in neither course from the total surveyed gives the union of the two sets,

|C \cup M|

.

2

Apply the Principle of Inclusion-Exclusion for two sets.

|C \cup M| = |C| + |M| - |C \cap M|

, so

95 = 70 + 65 - |C \cap M|

.

Adding

|C|

and

|M|

double-counts the students taking both courses, so subtracting the union yields the intersection.

3

Solve for the intersection

|C \cap M|

.

|C \cap M| = 135 - 95 = 40

.

Direct arithmetic evaluation gives the required count of students enrolled in both subjects.

Anahtar Kavram

Principle of Inclusion-Exclusion for Two Sets

Tahmini Süre:1m 0s

Soru 163Soru →

In a department of $90$ employees, $55$ speak French, $45$ speak Spanish, and $10$ speak neither French nor Spanish. How many employees speak both French and Spanish?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20

Subtracting the 10 employees who speak neither language from the total of 90 leaves 80 employees who speak at least one language. By the principle of inclusion-exclusion, Total(At least one) = French + Spanish - Both. Substituting the known values gives 80 = 55 + 45 - Both, which simplifies to 80 = 100 - Both, so Both = 20.

Adım Adım Çözüm

1

Find the number of employees who speak at least one of the languages

80 employees

Subtract the 10 employees who speak neither language from the total department size of 90.

2

Sum the total counts for each language group

100

Add the number of French speakers (55) to Spanish speakers (45).

3

Calculate the overlap (intersection) using inclusion-exclusion

20 employees

Subtract the count of employees speaking at least one language (80) from the sum of the individual language groups (100).

Anahtar Kavram

Principle of Inclusion-Exclusion for Two Sets

Tahmini Süre:45s

Soru 164Soru →

Dataset $D$ consists of $40$ distinct real numbers arranged in strictly increasing order: $x_1 < x_2 < \dots < x_{40}$ . Let $R_D$ , $Q_D$ , and $\sigma_D$ denote the range, interquartile range, and standard deviation of Dataset $D$ , respectively.

A new dataset, Dataset $E$ , is created by replacing each of the $10$ largest numbers ( $x_{31}, x_{32}, \dots, x_{40}$ ) with the value of $x_{30}$ , which represents the $75\text{th}$ percentile of Dataset $D$ . Let $R_E$ , $Q_E$ , and $\sigma_E$ denote the range, interquartile range, and standard deviation of Dataset $E$ , respectively.

Which of the following statements MUST be true? Select all such statements.

Geçerli olan tümünü seçin

The range of Dataset

E

is strictly less than the range of Dataset

D

(

R_E < R_D

).

The interquartile range of Dataset

E

is equal to the interquartile range of Dataset

D

(

Q_E = Q_D

).

The standard deviation of Dataset

E

is strictly less than the standard deviation of Dataset

D

(

\sigma_E < \sigma_D

).

The median of Dataset

E

is strictly less than the median of Dataset

D

.

The interquartile range of Dataset

E

is strictly less than the interquartile range of Dataset

D

(

Q_E < Q_D

).

Cevabı ve açıklamayı göster

Cevap: The range of Dataset $E$ is strictly less than the range of Dataset $D$ ( $R_E < R_D$ ).; The interquartile range of Dataset $E$ is equal to the interquartile range of Dataset $D$ ( $Q_E = Q_D$ ).; The standard deviation of Dataset $E$ is strictly less than the standard deviation of Dataset $D$ ( $\sigma_E < \sigma_D$ ).

Cevap

The statements asserting that the range of Dataset E is strictly less than the range of Dataset D (

R_E < R_D

), the interquartile range of Dataset E is equal to the interquartile range of Dataset D (

Q_E = Q_D

), and the standard deviation of Dataset E is strictly less than the standard deviation of Dataset D (

\sigma_E < \sigma_D

) are all correct.

The range of Dataset E decreases because its maximum value is reduced from

x_{40}

to

x_{30}

while its minimum value

x_1

remains the same. The interquartile range remains unchanged because the boundary values defining the

25\text{th}

percentile (

x_{10}

) and

75\text{th}

percentile (

x_{30}

) are not altered. The standard deviation strictly decreases because reducing the

10

largest values brings them closer to the rest of the distribution, decreasing total variation.

Adım Adım Çözüm

1

Analyze the change in the maximum and minimum values to evaluate the Range.

In Dataset

D

,

\text{Range } R_D = x_{40} - x_1

. In Dataset

E

, the maximum element becomes

x_{30}

while the minimum remains

x_1

. Since all numbers are distinct and ordered,

x_{30} < x_{40}

, so

R_E = x_{30} - x_1 < R_D

. Thus,

R_E < R_D

is true.

Range is defined as

\text{Maximum} - \text{Minimum}

.

2

Analyze the

25\text{th}

percentile (

Q_1

) and

75\text{th}

percentile (

Q_3

) to evaluate the Interquartile Range.

For

N=40

,

Q_1 = x_{10}

and

Q_3 = x_{30}

. Dataset

E

modifies only the values strictly above

x_{30}

(

x_{31}

through

x_{40}

). Thus

x_{10}

and

x_{30}

retain their exact values in Dataset

E

. Therefore,

Q_E = x_{30} - x_{10} = Q_D

.

Interquartile range depends solely on

Q_3 - Q_1

, which relies on position cutoffs that were unaffected.

3

Analyze the effect of pulling extreme upper values inward on the Standard Deviation.

Standard deviation measures overall spread around the mean and can be calculated via pairwise differences

\text{Var}(X) = \frac{1}{2N^2} \sum_{i,j} (x_i - x_j)^2

. Replacing

x_{31}, \dots, x_{40}

with

x_{30}

strictly reduces the distances between these upper points and all other points, ensuring

\sigma_E < \sigma_D

.

