All practice questions

Question 1721Question →

An aerospace engineering team evaluated telemetry logs from a constellation of $150$ satellites to monitor three types of sensor anomalies: Power Fluctuation ( $P$ ), Thermal Spike ( $T$ ), and Signal Attenuation ( $S$ ). The evaluation revealed the following:

- $65$ satellites exhibited Power Fluctuation.
- $58$ satellites exhibited Thermal Spike.
- $42$ satellites exhibited Signal Attenuation.
- $22$ satellites exhibited both Power Fluctuation and Thermal Spike.
- $18$ satellites exhibited both Thermal Spike and Signal Attenuation.
- $15$ satellites exhibited both Power Fluctuation and Signal Attenuation.
- $8$ satellites exhibited all three anomalies.

How many satellites exhibited exactly one of these three anomalies?

55

79

110

118

125

Show answer & explanation

Answer: 79

Answer

79 satellites exhibited exactly one anomaly.

To find the number of satellites exhibiting exactly one anomaly, break down each set into its mutually exclusive regions. First, find the satellites in exactly two sets by subtracting the triple intersection (8) from each pairwise intersection: Power and Thermal only = 22 - 8 = 14; Thermal and Signal only = 18 - 8 = 10; Power and Signal only = 15 - 8 = 7. Next, subtract all overlapping regions from each individual set total: Power only = 65 - (14 + 7 + 8) = 36; Thermal only = 58 - (14 + 10 + 8) = 26; Signal only = 42 - (7 + 10 + 8) = 17. Adding these single-anomaly counts together gives 36 + 26 + 17 = 79.

Step-by-Step Solution

1

Determine the number of satellites exhibiting ONLY pairwise anomalies (exactly two anomalies).

P and T only = 22 - 8 = 14; T and S only = 18 - 8 = 10; P and S only = 15 - 8 = 7.

The given pairwise intersection values include the 8 satellites that exhibited all three anomalies.

2

Calculate the number of satellites exhibiting exactly one anomaly for each set.

P only = 65 - (14 + 7 + 8) = 36; T only = 58 - (14 + 10 + 8) = 26; S only = 42 - (7 + 10 + 8) = 17.

Subtracting all elements that belong to two or three sets from each total set size leaves only those in the single set.

3

Sum the single-anomaly totals.

36 + 26 + 17 = 79.

The regions corresponding to exactly one anomaly are mutually exclusive.

Key Concept

Principle of Inclusion-Exclusion and Venn Diagram Region Decomposition

Question 1722Question →

A software security audit evaluated $500$ open-source repositories for three specific vulnerability types: SQL Injection ( $S$ ), Cross-Site Scripting ( $X$ ), and Buffer Overflow ( $B$ ). Exactly $80$ repositories had none of these vulnerabilities. The audit revealed the following data:

- $220$ repositories contained SQL Injection vulnerabilities.
- $190$ repositories contained Cross-Site Scripting vulnerabilities.
- $210$ repositories contained Buffer Overflow vulnerabilities.
- $75$ repositories contained both SQL Injection and Cross-Site Scripting vulnerabilities.
- $80$ repositories contained both Cross-Site Scripting and Buffer Overflow vulnerabilities.
- $30$ repositories contained all three vulnerability types.

How many of the audited repositories contained Buffer Overflow vulnerabilities ONLY?

55

85

100

130

155

Show answer & explanation

Answer: $85$

Answer

85 repositories contained Buffer Overflow vulnerabilities only.

The total number of repositories containing at least one vulnerability is

500 - 80 = 420

. Using the inclusion-exclusion formula

|S \cup X \cup B| = |S| + |X| + |B| - (|S \cap X| + |X \cap B| + |S \cap B|) + |S \cap X \cap B|

, we substitute the known values:

420 = 220 + 190 + 210 - (75 + 80 + |S \cap B|) + 30

. Solving gives

420 = 650 - 155 - |S \cap B|

, so

|S \cap B| = 75

. The repositories containing Buffer Overflow ONLY are given by subtracting the overlapping regions from the total Buffer Overflow set:

|B| - (|S \cap B| \text{ only}) - (|X \cap B| \text{ only}) - |S \cap X \cap B| = 210 - (75 - 30) - (80 - 30) - 30 = 210 - 45 - 50 - 30 = 85

.

Step-by-Step Solution

1

Calculate the total number of repositories containing at least one vulnerability

|S \cup X \cup B| = 500 - 80 = 420

The total population is 500 and 80 repositories have no vulnerabilities.

2

Apply the 3-Set Inclusion-Exclusion Principle to determine the sum of pairwise intersections

|S \cap X| + |X \cap B| + |S \cap B| = 230

From

|S \cup X \cup B| = |S| + |X| + |B| - (|S \cap X| + |X \cap B| + |S \cap B|) + |S \cap X \cap B|

, we get

420 = 220 + 190 + 210 - \sum |\text{intersections}| + 30

, so

\sum |\text{intersections}| = 650 - 420 = 230

.

3

Find the missing pairwise intersection

|S \cap B|

|S \cap B| = 75

Since

|S \cap X| = 75

and

|X \cap B| = 80

, we have

75 + 80 + |S \cap B| = 230

, which yields

|S \cap B| = 75

.

4

Calculate the number of repositories with Buffer Overflow ONLY

|B \text{ only}| = 85

Subtract the exclusive two-set overlaps and the three-set overlap from

|B|

:

|B \text{ only}| = |B| - (|S \cap B| - |S \cap X \cap B|) - (|X \cap B| - |S \cap X \cap B|) - |S \cap X \cap B| = 210 - (75 - 30) - (80 - 30) - 30 = 210 - 45 - 50 - 30 = 85

.

Key Concept

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Analysis

Question 1723Question →

A quality control manager at a pharmaceutical manufacturing facility evaluated the disintegration time, $t$ (in seconds), for a batch of $400$ coated tablets. The results are summarized in the frequency distribution table below:

Disintegration Time $t$ (seconds)	Frequency
$10 \le t < 20$	$40$
$20 \le t < 30$	$110$
$30 \le t < 40$	$160$
$40 \le t < 50$	$60$
$50 \le t < 60$	$30$

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

Select all that apply

The estimated mean disintegration time of the batch, calculated using class midpoints, is

33.25

seconds.

The median disintegration time of the batch lies within the interval

30 \le t < 40

.

Exactly

40\%

of the tablets in the batch have a disintegration time of less than

30

seconds.

The ratio of the number of tablets with a disintegration time of at least

30

seconds to the number of tablets with a disintegration time of less than

30

seconds is

5

to

3

.

More than

25\%

of the tablets in the batch have a disintegration time of

50

seconds or greater.

Show answer & explanation

Answer: The estimated mean disintegration time of the batch, calculated using class midpoints, is $33.25$ seconds.; The median disintegration time of the batch lies within the interval $30 \le t < 40$ .; The ratio of the number of tablets with a disintegration time of at least $30$ seconds to the number of tablets with a disintegration time of less than $30$ seconds is $5$ to $3$ .

Answer

The statements asserting that the estimated mean is

33.25

seconds, that the median lies in the interval

30 \le t < 40

, and that the ratio of tablets taking at least

30

seconds to those taking less than

30

seconds is

5

to

3

are all correct.

The estimated mean of

33.25

seconds is correctly computed from class midpoints

(13,300 / 400)

. The median interval

30 \le t < 40

correctly encompasses the middle values (

200^{\text{th}}

and

201^{\text{st}}

observations out of

400

). The part-to-part ratio of tablets with disintegration time

\ge 30

seconds (

250

) to

< 30

seconds (

150

) simplifies to

250:150 = 5:3

.

Step-by-Step Solution

1

Calculate the estimated mean using class midpoints.

Midpoints are

15, 25, 35, 45, 55

. Total product sum

\sum f \cdot m = (40 \times 15) + (110 \times 25) + (160 \times 35) + (60 \times 45) + (30 \times 55) = 13,300

. Mean

= 13,300 / 400 = 33.25

seconds.

The mean of grouped data is estimated by taking the weighted sum of interval midpoints divided by total sample size.

2

Determine the interval containing the sample median.

Cumulative frequencies:

10 \le t < 20 \rightarrow 40

;

20 \le t < 30 \rightarrow 150

;

30 \le t < 40 \rightarrow 310

. The

200^{\text{th}}

and

201^{\text{st}}

data points fall between cumulative counts

150

and

310

, putting the median in

30 \le t < 40

.

The median corresponds to the middle position (

N/2 = 200

) of ordered data.

3

Evaluate the percentage of tablets with disintegration time less than

30

seconds.

Count

= 40 + 110 = 150

. Percentage

= (150 / 400) \times 100\% = 37.5\%

.

Part-to-whole percentage requires dividing the sum of frequencies below

30

seconds by the total sample size

400

.

4

Compute the part-to-part ratio of tablets with

t \ge 30

seconds versus

t < 30

seconds.

Tablets with

t \ge 30

:

160 + 60 + 30 = 250

. Tablets with

t < 30

:

40 + 110 = 150

. Ratio

= 250 : 150 = 5 : 3

.

Formulating the part-to-part ratio comparing the upper three intervals to the lower two intervals.

5

Evaluate the percentage of tablets with disintegration time

t \ge 50

seconds.

Count

= 30

. Percentage

= (30 / 400) \times 100\% = 7.5\%

.

Checking if

7.5\%

exceeds

25\%

shows the statement is false.

Key Concept

Grouped Data Analysis: Mean Estimation, Median Interval Identification, and Relative Frequency Calculations

Question 1724Question →

An investment firm allocated a total principal of $\$120,000$ between two accounts, Account X and Account Y. Account X earned a simple annual interest rate of $6\%$ , and Account Y earned a simple annual interest rate of $10\%$ . At the end of one year, the combined total interest earned from both accounts was $\$9,200$ . An annual administrative fee was then deducted from the interest: a $25\%$ fee on the interest earned from Account X, and a $10\%$ fee on the interest earned from Account Y. What was the net amount of interest, in dollars, remaining after these administrative fees were deducted?

\$7,530

\$7,650

\$7,750

\$8,775

\$10,750

Show answer & explanation

Answer: $\$7,650$

Answer

The net amount of interest remaining after administrative fees were deducted is

\$7,650

.

The correct answer is

\$7,650

. Solving the system

x + y = 120,000

and

0.06x + 0.10y = 9,200

gives

\$70,000

invested in Account X and

\$50,000

in Account Y. Account X produces

\$4,200

in interest, from which a

25\%

fee (

\$1,050

) is deducted. Account Y produces

\$5,000

in interest, from which a

10\%

fee (

\$500

) is deducted. Subtracting total fees of

\$1,550

from total interest of

\$9,200

yields

\$7,650

.

Step-by-Step Solution

1

Set up a system of linear equations for the principal amounts invested in Account X and Account Y.

Let

x

be the amount invested in Account X and

y

be the amount invested in Account Y. The total principal is

x + y = 120,000

. The total interest earned is

0.06x + 0.10y = 9,200

.

Modeling the word problem using two variables captures both total principal and total combined interest.

2

Solve the system of equations for

x

and

y

.

Multiplying

x + y = 120,000

by

0.06

gives

0.06x + 0.06y = 7,200

. Subtracting this equation from

0.06x + 0.10y = 9,200

yields

0.04y = 2,000

, so

y = 50,000

. Substituting back gives

x = 70,000

.

Determining individual account principals allows calculation of interest generated by each account.

3

Calculate the interest earned from each account and the corresponding administrative fees.

Interest from X =

0.06 \times 70,000 = \$4,200

. Interest from Y =

0.10 \times 50,000 = \$5,000

. Fee for X =

0.25 \times 4,200 = \$1,050

. Fee for Y =

0.10 \times 5,000 = \$500

. Total fees =

1,050 + 500 = \$1,550

.

Each fee rate must be multiplied by the specific interest amount earned by that account.

4

Subtract the total administrative fees from the total interest earned.

Net Interest =

\$9,200 - \$1,550 = \$7,650

.

Deducting fees yields the net interest remaining.

Key Concept

System of Linear Equations for Mixture and Interest Word Problems

Estimated Time:2m 0s

Question 1725Question →

A machine operates using two independent components, Component $A$ and Component $B$ . The probability that Component $A$ functions properly on a given day is $0.90$ , and the probability that Component $B$ functions properly on that same day is $0.80$ . What is the probability that at least one of the components functions properly on a given day?

Show answer & explanation

Answer: 0.98

Answer

The probability that at least one component functions properly is

0.98

.

To determine the probability that at least one component functions properly, use the complement rule:

P(\text{at least one}) = 1 - P(\text{neither})

. Since Component

A

and Component

B

operate independently, the probability that

A

fails is

1 - 0.90 = 0.10

and the probability that

B

fails is

1 - 0.80 = 0.20

. The probability of both components failing simultaneously is

0.10 \times 0.20 = 0.02

. Subtracting this probability from

1

gives

1 - 0.02 = 0.98

.