Reducing the distance of tail observations from the rest of the dataset strictly decreases overall variance and standard deviation.

Anahtar Kavram

Impact of Data Transformations on Measures of Dispersion and Position

Soru 165Soru →

The frequency distribution table below summarizes the processing times, in minutes, for a sample of 100 health insurance claims processed by an administrative team.

Claim Processing Time (minutes)	Frequency
$10 \le t < 20$	15
$20 \le t < 30$	25
$30 \le t < 40$	35
$40 \le t < 50$	20
$50 \le t < 60$	5

Which of the following statements regarding this frequency distribution must be true? Select all that apply.

Geçerli olan tümünü seçin

The interval

30 \le t < 40

contains the median processing time for the dataset.

Exactly 60 percent of the claims required 30 minutes or more to process.

The lower bound estimate for the overall mean processing time is 31.0 minutes.

The number of claims in the interval

20 \le t < 30

represents 25 percent of all claims.

The ratio of the number of claims completed in under 30 minutes to the number of claims taking 40 minutes or more is 2 to 1.

Cevabı ve açıklamayı göster

Cevap: The interval $30 \le t < 40$ contains the median processing time for the dataset.; Exactly 60 percent of the claims required 30 minutes or more to process.; The number of claims in the interval $20 \le t < 30$ represents 25 percent of all claims.

Cevap

The true statements are: the median processing time lies in the interval

30 \le t < 40

; exactly 60 percent of the claims required 30 minutes or more to process; and the number of claims in the interval

20 \le t < 30

represents 25 percent of all claims.

The statement regarding the median is correct because the 50th and 51st values lie within the cumulative frequency span of 41 to 75, corresponding to

30 \le t < 40

. The statement regarding claims requiring 30 minutes or more is correct because

35 + 20 + 5 = 60

out of 100 claims is 60 percent. The statement regarding the

20 \le t < 30

interval is correct because 25 out of 100 claims is exactly 25 percent.

Adım Adım Çözüm

1

Calculate cumulative frequencies to locate the median class interval.

Cumulative frequencies are 15 (

10 \le t < 20

), 40 (

20 \le t < 30

), 75 (

30 \le t < 40

), 95 (

40 \le t < 50

), and 100 (

50 \le t < 60

). For 100 observations, the 50th and 51st values fall in the

30 \le t < 40

interval.

The median of an even number

N=100

of sorted values is the average of the 50th and 51st values.

2

Calculate the percentage of claims taking 30 minutes or more.

Sum the frequencies for intervals

\ge 30

:

35 + 20 + 5 = 60

. As a percentage of 100 total claims, this is

\frac{60}{100} \times 100\% = 60\%

.

To find the proportion meeting or exceeding a threshold, sum all class frequencies with lower limits at or above 30 minutes.

3

Evaluate the percentage representation of the

20 \le t < 30

interval.

Frequency is 25 out of 100 total claims, which equals

\frac{25}{100} = 25\%

.

The relative frequency of a grouped class is its frequency divided by the total frequency.

4

Determine the lower bound for the estimated mean processing time to evaluate incorrect claims.

Minimum sum =

15(10) + 25(20) + 35(30) + 20(40) + 5(50) = 2750

. Lower bound mean =

2750 / 100 = 27.5

minutes.

The absolute minimum mean occurs when every observation equals the lower boundary of its respective interval.

5

Calculate the ratio of claims under 30 minutes to claims 40 minutes or more.

Claims

< 30

min =

15 + 25 = 40

. Claims

\ge 40

min =

20 + 5 = 25

. Ratio =

40 : 25 = 8 : 5

.

Must include all relevant intervals (including

50 \le t < 60

) when computing the denominator for

\ge 40

minutes.

Anahtar Kavram

Grouped Frequency Distributions, Median Class Identification, and Relative Frequencies

Soru 166Soru →

A university archive uses two high-speed scanners, Machine A and Machine B, to digitize manuscript pages. Machine A operates at a constant rate of $60$ pages per hour, and Machine B operates at a constant rate of $80$ pages per hour. On Monday, Machine A starts scanning at 8:00 AM. At 9:30 AM, Machine B is brought online to work alongside Machine A. If both machines continue scanning together at their respective constant rates until a total of $510$ pages have been scanned, at what time will the scanning project be completed?

11:00 AM

11:38 AM

12:30 PM

2:45 PM

3:30 PM

Cevabı ve açıklamayı göster

Cevap: 12:30 PM

Cevap

12:30 PM

The option stating 12:30 PM is correct. Machine A operates alone from 8:00 AM to 9:30 AM (1.5 hours), completing

60 \times 1.5 = 90

pages. This leaves

510 - 90 = 420

pages to be scanned. From 9:30 AM onward, both machines work together at a combined rate of

60 + 80 = 140

pages per hour. The remaining 420 pages require

\frac{420}{140} = 3

hours. Adding 3 hours to 9:30 AM gives a completion time of 12:30 PM.

Adım Adım Çözüm

1

Calculate the number of pages scanned by Machine A during its head start

60 \text{ pages/hr} \times 1.5 \text{ hours} = 90 \text{ pages}

Machine A operates alone from 8:00 AM to 9:30 AM, which is

1.5

hours.

2

Determine the remaining number of pages to be scanned

510 - 90 = 420 \text{ pages}

Subtract the pages already completed from the total requirement of 510 pages.

3

Calculate the combined scanning rate of both machines

60 + 80 = 140 \text{ pages/hr}

When both machines operate simultaneously, their individual rates add together.