Step-by-Step Solution

1

Find the probability of failure for each component.

P(A^c) = 1 - 0.90 = 0.10

and

P(B^c) = 1 - 0.80 = 0.20

The event that a component fails is the complement of the event that it functions properly.

2

Calculate the joint probability of both components failing.

P(A^c \text{ and } B^c) = 0.10 \times 0.20 = 0.02

Because the components operate independently, their failure events are independent, so their individual probabilities are multiplied.

3

Calculate the probability that at least one component functions properly.

P(\text{at least one functions}) = 1 - 0.02 = 0.98

The event 'at least one component functions' is the exact complement of 'both components fail'.

Key Concept

Probability of Independent Events and Complement Rule

Estimated Time:45s

Question 1726Question →

A commercial print shop uses two high-speed printing presses, Press Alpha and Press Beta. Press Alpha operates at a constant rate of $120$ pages per minute, while Press Beta operates at a constant rate of $180$ pages per minute. Press Alpha begins printing a job of $15,000$ pages at 9:00 AM. At 9:15 AM, Press Beta is turned on to assist Press Alpha, and both presses continue printing simultaneously at their respective constant rates until the job is completed. How many total minutes after 9:00 AM will the entire $15,000$ -page job be finished?

Show answer & explanation

Answer: 59

Answer

The entire 15,000-page job will be finished 59 minutes after 9:00 AM.

Press Alpha operates alone for the first 15 minutes, completing

15 \times 120 = 1,800

pages. That leaves

15,000 - 1,800 = 13,200

pages. Once Press Beta joins at 9:15 AM, the combined rate becomes

120 + 180 = 300

pages per minute. The remaining pages require

13,200 / 300 = 44

minutes. Summing the 15-minute initial period and the 44-minute joint period gives a total of 59 minutes after 9:00 AM.

Step-by-Step Solution

1

Find the work completed by Press Alpha during the 15-minute staggered start period.

1,800 pages completed.

Press Alpha ran alone for 15 minutes at 120 pages per minute.

2

Determine the remaining work to be done after 9:15 AM.

13,200 pages remaining.

Subtract the completed pages from the total batch size of 15,000 pages.

3

Calculate the combined work rate of Press Alpha and Press Beta.

300 pages per minute.

When working together, rates add linearly: 120 + 180 = 300.

4

Calculate time needed to complete the remaining pages.

44 minutes.

Divide remaining work (13,200 pages) by combined rate (300 pages/min).

5

Calculate total elapsed time from 9:00 AM.

59 minutes.

Combine the 15 initial minutes with the 44 subsequent minutes.

Key Concept

Linear work-rate equations with staggered initial start times

Estimated Time:1m 30s

Question 1727Question →

A dataset $W$ consists of $10$ distinct positive integers. The arithmetic mean of the dataset is $24$ , and its median is $22$ . A new dataset $V$ is created by replacing every integer $x$ in $W$ that is strictly less than the median of $W$ with $2x + 1$ , while leaving all other integers in $W$ unchanged. If the sum of the $5$ smallest integers in $W$ is $60$ , what is the arithmetic mean of the integers in dataset $V$ ?

28.5

30.0

30.5

31.0

36.5

Show answer & explanation

Answer: 30.5

Answer

30.5

The total sum of the original dataset

W

is

10 \times 24 = 240

. Since

W

consists of

10

distinct integers, exactly

5

elements are strictly less than the median. The sum of these

5

elements is given as

60

, which leaves the sum of the remaining

5

elements as

240 - 60 = 180

. When each of the

5

lower elements

x

is replaced with

2x + 1

, their new sum becomes

2(60) + 5(1) = 125

. The sum of the new dataset

V

is

125 + 180 = 305

, and its mean is

305 / 10 = 30.5

.

Step-by-Step Solution

1

Calculate the total sum of the original dataset

W

.

Since dataset

W

has

10

elements with a mean of

24

, its total sum is

10 \times 24 = 240

.

The sum of elements in any set is equal to the number of elements multiplied by the arithmetic mean.

2

Determine how many elements are strictly less than the median of

W

.

Because all

10

elements in

W

are distinct positive integers, ordering them as

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9 < x_{10}

places the median between

x_5

and

x_6

. Thus, exactly

5

elements (

x_1

through

x_5

) are strictly less than the median.

In an even-sized set of distinct values, exactly half of the elements lie strictly below the median position.

3

Find the sum of the remaining

5

elements (the upper half) in

W

.

Given that the sum of the

5

smallest elements is

60

, the sum of the remaining

5

elements is

240 - 60 = 180

.

The total sum of the dataset is the sum of its lower

5

elements plus the sum of its upper

5

elements.

4

Calculate the new sum of the transformed

5

smallest elements in dataset

V

.

Each of the

5

elements

x_i

is replaced by

2x_i + 1

. The new sum is

\sum_{i=1}^5 (2x_i + 1) = 2 \sum_{i=1}^5 x_i + 5(1) = 2(60) + 5 = 125

.

Linear transformations applied to individual elements scale their sum by the multiplier and add the constant term multiplied by the number of elements.

5

Compute the total sum and arithmetic mean of dataset

V

.

The total sum of

V

is

125 + 180 = 305

. The new mean is

\frac{305}{10} = 30.5

.

The mean of

V

is its total sum divided by the number of elements (

10

).

Key Concept

Linear transformation of dataset subsets and median position in distinct ordered sets

Question 1728Question →

In the $xy$ -plane, the graph of the function $g(x)$ is obtained by taking the graph of $f(x) = (x + 2)^3 - 1$ , reflecting it across the $y$ -axis, translating it $4$ units to the right, and then translating it $3$ units upward. What is the $y$ -intercept of the graph of $y = g(x)$ ?

-214

-6

12

212

218

Show answer & explanation

Answer: $218$

Answer

The

y

-intercept of the graph of

y = g(x)

is

218

.

Applying the transformations step-by-step to

f(x) = (x + 2)^3 - 1

gives

g(x) = (-x + 6)^3 + 2

. Substituting

x = 0

yields

g(0) = 6^3 + 2 = 218

, making

218

the correct

y

-intercept.

Step-by-Step Solution

1

Reflect the function

f(x) = (x + 2)^3 - 1

across the

y

-axis.

Replacing

x

with

-x

yields

y_1 = f(-x) = (-x + 2)^3 - 1

.

Reflecting a graph across the

y

-axis corresponds to replacing

x

with

-x

in the function rule.

2

Translate the reflected graph

4

units to the right.

Replacing

x

with

x - 4

yields

y_2 = (-(x - 4) + 2)^3 - 1 = (-x + 4 + 2)^3 - 1 = (-x + 6)^3 - 1

.

Translating a graph

h

units to the right replaces

x

with

x - h

.

3

Translate the graph

3

units upward to form

g(x)

.

g(x) = (-x + 6)^3 - 1 + 3 = (-x + 6)^3 + 2

.

Translating a graph

k

units upward adds

k

to the expression.

4

Find the

y

-intercept by evaluating

g(0)

.

g(0) = (-0 + 6)^3 + 2 = 6^3 + 2 = 216 + 2 = 218

.

The

y

-intercept occurs where

x = 0

.

Key Concept

Function Transformations in Coordinate Geometry

Estimated Time:1m 30s

Question 1729Question →

The ratio of the measure of each interior angle of a regular $n$ -sided polygon $P$ to the measure of each interior angle of a regular $(n+2)$ -sided polygon is $\frac{24}{25}$ . Which of the following statements about polygon $P$ must be true? Select all such statements.

Select all that apply

Polygon

P

has

10

sides.

Polygon

P

has

35

diagonals.

The measure of each exterior angle of polygon

P

is

40^\circ

.

The sum of the measures of the interior angles of polygon

P

is

1{}800^\circ

.

The measure of each interior angle of polygon

P

is

144^\circ

.

Show answer & explanation

Answer: Polygon $P$ has $10$ sides.; Polygon $P$ has $35$ diagonals.; The measure of each interior angle of polygon $P$ is $144^\circ$ .

Answer

The statements asserting that polygon P has 10 sides, polygon P has 35 diagonals, and each interior angle of polygon P measures 144 degrees are all correct.

To determine which statements are true, we set up the ratio of the interior angle of a regular

n

-sided polygon to that of a regular

(n+2)

-sided polygon:

\frac{(n-2)(n+2)}{n^2} = \frac{24}{25}

. Simplifying gives

1 - \frac{4}{n^2} = \frac{24}{25}

, which leads to

n^2 = 100

, so

n = 10

. Therefore, polygon

P

is a regular decagon (10 sides). Evaluating the properties of a regular 10-gon shows that the number of diagonals is

\frac{10(10-3)}{2} = 35

, each interior angle measures

\frac{8 \cdot 180^\circ}{10} = 144^\circ

, each exterior angle measures

\frac{360^\circ}{10} = 36^\circ

, and the sum of the interior angles is

8 \cdot 180^\circ = 1{}440^\circ

. Consequently, the options stating that the polygon has 10 sides, has 35 diagonals, and has interior angles measuring

144^\circ

are correct.

Step-by-Step Solution

1

Set up the algebraic equation comparing the interior angles of an

n

-gon and an

(n+2)

-gon.

\frac{\frac{(n-2) \cdot 180^\circ}{n}}{\frac{n \cdot 180^\circ}{n+2}} = \frac{24}{25} \implies \frac{(n-2)(n+2)}{n^2} = \frac{24}{25}

The formula for each interior angle of a regular polygon with

k

sides is

\frac{(k-2) \cdot 180^\circ}{k}

.

2

Solve for

n

.

1 - \frac{4}{n^2} = \frac{24}{25} \implies \frac{4}{n^2} = \frac{1}{25} \implies n^2 = 100 \implies n = 10

Expanding

(n-2)(n+2) = n^2 - 4

allows simplifying the algebraic ratio.

3

Evaluate polygon properties for

n = 10

.

Diagonals:

\frac{10(10-3)}{2} = 35

; Interior angle:

\frac{8 \cdot 180^\circ}{10} = 144^\circ

; Exterior angle:

\frac{360^\circ}{10} = 36^\circ

; Interior angle sum:

8 \cdot 180^\circ = 1{}440^\circ

.

Apply standard formulas for diagonal count, exterior angle measure, and interior angle sum for a regular decagon.

Key Concept

Interior and exterior angle formulas of regular polygons and diagonal counting formulas

Question 1730Question →

What is the sum of all real solutions to the equation

(x^2 - 3x)^2 - 8(x^2 - 3x) + 12 = 0

?

3

6

8

12

-6

Show answer & explanation

Answer: $6$

Answer

The sum of all real solutions to the given equation is 6.

By substituting

u = x^2 - 3x

, the quartic equation reduces to the quadratic equation

u^2 - 8u + 12 = 0

, which factors as

(u - 2)(u - 6) = 0

. Setting

x^2 - 3x = 2

and

x^2 - 3x = 6

yields two distinct quadratic equations:

x^2 - 3x - 2 = 0

and

x^2 - 3x - 6 = 0

. Since both discriminants (

17

and

33

) are strictly positive, each equation has two distinct real solutions. By Vieta's formulas, the sum of the roots for a quadratic equation

ax^2 + bx + c = 0

is

-b/a

. Thus, the sum of the roots for

x^2 - 3x - 2 = 0

is

3

, and the sum of the roots for

x^2 - 3x - 6 = 0

is

3

. Adding these together gives a total sum of

3 + 3 = 6

.

Step-by-Step Solution

1

Apply algebraic substitution to reduce the equation to a standard quadratic form.

Let

u = x^2 - 3x

. The original equation becomes

u^2 - 8u + 12 = 0

.

Recognizing repeated quadratic expressions allows for simplification into a single quadratic in terms of

u

.

2

Factor the quadratic equation in

u

to find its roots.

(u - 2)(u - 6) = 0

, which gives

u = 2

and

u = 6

.

The factors of

12

that sum to

-8

are

-2

and

-6

.

3

Substitute

x^2 - 3x

back for

u

and check the discriminant of each resulting quadratic equation.

For

u = 2

:

x^2 - 3x - 2 = 0

has discriminant

\Delta_1 = (-3)^2 - 4(1)(-2) = 17 > 0

(2 distinct real roots).
For

u = 6

:

x^2 - 3x - 6 = 0

has discriminant

\Delta_2 = (-3)^2 - 4(1)(-6) = 33 > 0

(2 distinct real roots).

Verifying that the discriminant is positive ensures that all four roots are real numbers.

4

Calculate the sum of the roots for each quadratic equation using Vieta's formulas.

For

x^2 - 3x - 2 = 0

, the sum of roots is

-\frac{-3}{1} = 3

.
For

x^2 - 3x - 6 = 0

, the sum of roots is

-\frac{-3}{1} = 3

.