4

Calculate the time required to scan the remaining pages

\frac{420 \text{ pages}}{140 \text{ pages/hr}} = 3 \text{ hours}

Divide the remaining work by the combined rate.

5

Determine the final completion time

9:30 \text{ AM} + 3 \text{ hours} = 12:30 \text{ PM}

Add the combined work duration to the time when both machines began working together (9:30 AM).

Anahtar Kavram

Combined Work Rates with Staggered Start Times

Tahmini Süre:1m 30s

Soru 167Soru →

A IT consultant charges a flat setup fee of $\$150$ plus a standard hourly rate of $\$80$ for regular hours worked on a project. For any rush hours worked, the hourly rate increases by $50\%$ . On a recently completed project, the consultant worked a total of $t$ hours, of which $r$ hours were rush hours (where $0 \leq r \leq t$ ). Which of the following statements must be true regarding the total charge $C$ , in dollars, for this project? Select all such statements.

Geçerli olan tümünü seçin

The total charge, in dollars, can be expressed as

C = 150 + 80t + 40r

.

If the consultant worked a total of

20

hours and the total charge was

\$2,150

, then exactly

10

of those hours were rush hours.

The total charge, in dollars, can be expressed as

C = 150 + 120t - 40r

.

If no rush hours were worked (

r = 0

), the average charge per hour, including the setup fee, is

\frac{150}{t} + 80

dollars per hour.

If all hours worked were rush hours (

r = t

), the total charge is given by

C = 150 + 160t

.

Cevabı ve açıklamayı göster

Cevap: The total charge, in dollars, can be expressed as $C = 150 + 80t + 40r$ .; If the consultant worked a total of $20$ hours and the total charge was $\$2,150$ , then exactly $10$ of those hours were rush hours.; If no rush hours were worked ( $r = 0$ ), the average charge per hour, including the setup fee, is $\frac{150}{t} + 80$ dollars per hour.

Cevap

The correct statements are those expressing the total charge as

C = 150 + 80t + 40r

, determining that

10

rush hours were worked when total cost is

\$2,150

for

20

hours, and calculating the average hourly cost as

\frac{150}{t} + 80

dollars per hour when no rush hours are worked.

The model correctly partitions total hours

t

into

(t - r)

regular hours at

\$80

/hr and

r

rush hours at

\$120

/hr, which simplifies algebraically to

C = 150 + 80t + 40r

. Solving this equation for

t = 20

and

C = 2150

yields

r = 10

. Furthermore, when

r = 0

, dividing total charge

150 + 80t

by

t

gives

\frac{150}{t} + 80

.

Adım Adım Çözüm

1

Calculate the hourly rush rate from the given percentage increase.

The standard rate is

\$80

/hr. The rush rate is

\$80 \times (1 + 0.50) = \$120

/hr.

Rush hours cost

50\%

more than standard regular hours.

2

Set up the algebraic model for total charge

C

using standard hours

(t - r)

and rush hours

r

.

C = 150 + 80(t - r) + 120r = 150 + 80t - 80r + 120r = 150 + 80t + 40r

.

This combines the fixed setup fee with variable costs from regular and rush hours.

3

Evaluate the specific case where total time

t = 20

and total cost

C = 2150

.

2150 = 150 + 80(20) + 40r \implies 2150 = 1750 + 40r \implies 40r = 400 \implies r = 10

.

Plugging given values into the algebraic model allows solving for the unknown number of rush hours.

4

Calculate average hourly cost when

r = 0

.

Average cost

= \frac{150 + 80t}{t} = \frac{150}{t} + 80

.

Average hourly rate is total charge divided by total hours worked.

Anahtar Kavram

Linear algebraic modeling of rate problems with multiple rate components

Soru 168Soru →

A university research department needs to form a 4-person delegation consisting of exactly 2 senior researchers and 2 junior researchers. The pool of available candidates contains 5 senior researchers and 6 junior researchers. However, 2 specific senior researchers refuse to serve on the delegation together. How many different 4-person delegations can be formed under these conditions?

120

135

150

270

300

Cevabı ve açıklamayı göster

Cevap: 135

Cevap

135 different 4-person delegations can be formed.

To calculate the number of valid delegations, first find the unrestricted number of delegations by multiplying the combinations of senior researchers

\binom{5}{2} = 10

and junior researchers

\binom{6}{2} = 15

, which equals 150. Next, calculate the number of invalid delegations where both restricted senior researchers are chosen together, which is

\binom{2}{2} \times \binom{6}{2} = 15

. Subtracting the invalid delegations from the total gives

150 - 15 = 135

.

Adım Adım Çözüm

1

Calculate the total number of possible delegations without any restrictions.

\binom{5}{2} \times \binom{6}{2} = 10 \times 15 = 150

Choosing 2 senior researchers out of 5 requires combinations,

\binom{5}{2} = 10

. Choosing 2 junior researchers out of 6 requires combinations,

\binom{6}{2} = 15

. By the Fundamental Counting Principle, the total number of unrestricted delegations is

10 \times 15 = 150

.

2

Calculate the number of invalid delegations that include both restricted senior researchers.

\binom{2}{2} \times \binom{6}{2} = 1 \times 15 = 15

If both conflicting senior researchers are placed on the delegation, there is only

\binom{2}{2} = 1

way to select the senior members. The 2 junior researchers can still be selected in

\binom{6}{2} = 15

ways, giving 15 invalid delegations.

3

Subtract the number of invalid delegations from the total possible delegations.

150 - 15 = 135

Subtracting the 15 restricted delegations from the 150 total delegations leaves 135 valid delegations.