According to Vieta's formulas, for a quadratic

ax^2 + bx + c = 0

, the sum of the roots is

-b/a

.

5

Add the sums of the roots from both equations to find the total sum of all real solutions.

Total sum

= 3 + 3 = 6

.

Since all four roots are real and distinct, the overall sum is the sum of the roots of the two constituent quadratics.

Key Concept

Quadratic Substitution and Vieta's Formulas for Root Sums

Question 1731Question →

For all non-zero real numbers $a$ and $b$ , the custom binary operation $\diamondsuit$ is defined by $a \diamondsuit b = \frac{a^2 - b^2}{ab}$ . If the function $f$ is defined for all $x \neq 0$ by $f(x) = x \diamondsuit 2$ , what is the value of $f(4) - f(1)$ ?

Show answer & explanation

Answer: 3

Answer

3

Evaluating

f(4) = 4 \diamondsuit 2 = \frac{16 - 4}{8} = 1.5

and

f(1) = 1 \diamondsuit 2 = \frac{1 - 4}{2} = -1.5

, the required difference is

f(4) - f(1) = 1.5 - (-1.5) = 3

.

Step-by-Step Solution

1

Evaluate

f(4)

using the custom operation definition

f(4) = \frac{4^2 - 2^2}{4 \cdot 2} = \frac{12}{8} = 1.5

Substitute

a = 4

and

b = 2

into

a \diamondsuit b = \frac{a^2 - b^2}{ab}

.

2

Evaluate

f(1)

using the custom operation definition

f(1) = \frac{1^2 - 2^2}{1 \cdot 2} = -1.5

Substitute

a = 1

and

b = 2

into

a \diamondsuit b = \frac{a^2 - b^2}{ab}

.

3

Compute the difference

f(4) - f(1)

3

Subtracting

-1.5

from

1.5

yields

1.5 - (-1.5) = 1.5 + 1.5 = 3

.

Key Concept

Custom Symbol Operations and Function Evaluation

Estimated Time:1m 30s

Question 1732Question →

A research committee must select a delegation of $6$ members from a pool of $4$ senior fellows and $4$ junior analysts to sit around a circular conference table with $6$ evenly spaced seats. The delegation must consist of exactly $3$ senior fellows and $3$ junior analysts. If no two senior fellows may sit in adjacent seats around the table, how many distinct delegation seating arrangements are possible? (Two seating arrangements are considered identical if one can be rotated to match the other.)

Show answer & explanation

Answer: 192

Answer

The total number of distinct delegation seating arrangements is 192.

To find the total number of distinct delegation seating arrangements, we first determine how many ways the 6 delegates can be selected, and then multiply by the number of valid circular seating arrangements for those delegates.

1. Selection of Delegates:
- Ways to select 3 senior fellows from 4:

\binom{4}{3} = 4

- Ways to select 3 junior analysts from 4:

\binom{4}{3} = 4

- Total delegate combinations:

4 \times 4 = 16

2. Circular Seating Arrangements:
- With 3 seniors and 3 juniors at a 6-seat table, no two seniors can sit adjacent if and only if seniors and juniors alternate seats.
- Fix one senior fellow to eliminate rotational symmetry.
- The remaining 2 seniors can be arranged in

2! = 2

ways.
- The 3 junior analysts can be arranged in the 3 intermediate seats in

3! = 6

ways.
- Seating arrangements per delegation =

2 \times 6 = 12

3. Total Arrangements:
- Total =

16 \times 12 = 192

.

Step-by-Step Solution

1

Calculate combinations of senior fellows and junior analysts to form the 6-person delegation.

Number of ways to choose 3 seniors out of 4 is

\binom{4}{3} = 4

. Number of ways to choose 3 juniors out of 4 is

\binom{4}{3} = 4

. Total selection combinations =

4 \times 4 = 16

.

Choosing members from distinct pools uses combinations because member order within the selection does not matter.

2

Analyze the seating constraint for 3 seniors and 3 juniors around a 6-seat circular table.

The senior fellows must occupy alternating seats around the table (e.g., seats 1, 3, 5), leaving seats 2, 4, 6 for the junior analysts.

Placing 3 seniors among 6 circular seats with no two adjacent forces seniors to occupy every second seat.

3

Calculate the number of distinct circular seating arrangements for any specific set of 6 selected people.

Arrangements =

(3 - 1)! \times 3! = 2 \times 6 = 12

.

To account for rotational symmetry at a circular table, fix one senior fellow's seat. The remaining 2 senior fellows can be seated in

2! = 2

ways, and the 3 junior analysts can be seated in

3! = 6

ways in the remaining open seats.

4

Apply the Fundamental Counting Principle to combine selection and seating steps.

Total arrangements =

16 \text{ (selections)} \times 12 \text{ (seating arrangements)} = 192

.

Each of the 16 unique delegations can be seated around the circular table in 12 distinct relative orders.

Key Concept

Combinations and Circular Permutations with Adjacency Restrictions

Estimated Time:2m 0s

Question 1733Question →

In the $xy$ -coordinate plane, point $P$ has coordinates $(0, 0)$ and point $Q$ has coordinates $(6, 8)$ . Point $R$ is positioned such that triangle $PQR$ is a right triangle with hypotenuse $PQ$ . Which of the following could be the coordinates of point $R$ ? Select all such points.

Select all that apply

(8, 4)

(-2, 4)

(3, -1)

(4, 8)

(6, 6)

Show answer & explanation

Answer: $(8, 4)$ ; $(-2, 4)$ ; $(3, -1)$

Answer

The valid coordinates for point

R

are

(8, 4)

,

(-2, 4)

, and

(3, -1)

.

According to the Pythagorean theorem and Thales's theorem, any point forming a right angle with segment

PQ

as hypotenuse must satisfy

PR^2 + RQ^2 = PQ^2 = 100

, placing it on a circle centered at

(3, 4)

with radius

5

. The points

(8, 4)

,

(-2, 4)

, and

(3, -1)

each lie on this circle because their squared distances to

P

and

Q

sum to

100

.

Step-by-Step Solution

1

Calculate the square of hypotenuse

PQ

using the distance formula.

PQ^2 = (6 - 0)^2 + (8 - 0)^2 = 36 + 64 = 100

.

Since

PQ

is given as the hypotenuse of right triangle

PQR

, the Pythagorean theorem requires

PR^2 + RQ^2 = PQ^2 = 100

.

2

Express the condition

PR^2 + RQ^2 = 100

in terms of coordinates

(x, y)

of point

R

.