Anahtar Kavram

Combinations with Exclusion Restrictions and Fundamental Counting Principle

Soru 169Soru →

A boutique perfume workshop creates a signature fragrance blend by combining two fragrance oils, Oil A and Oil B. Oil A costs $12 per ounce and contains 40% pure essential oil by volume. Oil B costs$ 20 per ounce and contains 80% pure essential oil by volume. The perfumer creates a 30-ounce batch of the signature blend at a total cost that averages exactly $15 per ounce. What is the total volume, in ounces, of pure essential oil contained in this 30-ounce blend?

Cevabı ve açıklamayı göster

Cevap: 16.5

Cevap

The total volume of pure essential oil contained in the 30-ounce blend is 16.5 ounces.

To find the total essential oil content, first set up a system of equations for the volumes of Oil A (

x

) and Oil B (

y

):

x + y = 30

and

12x + 20y = 450

. Solving this system yields

x = 18.75

ounces and

y = 11.25

ounces. Multiplying each by its respective essential oil concentration gives

0.40 \times 18.75 = 7.5

ounces from Oil A and

0.80 \times 11.25 = 9

ounces from Oil B, totaling 16.5 ounces of essential oil.

Adım Adım Çözüm

1

Set up linear equations representing the total volume and total cost of the mixture.

x + y = 30

and

12x + 20y = 450

, where

x

is ounces of Oil A and

y

is ounces of Oil B.

The total cost of the 30-ounce blend at

15 per ounce is

15 \times 30 = 450$ dollars.

2

Solve for the quantities of Oil A and Oil B used in the mixture.

x = 18.75

ounces of Oil A and

y = 11.25

ounces of Oil B.

Substituting

y = 30 - x

into

12x + 20(30 - x) = 450

yields

-8x = -150

, giving

x = 18.75

.

3

Compute the amount of pure essential oil contributed by each component and sum them.

0.40(18.75) + 0.80(11.25) = 7.5 + 9 = 16.5

ounces.

Oil A contains 40% essential oil by volume and Oil B contains 80% essential oil by volume.

Anahtar Kavram

Systems of Linear Equations and Mixture Modeling

Tahmini Süre:2m 0s

Soru 170Soru →

In a reliability study of a power station, Event $A$ is defined as an inverter system malfunction and Event $B$ is defined as a battery backup failure during peak hours. The probability that at least one of these two malfunctions occurs is $0.80$ , and the probability that an inverter system malfunction occurs without a battery backup failure is $0.35$ . If Event $A$ and Event $B$ are independent events, what is the probability of an inverter system malfunction, $P(A)$ ?

\frac{7}{20}

\frac{9}{20}

\frac{7}{11}

\frac{11}{20}

\frac{4}{5}

Cevabı ve açıklamayı göster

Cevap: $\frac{7}{11}$

Cevap

The probability of an inverter system malfunction,

P(A)

, is

\frac{7}{11}

.

The probability of the union of two events can be decomposed as

P(A \cup B) = P(A \cap B^c) + P(B)

. Given

P(A \cup B) = 0.80

and

P(A \cap B^c) = 0.35

, solving yields

P(B) = 0.45

. Because Event

A

and Event

B

are independent,

P(A \cap B) = P(A)P(B) = 0.45 P(A)

. Since

P(A \cap B^c) = P(A) - P(A \cap B)

, we substitute to get

0.35 = P(A) - 0.45 P(A) = 0.55 P(A)

. Solving for

P(A)

gives

P(A) = \frac{0.35}{0.55} = \frac{7}{11}

.

Adım Adım Çözüm

1

Relate the union probability to the difference event

A \cap B^c

and event

B

.

P(A \cup B) = P(A \cap B^c) + P(B)

.

The union of two events

A

and

B

can be partitioned into the region where only

A

occurs (

A \cap B^c

) and all outcomes in

B

.

2

Calculate

P(B)

using the given values.

P(B) = P(A \cup B) - P(A \cap B^c) = 0.80 - 0.35 = 0.45 = \frac{9}{20}

.

Subtracting

P(A \cap B^c)

from

P(A \cup B)

yields

P(B)

.

3

Apply the definition of independent events to express

P(A \cap B)

in terms of

P(A)

.

P(A \cap B) = P(A) \cdot P(B) = 0.45 P(A)

.

For independent events, the joint probability is the product of their individual probabilities.

4

Set up and solve the equation for

P(A)

.

P(A \cap B^c) = P(A) - P(A \cap B) \implies 0.35 = P(A) - 0.45 P(A) = 0.55 P(A) \implies P(A) = \frac{0.35}{0.55} = \frac{7}{11}

.

Dividing

0.35

by

0.55

yields the exact value of

P(A)

.

Anahtar Kavram

Probability rules for independent events and set operations on sample spaces

Tahmini Süre:2m 0s

Soru 171Soru →

While the ethnomusicologist's fieldwork in the remote highlands was initially dismissed by academics as mere amateur collection, her subsequent analytical monographs proved remarkably __________, uncovering structural patterns in folk melodies that fundamentally transformed scholars' understanding of oral traditions. Which two of the following answer choices, when inserted into the blank, produce completed sentences that are alike in meaning?

Geçerli olan tümünü seçin

trenchant

incisive

meticulous

pugnacious

belligerent

harmonious

Cevabı ve açıklamayı göster

Cevap: trenchant; incisive

Cevap

The words 'trenchant' and 'incisive' both complete the sentence to convey that the monographs were exceptionally insightful and penetrating in their analysis.

Both 'trenchant' and 'incisive' signify sharp, penetrating intellectual insight. Inserting either word emphasizes how the monographs provided keen analysis that overturned previous academic dismissal.

Adım Adım Çözüm

1

Analyze structural signals and context clues in the stem

The contrast signal 'While... initially dismissed' indicates that the subsequent monographs were not superficial or amateurish, but rather highly effective and revealing ('uncovering structural patterns... fundamentally transformed').

Establishing sentence polarity ensures the selected words match the required positive, analytical tone.

2

Identify the target meaning required for the blank

The blank requires an adjective meaning insightful, sharp, penetrating, or highly analytical.

Defining the required contextual meaning allows for targeted option filtering.

3

Evaluate choices for contextual fit and synonymy

'Trenchant' and 'incisive' both mean sharp, clear-cut, and intellectually penetrating. Using either word creates sentences with equivalent meaning.

Sentence Equivalence requires selecting two words that produce identical overall sentence meaning.

Anahtar Kavram

Contextual Synonym Pair Selection in Sentence Equivalence

Soru 172Soru →

The behavioral economist observed that the new financial incentives did not merely encourage short-term compliance; furthermore, by altering long-term risk perception, the policy ________ a profound shift in household savings strategies.

Which of the following words, when inserted into the blank, produce sentences that are equivalent in meaning? Select exactly two choices.

Geçerli olan tümünü seçin

induced

engendered

stifled

curbed

ameliorated

validated

Cevabı ve açıklamayı göster

Cevap: induced; engendered

Cevap

The two choices that produce equivalent, logically coherent sentences are 'induced' and 'engendered'.

The sentence relies on the additive continuation transition 'furthermore' and the causal clause 'by altering long-term risk perception'. The initial clause states that the policy encouraged compliance; the second clause expands on this by stating that the policy also brought about a shift in savings strategies. Both 'induced' and 'engendered' mean to cause, produce, or give rise to, making them exact functional synonyms in this context.

Adım Adım Çözüm

1

Analyze the sentence structure and identify transition signals.

The clause opens with 'did not merely encourage...', followed by the continuation signal 'furthermore' and a causal modifier 'by altering long-term risk perception'.

Structural signals dictate whether the blank requires a word that reinforces, elaborates on, or reverses the direction of the preceding clause.

2

Determine the required meaning and tone for the blank.

The blank must mean 'brought about', 'caused', or 'produced' to show how altering risk perception resulted in a profound shift.

The transition 'furthermore' combined with the causal preposition 'by' indicates that the policy created or generated a secondary, broader outcome.

3

Evaluate the choices to find a pair of near-synonyms that both fit the contextual requirement.

'induced' and 'engendered' both mean to bring about or give rise to a condition/action, yielding identical sentence meanings.

Sentence Equivalence requires selecting two distinct options that both fit the syntax and produce logically identical sentences.

Anahtar Kavram

Interpreting Continuation and Causal Transitions in Sentence Equivalence

Soru 173Soru →

Passage:
In 1898, Dutch microbiologist Martinus Beijerinck published findings demonstrating that the agent responsible for tobacco mosaic disease was fundamentally distinct from bacteria. Unlike bacterial pathogens, which could be removed from liquid suspension by filtration through porcelain Chamberland filters, the mosaic agent passed unimpeded through the fine pores. Furthermore, Beijerinck observed that while the agent failed to multiply in nutrient broth or cell-free culture media, it proliferated readily when introduced to living, actively growing plant tissues. To describe this novel entity, he coined the term 'contagium vivum fluidum' (contagious living fluid). Although Beijerinck initially hypothesized that the agent was liquid in nature rather than particulate, his crucial insight lay in establishing its absolute dependence on host cellular machinery for replication—a foundational distinction that separated viruses from known cellular microorganisms.

According to the passage, which of the following statements regarding Martinus Beijerinck's research on the tobacco mosaic agent are explicitly supported?

Select all that apply.

Geçerli olan tümünü seçin

The tobacco mosaic agent was capable of passing through porcelain Chamberland filters.

The mosaic agent multiplied efficiently in nutrient broth solutions without host tissue.

The mosaic agent required living, actively growing host tissue in order to replicate.

Cevabı ve açıklamayı göster

Cevap: The tobacco mosaic agent was capable of passing through porcelain Chamberland filters.; The mosaic agent required living, actively growing host tissue in order to replicate.

Cevap

The statements establishing that the mosaic agent passed through porcelain Chamberland filters and that it required living, actively growing host tissue to replicate are both explicitly supported by the passage.

The passage directly supports two statements: first, that the mosaic agent passed unimpeded through the fine pores of porcelain Chamberland filters; and second, that it proliferated in living plant tissues and depended absolutely on host cellular machinery to replicate.

Adım Adım Çözüm

1

Scan the passage for keywords related to filtration and Chamberland filters.

Located sentence: 'Unlike bacterial pathogens... the mosaic agent passed unimpeded through the fine pores.'

Verifies whether the statement regarding passage through porcelain filters is explicitly supported.

2

Scan the passage for references to nutrient broth and cell-free culture media.

Located sentence: 'Beijerinck observed that while the agent failed to multiply in nutrient broth...'

Evaluates whether the agent could replicate in broth solutions without host tissue.

3

Scan the passage for references to living host tissue requirements.

Located sentences detailing that the agent 'proliferated readily when introduced to living, actively growing plant tissues' and had an 'absolute dependence on host cellular machinery for replication.'

Confirms whether host tissue was required for replication.

Anahtar Kavram

Explicit Detail Retrieval

Soru 174Soru →

If

x

is a positive integer such that

\frac{\sqrt{54 \cdot 4^{2x+1} + 9 \cdot 4^{2x}}}{\sqrt{2^{x+3} + 2^x}} = 2560

what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

x

is

6

.

Factoring out common exponential terms inside both radicals yields

\sqrt{225 \cdot 2^{4x}} = 15 \cdot 2^{2x}

for the numerator and

\sqrt{9 \cdot 2^x} = 3 \cdot 2^{x/2}

for the denominator. Dividing these gives

5 \cdot 2^{3x/2}

. Equating this to

2560

results in

2^{3x/2} = 512 = 2^9

, which simplifies to

\frac{3x}{2} = 9

, or

x = 6

.

Adım Adım Çözüm

1

Simplify the numerator inside the radical expression.

\sqrt{54 \cdot 4^{2x+1} + 9 \cdot 4^{2x}} = 15 \cdot 2^{2x}

Rewrite

4^{2x+1}

as

4 \cdot 4^{2x}

. Then factor out

4^{2x}

:

54(4 \cdot 4^{2x}) + 9 \cdot 4^{2x} = (216 + 9)4^{2x} = 225 \cdot 4^{2x}

. Taking the square root gives

\sqrt{225} \cdot \sqrt{(2^2)^{2x}} = 15 \cdot 2^{2x}

.

2

Simplify the denominator inside the radical expression.

\sqrt{2^{x+3} + 2^x} = 3 \cdot 2^{x/2}

Rewrite

2^{x+3}

as

2^3 \cdot 2^x = 8 \cdot 2^x

. Factoring out

2^x

gives

(8 + 1)2^x = 9 \cdot 2^x

. Taking the square root gives

\sqrt{9} \cdot \sqrt{2^x} = 3 \cdot 2^{x/2}

.

3

Simplify the quotient of the two radical expressions.

\frac{15 \cdot 2^{2x}}{3 \cdot 2^{x/2}} = 5 \cdot 2^{3x/2}

Divide the constants

\frac{15}{3} = 5

and subtract exponents with the same base:

2x - \frac{x}{2} = \frac{3x}{2}

.

4

Equate to 2560 and solve for

x

.

x = 6

Divide both sides by 5:

2^{3x/2} = \frac{2560}{5} = 512

. Express 512 as a power of 2:

512 = 2^9

. Therefore,

\frac{3x}{2} = 9 \implies 3x = 18 \implies x = 6

.

Anahtar Kavram

Exponent and Radical Simplification using Base Prime Factorization

Tahmini Süre:2m 0s

Soru 175Soru →

On the real number line, the set of all real numbers $x$ that satisfy the inequality $|3x - 7| \le 11$ forms a closed interval $[a, b]$ . What is the value of $|a + b|$ ?

\frac{14}{3}

\frac{22}{3}

6

\frac{10}{3}

\frac{26}{3}

Cevabı ve açıklamayı göster

Cevap: $\frac{14}{3}$

Cevap

The value of

|a + b|

is

\frac{14}{3}

.

Rewriting the inequality

|3x - 7| \le 11

as

-11 \le 3x - 7 \le 11

and solving yields

-\frac{4}{3} \le x \le 6

. Thus, the endpoints are

a = -\frac{4}{3}

and

b = 6

. Summing these values gives

a + b = \frac{14}{3}

, and taking the absolute value yields

\frac{14}{3}

.

Adım Adım Çözüm

1

Express the absolute value inequality as a compound inequality.

-11 \le 3x - 7 \le 11

An inequality of the form

|u| \le k

for

k \ge 0

is equivalent to

-k \le u \le k

.

2

Isolate

3x

by adding

7

to all parts of the inequality.

-11 + 7 \le 3x \le 11 + 7 \implies -4 \le 3x \le 18

Adding a constant to all parts preserves the direction of the inequality.

3

Divide all parts by

3

to solve for

x

.

-\frac{4}{3} \le x \le 6

Dividing by a positive number isolates

x

without flipping inequality signs.

4

Identify interval bounds

a

and

b

, then calculate

|a + b|

.

a = -\frac{4}{3}

,

b = 6 \implies a + b = -\frac{4}{3} + \frac{18}{3} = \frac{14}{3} \implies \left|\frac{14}{3}\right| = \frac{14}{3}

The question asks for the absolute value of the sum of the endpoints of interval

[a, b]

.

Anahtar Kavram

Absolute value inequalities on the number line and interval endpoints

Soru 176Soru →

A quality control inspector evaluates a batch of $16$ precision components. Exactly $10$ of the components meet all engineering specifications, while $6$ have minor surface defects. If the inspector randomly selects $2$ components from the batch one after another without replacement, what is the probability that both selected components meet all engineering specifications?

Cevabı ve açıklamayı göster

Cevap: 0.375

Cevap

The probability that both selected components meet all engineering specifications is

0.375

(or

\frac{3}{8}

).

Since the components are selected without replacement, the outcome of the second draw depends on the outcome of the first draw. The probability of selecting a qualifying component first is

\frac{10}{16}

. Given that a qualifying component was drawn first,

9

qualifying components remain out of

15

total components. The joint probability of both events occurring is

\frac{10}{16} \times \frac{9}{15} = \frac{90}{240} = \frac{3}{8} = 0.375

.

Adım Adım Çözüm

1

Determine the probability of selecting a component meeting specifications on the first draw.

P(E_1) = \frac{10}{16} = \frac{5}{8}

There are

10

qualifying components out of

16

total components.

2

Determine the conditional probability of selecting a second component meeting specifications, given that the first component selected also met specifications.

P(E_2 \mid E_1) = \frac{9}{15} = \frac{3}{5}

Because sampling is done without replacement,

9

qualifying components remain out of

15

total remaining components.

3

Apply the multiplication rule for dependent events to calculate the probability of both events occurring.

P(E_1 \cap E_2) = P(E_1) \times P(E_2 \mid E_1) = \frac{5}{8} \times \frac{3}{5} = \frac{3}{8} = 0.375

For dependent events, the joint probability is the product of the first event's probability and the conditional probability of the second event.

Anahtar Kavram

Probability of Dependent Events without Replacement

Tahmini Süre:1m 15s

Soru 177Soru →

If

k

is a positive integer, which of the following expressions is equivalent to

\frac{2^{k+2} \cdot 3^{2k-1} + 6^k \cdot 3^{k-1}}{12^k \cdot 2^{-1}}

for all values of

k

?

\frac{5}{3} \left(\frac{3}{2}\right)^k

\frac{10}{3} \left(\frac{3}{2}\right)^k

\frac{5}{3} \cdot 6^k

5 \left(\frac{3}{2}\right)^k

\frac{10}{3} \cdot 3^k

Cevabı ve açıklamayı göster

Cevap: $\frac{10}{3} \left(\frac{3}{2}\right)^k$

Cevap

The expression simplifies to

\frac{10}{3} \left(\frac{3}{2}\right)^k

.

Factoring the terms in the numerator into base

18^k

gives

\frac{4}{3} \cdot 18^k + \frac{1}{3} \cdot 18^k = \frac{5}{3} \cdot 18^k

. The denominator equals

\frac{1}{2} \cdot 12^k

. Dividing numerator by denominator yields

\frac{5/3}{1/2} \cdot \left(\frac{18}{12}\right)^k = \frac{10}{3} \left(\frac{3}{2}\right)^k

.

Adım Adım Çözüm

1

Rewrite each term in the numerator using prime base factorization

2^{k+2} \cdot 3^{2k-1} = 2^k \cdot 2^2 \cdot (3^2)^k \cdot 3^{-1} = 4 \cdot 2^k \cdot 9^k \cdot \frac{1}{3} = \frac{4}{3} \cdot 18^k

, and

6^k \cdot 3^{k-1} = (2 \cdot 3)^k \cdot 3^k \cdot 3^{-1} = 2^k \cdot 9^k \cdot \frac{1}{3} = \frac{1}{3} \cdot 18^k

.

Converting all powers to base 18 allows terms with identical exponential factors to be combined.

2

Combine the terms in the numerator

\frac{4}{3} \cdot 18^k + \frac{1}{3} \cdot 18^k = \left(\frac{4}{3} + \frac{1}{3}\right) \cdot 18^k = \frac{5}{3} \cdot 18^k

.

Adding coefficients of like exponential terms.

3

Simplify the denominator expression

12^k \cdot 2^{-1} = \frac{1}{2} \cdot 12^k

.

Applying the negative exponent rule

a^{-n} = \frac{1}{a^n}

.

4

Divide the numerator by the denominator

\frac{\frac{5}{3} \cdot 18^k}{\frac{1}{2} \cdot 12^k} = \frac{5/3}{1/2} \cdot \left(\frac{18}{12}\right)^k = \left(\frac{5}{3} \cdot 2\right) \cdot \left(\frac{3}{2}\right)^k = \frac{10}{3} \left(\frac{3}{2}\right)^k

Dividing fractions by multiplying by the reciprocal and applying quotient rule for powers with the same exponent.

Anahtar Kavram

Prime base factorization and laws of exponents

Soru 178Soru →

An index value $I_0 = 1{}000$ increases over a 6-month period. For each month $n$ from 1 to 6, the index value $I_n$ is calculated by increasing the previous month's value $I_{n-1}$ by $5\%$ and then rounding the result to the nearest integer (with half-integers rounded up). Let $U_6 = 1{}000 \times (1.05)^6$ represent the exact unrounded compounded value at month 6.

Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

I_2 = 1{}103

The exact unrounded value

U_6 = 1{}000 \times (1.05)^6

is strictly greater than the rounded index value

I_6

.

The sequence of monthly increments

I_n - I_{n-1}

for

n = 1, 2, \dots, 6

is strictly increasing.

The sum of the first two monthly fractional growth rates,

\frac{I_1 - I_0}{I_0} + \frac{I_2 - I_1}{I_1}

, is equal to

\frac{I_2 - I_0}{I_0 + I_1}

.

The exponential growth factor

(1 + 0.05)^6

is equivalent to

1^6 + (0.05)^6

.

Cevabı ve açıklamayı göster

Cevap: $I_2 = 1{}103$ ; The sequence of monthly increments $I_n - I_{n-1}$ for $n = 1, 2, \dots, 6$ is strictly increasing.

Cevap

The statement specifying that

I_2 = 1{}103

and the statement asserting that the sequence of monthly increments

I_n - I_{n-1}

is strictly increasing are both correct.

The statement giving

I_2 = 1{}103

is correct because

1{}050 \times 1.05 = 1{}102.5

, which rounds up to

1{}103

. The statement regarding the sequence of monthly increments is correct because the increments

50, 53, 55, 58, 61, 64

strictly increase.

Adım Adım Çözüm

1

Calculate each term of the sequence

I_n

by applying a 5% increase and rounding to the nearest integer.

I_0 = 1{}000

;

I_1 = \text{round}(1{}000 \times 1.05) = 1{}050

;

I_2 = \text{round}(1{}050 \times 1.05) = \text{round}(1{}102.5) = 1{}103

;

I_3 = \text{round}(1{}103 \times 1.05) = \text{round}(1{}158.15) = 1{}158

;

I_4 = \text{round}(1{}158 \times 1.05) = \text{round}(1{}215.9) = 1{}216

;

I_5 = \text{round}(1{}216 \times 1.05) = \text{round}(1{}276.8) = 1{}277

;

I_6 = \text{round}(1{}277 \times 1.05) = \text{round}(1{}340.85) = 1{}341

.

This establishes the exact sequence of rounded monthly values.