(x - 0)^2 + (y - 0)^2 + (x - 6)^2 + (y - 8)^2 = 100 \implies (x - 3)^2 + (y - 4)^2 = 25

.

By Thales's Theorem, any point

R

that forms a right triangle with hypotenuse

PQ

lies on a circle whose diameter is

PQ

, centered at the midpoint

(3, 4)

with radius

R_{circle} = 5

.

3

Test each candidate coordinate pair to verify if it satisfies

(x - 3)^2 + (y - 4)^2 = 25

.

Points

(8,4)

,

(-2,4)

, and

(3,-1)

satisfy

(5)^2 + 0^2 = 25

,

(-5)^2 + 0^2 = 25

, and

0^2 + (-5)^2 = 25

, respectively. Points

(4,8)

and

(6,6)

yield

(4-3)^2 + (8-4)^2 = 17 \neq 25

and

(6-3)^2 + (6-4)^2 = 13 \neq 25

.

Only coordinates located on the circle of diameter

PQ

form a right angle

\angle PRQ = 90^\circ

.

Key Concept

Pythagorean Theorem and Right Triangles in Coordinate Geometry

Question 1734Question →

A survey of $280$ culinary school graduates evaluated their expertise across three specialized culinary disciplines: Molecular Gastronomy ( $M$ ), Pastry Arts ( $P$ ), and Sous-Vide Cooking ( $S$ ). The survey gathered the following data:
- $130$ graduates have expertise in Molecular Gastronomy.
- $140$ graduates have expertise in Pastry Arts.
- $120$ graduates have expertise in Sous-Vide Cooking.
- $50$ graduates have expertise in both Molecular Gastronomy and Pastry Arts.
- $45$ graduates have expertise in both Pastry Arts and Sous-Vide Cooking.
- $55$ graduates have expertise in both Molecular Gastronomy and Sous-Vide Cooking.
- $20$ graduates have expertise in all three disciplines.
- $20$ graduates have expertise in none of the three disciplines.

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

Select all that apply

Exactly

45

graduates have expertise in Molecular Gastronomy only.

The total number of graduates with expertise in exactly two disciplines is

150

.

The number of graduates who have expertise in Sous-Vide Cooking but not Molecular Gastronomy is

65

.

More than

25\%

of all surveyed graduates have expertise in Pastry Arts only.

The total number of graduates who have expertise in at least one of the three disciplines is

240

.

Show answer & explanation

Answer: Exactly $45$ graduates have expertise in Molecular Gastronomy only.; The number of graduates who have expertise in Sous-Vide Cooking but not Molecular Gastronomy is $65$ .

Answer

The statements confirming that exactly 45 graduates have expertise in Molecular Gastronomy only and that 65 graduates have expertise in Sous-Vide Cooking but not Molecular Gastronomy are correct.

The calculated region for Molecular Gastronomy only yields

130 - (30 + 35 + 20) = 45

, making the statement about Molecular Gastronomy only correct. Furthermore, the number of graduates with expertise in Sous-Vide Cooking but not Molecular Gastronomy equals the sum of the Sous-Vide only region (

40

) and the Pastry-Sous-Vide only region (

25

), which totals

65

, making that statement correct as well.

Step-by-Step Solution

1

Determine the number of graduates in all three disciplines and in exactly two disciplines.

Triple intersection

|M \cap P \cap S| = 20

. Pairwise intersections excluding the triple intersection:

M \cap P \text{ only} = 50 - 20 = 30

,

P \cap S \text{ only} = 45 - 20 = 25

,

M \cap S \text{ only} = 55 - 20 = 35

. Total in exactly two disciplines =

30 + 25 + 35 = 90

.

Each given pairwise intersection contains the central triple intersection, so we must subtract 20 to find the exclusive double-overlap regions.

2

Calculate single-discipline exclusive counts.

Molecular Gastronomy only =

130 - (30 + 35 + 20) = 45

. Pastry Arts only =

140 - (30 + 25 + 20) = 65

. Sous-Vide Cooking only =

120 - (35 + 25 + 20) = 40

.

Subtracting all double-overlap and triple-overlap members from each set total yields the number of people belonging exclusively to that single set.

3

Evaluate the individual option statements.

Molecular Gastronomy only = 45 (True). Exactly two disciplines = 90 (Statement B claims 150, False). Sous-Vide Cooking without Molecular Gastronomy =

40 + 25 = 65

(True). Pastry Arts only percentage =

\frac{65}{280} \approx 23.21\% < 25\%

(Statement D claims

>25\%

, False). Total with at least one discipline =

280 - 20 = 260

(Statement E claims 240, False).

Comparing calculated region sizes against each option statement verifies which conditions must hold true.

Key Concept

3-Set Inclusion-Exclusion Principle and Venn Diagram Region Analysis

Estimated Time:2m 0s

Question 1735Question →

A media analytics platform surveyed a group of $240$ subscribers regarding the genres of content they watched over the past month: Documentary ( $D$ ), Sci-Fi ( $S$ ), and Animation ( $A$ ). Among the surveyed subscribers:
- $110$ watched Documentary
- $120$ watched Sci-Fi
- $95$ watched Animation
- $45$ watched both Documentary and Sci-Fi
- $40$ watched both Sci-Fi and Animation
- $35$ watched both Documentary and Animation
- $15$ watched none of these three genres

How many of the surveyed subscribers watched content from exactly one of the three genres?

85

105

145

120

205

Show answer & explanation

Answer: 145

Answer

145 subscribers watched content from exactly one genre.

The correct answer is 145. First, subtract the 15 subscribers who watched no genres from the total population of 240 to get 225 subscribers in the union of all three sets. Applying the principle of inclusion-exclusion yields

225 = 110 + 120 + 95 - (45 + 40 + 35) + |D \cap S \cap A|

, which solves to

|D \cap S \cap A| = 20

. With 20 subscribers watching all three genres, subscribers watching strictly two genres are 25 (Documentary & Sci-Fi only), 20 (Sci-Fi & Animation only), and 15 (Documentary & Animation only). Isolating subscribers watching only one genre gives

110 - (25 + 15 + 20) = 50

for Documentary,

120 - (25 + 20 + 20) = 55

for Sci-Fi, and

95 - (15 + 20 + 20) = 40

for Animation. Summing these single-genre counts (

50 + 55 + 40

) gives 145.

Step-by-Step Solution

1

Find the number of subscribers who watched at least one genre and apply the 3-set Inclusion-Exclusion Principle to determine the number of subscribers who watched all three genres.