2

Evaluate the statement that

I_2 = 1{}103

.

From Step 1,

I_2 = 1{}103

.

This directly confirms the validity of the first statement.

3

Compute the sequence of monthly increments

I_n - I_{n-1}

for

n = 1, 2, \dots, 6

.

Increments:

I_1 - I_0 = 50

,

I_2 - I_1 = 53

,

I_3 - I_2 = 55

,

I_4 - I_3 = 58

,

I_5 - I_4 = 61

,

I_6 - I_5 = 64

.

Since

50 < 53 < 55 < 58 < 61 < 64

, the sequence of increments is strictly increasing.

4

Compare

I_6

with

U_6 = 1{}000 \times (1.05)^6

.

U_6 = 1{}000 \times 1.3400956... \approx 1{}340.10

. Since

I_6 = 1{}341

,

I_6 > U_6

.

The statement claiming

U_6 > I_6

is false.

Anahtar Kavram

Error propagation in recursive sequence rounding and non-distributivity of exponents over sums

Tahmini Süre:2m 30s

Soru 179Soru →

If $x$ is a negative real number such that $\sqrt{(-x)^3 \cdot x^2} = 32$ , what is the value of $x$ ?

-16

-4

-2

2

4

Cevabı ve açıklamayı göster

Cevap: $-4$

Cevap

-4

Simplifying the expression inside the radical gives

(-x)^3 \cdot x^2 = -x^5

. Setting

\sqrt{-x^5} = 32

and squaring both sides gives

-x^5 = 32^2 = 1024

, which means

x^5 = -1024

. The fifth root of

-1024

is

-4

. Since

-4

is a negative real number, it satisfies all conditions of the problem.

Adım Adım Çözüm

1

Simplify the expression under the square root

Since

(-x)^3 = -x^3

, we have

(-x)^3 \cdot x^2 = (-x^3) \cdot x^2 = -x^5

.

Applying exponent addition rules

x^a \cdot x^b = x^{a+b}

and odd power rules for negative quantities.

2

Square both sides of the equation to eliminate the square root

-x^5 = 32^2 = 1024

.

Squaring both sides of

\sqrt{-x^5} = 32

isolates the radicand.

3

Solve for x

x^5 = -1024 \implies x = \sqrt[5]{-1024} = -4

.

Taking the 5th root of

-1024 = (-2)^{10} = (-4)^5

yields

x = -4

, which satisfies the given condition

x < 0

.

Anahtar Kavram

Simplifying expressions with powers and square roots involving negative variables.

Tahmini Süre:1m 15s

Soru 180Soru →

For all real numbers $x$ and $y$ such that $x \neq y$ , $x \neq -y$ , and $x^2 + y^2 \neq 0$ , consider the algebraic expression:

E(x, y) = \frac{x^4 - y^4}{x^3 - x^2y + xy^2 - y^3}

Which of the following expressions are equivalent to $E(x, y)$ for all valid values of $x$ and $y$ ? Select all such expressions.

Geçerli olan tümünü seçin

\frac{x^2 - y^2}{x - y}

\frac{x^3 + y^3}{x^2 - xy + y^2}

\frac{x^2 + y^2}{x + y}

\frac{x^3 - y^3}{x^2 + xy + y^2}

\frac{x^2 - y^2}{x + y}

Cevabı ve açıklamayı göster

Cevap: $\frac{x^2 - y^2}{x - y}$ ; $\frac{x^3 + y^3}{x^2 - xy + y^2}$

Cevap

The expressions equivalent to

E(x, y)

are

\frac{x^2 - y^2}{x - y}

and

\frac{x^3 + y^3}{x^2 - xy + y^2}

.

Simplifying

E(x, y)

by factoring both numerator and denominator yields

x + y

. The option with

\frac{x^2 - y^2}{x - y}

simplifies directly to

x + y

by canceling

(x - y)

. The option with

\frac{x^3 + y^3}{x^2 - xy + y^2}

uses the sum of cubes identity to factor the numerator into

(x + y)(x^2 - xy + y^2)

, which also cancels down to

x + y

. Both of these options are mathematically identical to

E(x, y)

.

Adım Adım Çözüm

1

Factor the numerator of

E(x, y)

using the difference of squares identity twice.

x^4 - y^4 = (x^2 - y^2)(x^2 + y^2) = (x - y)(x + y)(x^2 + y^2)

Decomposing higher-power binomials into linear and quadratic factors enables algebraic simplification.

2

Factor the denominator of

E(x, y)

by grouping terms.

x^3 - x^2y + xy^2 - y^3 = x^2(x - y) + y^2(x - y) = (x - y)(x^2 + y^2)

Grouping adjacent terms with shared factors allows factoring out

(x - y)

.

3

Simplify the full expression

E(x, y)

by canceling common non-zero factors.

E(x, y) = \frac{(x - y)(x + y)(x^2 + y^2)}{(x - y)(x^2 + y^2)} = x + y

Since

x \neq y

and

x^2 + y^2 \neq 0

, the factors

(x - y)

and

(x^2 + y^2)

cancel completely.

4

Evaluate the given options to determine which simplify to

x + y

.

\frac{x^2 - y^2}{x - y} = x + y

and

\frac{x^3 + y^3}{x^2 - xy + y^2} = x + y

, whereas

\frac{x^3 - y^3}{x^2 + xy + y^2} = x - y

and

\frac{x^2 - y^2}{x + y} = x - y

.

Matching each option's fully simplified form to

x + y

identifies all valid equivalent expressions.

Anahtar Kavram

Simplifying rational expressions by polynomial factoring (grouping, difference of squares, and sum of cubes).

Cevap

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Sık Yapılan Hatalar

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