Number watching at least one genre:

240 - 15 = 225

. By inclusion-exclusion:

|D \cup S \cup A| = |D| + |S| + |A| - (|D \cap S| + |S \cap A| + |D \cap A|) + |D \cap S \cap A|

225 = 110 + 120 + 95 - (45 + 40 + 35) + |D \cap S \cap A|

225 = 325 - 120 + |D \cap S \cap A| = 205 + |D \cap S \cap A|

|D \cap S \cap A| = 225 - 205 = 20

Knowing the size of the central intersection region (

|D \cap S \cap A| = 20

) is necessary to isolate the individual non-overlapping Venn diagram regions.

2

Calculate the number of subscribers who watched EXCLUSIVELY two genres.

Documentary and Sci-Fi only:

45 - 20 = 25

Sci-Fi and Animation only:

40 - 20 = 20

Documentary and Animation only:

35 - 20 = 15

Subtracting the triple intersection from pairwise intersections isolates subscribers who belong to exactly two sets.

3

Calculate the number of subscribers who watched EXACTLY ONE genre and sum them.

Documentary only:

110 - (25 + 15 + 20) = 50

Sci-Fi only:

120 - (25 + 20 + 20) = 55

Animation only:

95 - (15 + 20 + 20) = 40

Total watching exactly one genre:

50 + 55 + 40 = 145

Subtracting all multi-set overlaps from each genre's total gives the unique single-genre viewers.

Key Concept

Three-Set Inclusion-Exclusion Principle and Regional Segmentation

Estimated Time:1m 45s

Question 1736Question →

In the $xy$ -plane, line $k$ passes through the points $(-3, 5)$ and $(1, -3)$ . Line $m$ is perpendicular to line $k$ at line $k$ 's $x$ -intercept. What is the $y$ -intercept of line $m$ ?

\frac{1}{4}

-\frac{1}{4}

10

-1

-4

Show answer & explanation

Answer: $\frac{1}{4}$

Answer

The

y

-intercept of line

m

is

\frac{1}{4}

.

The line

k

has a slope of

-2

and an

x

-intercept of

\left(-\frac{1}{2}, 0\right)

. A line perpendicular to line

k

must have a slope of

\frac{1}{2}

. Substituting the point

\left(-\frac{1}{2}, 0\right)

into the line equation yields

y = \frac{1}{2}x + \frac{1}{4}

, so the

y

-intercept is

\frac{1}{4}

.

Step-by-Step Solution

1

Calculate the slope of line

k

Slope

m_k = \frac{-3 - 5}{1 - (-3)} = \frac{-8}{4} = -2

.

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Find the equation of line

k

and determine its

x

-intercept

Line

k

equation:

y - (-3) = -2(x - 1) \implies y = -2x - 1

. Setting

y = 0

gives

0 = -2x - 1 \implies x = -\frac{1}{2}

. The

x

-intercept is

\left(-\frac{1}{2}, 0\right)

.

The

x

-intercept is the point where the line crosses the

x

-axis (

y = 0

).

3

Find the slope of line

m

Slope

m_m = -\frac{1}{m_k} = -\frac{1}{-2} = \frac{1}{2}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

4

Determine the equation of line

m

and its

y

-intercept

Using point-slope form with

\left(-\frac{1}{2}, 0\right)

and slope

\frac{1}{2}

:

y - 0 = \frac{1}{2}\left(x - \left(-\frac{1}{2}\right)\right) \implies y = \frac{1}{2}x + \frac{1}{4}

. Setting

x = 0

gives

y = \frac{1}{4}

.

The

y

-intercept is the constant term

b

when written in slope-intercept form

y = mx + b

.

Key Concept

Perpendicular line slopes and intercept calculations

Estimated Time:2m 0s

Question 1737Question →

In the $xy$ -plane, triangle $ABC$ has vertices $A(1, 2)$ , $B(4, 2)$ , and $C(1, 6)$ . The triangle is reflected across the line $y = x$ , and then translated $2$ units to the left and $3$ units downward. What is the $y$ -coordinate of the image of vertex $C$ ?

Show answer & explanation

Answer: -2

Answer

The y-coordinate of the image of vertex C is -2.

Reflecting a point

(x, y)

across the line

y = x

swaps the coordinates, transforming

C(1, 6)

into

(6, 1)

. Subsequently, translating the point

2

units to the left and

3

units downward subtracts

2

from the

x

-coordinate and

3

from the

y

-coordinate, resulting in

(6 - 2, 1 - 3) = (4, -2)

. The

y

-coordinate of this image point is

-2

.

Step-by-Step Solution

1

Apply reflection across the line

y = x

to point

C(1, 6)

.

The transformed point is

C'(6, 1)

.

Reflecting a point

(x, y)

across the line

y = x

swaps its coordinates to

(y, x)

.

2

Apply translation left by

2

units and down by

3

units to

C'(6, 1)

.

The final point is

C''(6 - 2, 1 - 3) = C''(4, -2)

.

Translating left subtracts from the

x

-coordinate, and translating downward subtracts from the

y

-coordinate.

3

Extract the

y

-coordinate of

C''(4, -2)

.

-2

The

y

-coordinate is the second entry in the coordinate pair

(x, y)

.

Key Concept

Coordinate Geometry Transformations: Reflection across y = x and Translation

Estimated Time:1m 15s

Question 1738Question →

Let $A$ and $B$ be two events in a sample space such that $0 < P(A) < 1$ and $0 < P(B) < 1$ . Which of the following statements must be true? Select all that apply.

Select all that apply

If

A

and

B

are mutually exclusive, then

A

and

B

cannot be independent.

If

P(A \cup B) = P(A) + P(B) - P(A)P(B)

, then

A

and

B

are independent events.

If

A

and

B

are independent events, then their complements

A^c

and

B^c

must be mutually exclusive.

If

P(A|B) > P(A)

, then

P(B|A) > P(B)

.

If

A

and

B

are independent events, then

P(A \cup B) = P(A) + P(B)

.

Show answer & explanation

Answer: If $A$ and $B$ are mutually exclusive, then $A$ and $B$ cannot be independent.; If $P(A \cup B) = P(A) + P(B) - P(A)P(B)$ , then $A$ and $B$ are independent events.; If $P(A|B) > P(A)$ , then $P(B|A) > P(B)$ .

Answer

The statements asserting that mutually exclusive non-impossible events cannot be independent, that

P(A \cup B) = P(A) + P(B) - P(A)P(B)

implies independence, and that

P(A|B) > P(A)

implies

P(B|A) > P(B)

are all true.

For events with probabilities strictly between 0 and 1: (1) Mutual exclusivity requires

P(A \cap B) = 0

, whereas independence requires

P(A \cap B) = P(A)P(B) > 0

, so mutually exclusive events cannot be independent. (2) Substituting

P(A \cup B) = P(A) + P(B) - P(A)P(B)

into the addition rule yields

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

, which defines independence. (3)

P(A|B) > P(A)

is mathematically equivalent to

P(A \cap B) > P(A)P(B)

, which in turn is equivalent to

P(B|A) > P(B)

.

Step-by-Step Solution

1

Analyze the relationship between mutual exclusivity and independence.

Mutually exclusive events satisfy

P(A \cap B) = 0

. For independent events,

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

. Since

P(A) > 0

and

P(B) > 0

,

P(A)P(B) > 0 \neq 0

. Thus, mutually exclusive non-impossible events can never be independent.

To evaluate structural compatibility between mutual exclusivity and independence.

2

Apply the addition rule of probability to check the union equation.

General addition rule:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

. Comparing to the given

P(A \cup B) = P(A) + P(B) - P(A)P(B)

shows

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

, which is the exact condition for independence.

To verify if the given expression for union probability forces independence.

3

Evaluate the joint probability of complementary events

A^c

and

B^c

.

Independence of

A

and

B

implies independence of

A^c

and

B^c

. Thus

P(A^c \cap B^c) = (1-P(A))(1-P(B)) > 0

. Because the joint probability is positive, the complements are not mutually exclusive.

To test whether independence of events implies mutual exclusivity of their complements.

4

Examine the symmetry of conditional probability inequalities.

P(A|B) > P(A) \implies P(A \cap B) > P(A)P(B) \implies P(B|A) = \frac{P(A \cap B)}{P(A)} > P(B)

.

To evaluate directional dependence between conditional probabilities.

5

Check the simple addition rule for independent events.

Simple addition

P(A \cup B) = P(A) + P(B)

applies only when

P(A \cap B) = 0

. Independent events have

P(A \cap B) = P(A)P(B) > 0

, so

P(A \cup B) < P(A) + P(B)

.

To distinguish between addition rules for mutually exclusive vs. independent events.

Key Concept

Theoretical relationships between independent, dependent, mutually exclusive, and conditional events.

Estimated Time:3m 0s

Question 1739Question →

A box contains $4$ red blocks and $6$ yellow blocks. A block is selected at random from the box, its color is noted, and it is returned to the box. A second block is then selected at random from the box. What is the probability that both selected blocks are red?

\frac{4}{25}

\frac{2}{15}

\frac{2}{5}

\frac{4}{9}

\frac{4}{5}

Show answer & explanation

Answer: $\frac{4}{25}$

Answer

The probability that both selected blocks are red is

\frac{4}{25}

.

Because the first block is returned to the box before the second selection, the two draws are independent events. The probability of selecting a red block on any single draw is

\frac{4}{10} = \frac{2}{5}

. Applying the multiplication rule for independent events gives

P(\text{Both red}) = \frac{2}{5} \times \frac{2}{5} = \frac{4}{25}

.

Step-by-Step Solution

1

Determine the total number of blocks in the box.

The total number of blocks is

4 + 6 = 10

.

Probability requires finding the ratio of favorable outcomes to total possible outcomes.

2

Calculate the probability of drawing a red block on the first selection.

P(\text{First is red}) = \frac{4}{10} = \frac{2}{5}

.

There are

4

red blocks out of

10

total blocks.

3

Calculate the probability of drawing a red block on the second selection.

Since the first block is returned to the box, the events are independent, so

P(\text{Second is red}) = \frac{4}{10} = \frac{2}{5}

.

Replacement preserves the original sample space composition.

4

Apply the multiplication rule for independent events.

P(\text{Both are red}) = P(\text{First is red}) \times P(\text{Second is red}) = \frac{2}{5} \times \frac{2}{5} = \frac{4}{25}

.

The probability of two independent events both occurring is the product of their individual probabilities.

Key Concept

Probability of Independent Events

Question 1740Question →

A dataset $S$ consists of 12 numbers listed in increasing order: $x_1, x_2, \dots, x_{12}$ . The median of dataset $S$ is 40. The arithmetic mean of the 6 smallest numbers in $S$ is 28, and the arithmetic mean of the 6 largest numbers in $S$ is 56. A new dataset $T$ is formed by subtracting 4 from each of the 6 smallest numbers in $S$ and adding 8 to each of the 6 largest numbers in $S$ . What is the positive difference between the arithmetic mean of dataset $T$ and the median of dataset $T$ ?

Show answer & explanation

Answer: 2

Answer

2

The total sum of dataset S is 504, giving a mean of 42. Transforming the elements adds a net total of 24 to the overall sum, so the mean of dataset T becomes 44. Because decreasing the lower half and increasing the upper half preserves the relative sorted order of all 12 numbers, the middle two elements of dataset T are x_6 - 4 and x_7 + 8. Thus, the new median is (x_6 + x_7)/2 + 2 = 40 + 2 = 42. The positive difference between the mean of 44 and the median of 42 is 2.

Step-by-Step Solution

1

Calculate the arithmetic mean of the original dataset

S

.

The sum of the 6 smallest numbers is

6 \times 28 = 168

, and the sum of the 6 largest numbers is

6 \times 56 = 336

. The total sum of dataset

S

is

168 + 336 = 504

. Thus, the mean of

S

is

\frac{504}{12} = 42

.

The mean of a dataset is the sum of all elements divided by the total number of elements.

2

Calculate the arithmetic mean of the new dataset

T

.

The sum of dataset

T

is

504 + 6(-4) + 6(8) = 504 - 24 + 48 = 528

. The mean of dataset

T

is

\frac{528}{12} = 44

.

Modifying each of the 12 elements changes the overall sum by the sum of individual changes.

3

Determine the median of the new dataset

T

.

Since

x_6 < x_7

, after transformations

x_6 - 4 < x_7 + 8

. The relative order of all elements is preserved. The median of

T

is

\frac{(x_6 - 4) + (x_7 + 8)}{2} = \frac{x_6 + x_7}{2} + 2 = 40 + 2 = 42

.

The median of an even number of ordered elements is the average of the two middle elements.

4

Calculate the positive difference between the mean and median of dataset

T

.

|44 - 42| = 2.

Subtract the median from the mean and take the absolute value.

Key Concept

Effect of linear transformations and subgroup operations on the mean and median of ordered datasets

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept