Tüm alıştırma soruları

For all real numbers

x

, the function

f

is defined by

f(x) = x^2 - 2x

. The custom binary operation

\otimes

is defined for all real numbers

a

and

b

by

a \otimes b = f(a + b) - f(a - b)

If

k

is a constant such that

(k \otimes 3) \otimes 2 = 88

, what is the value of

k

?

\frac{5}{12}

\frac{11}{12}

2

\frac{17}{6}

6

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

k

is 2.

Expanding the definition

a \otimes b = f(a+b) - f(a-b)

using

f(x) = x^2 - 2x

gives

[(a+b)^2 - 2(a+b)] - [(a-b)^2 - 2(a-b)] = (a^2 + 2ab + b^2 - 2a - 2b) - (a^2 - 2ab + b^2 - 2a + 2b) = 4ab - 4b = 4b(a-1)

. Evaluating

k \otimes 3

yields

12k - 12

. Substituting this into

(12k - 12) \otimes 2

yields

4(2)(12k - 12 - 1) = 96k - 104

. Setting

96k - 104 = 88

gives

96k = 192

, so

k = 2

.

Adım Adım Çözüm

1

Express the custom operation

a \otimes b

in simplified algebraic terms.

a \otimes b = f(a+b) - f(a-b) = [(a+b)^2 - 2(a+b)] - [(a-b)^2 - 2(a-b)] = 4ab - 4b = 4b(a - 1)

.

Expanding and canceling common terms simplifies the binary operation definition.

2

Evaluate the inner operation

k \otimes 3

.

k \otimes 3 = 4(3)(k - 1) = 12k - 12

.

Substitute

a = k

and

b = 3

into the simplified operation formula

4b(a - 1)

.

3

Evaluate the outer operation

(12k - 12) \otimes 2

.

(12k - 12) \otimes 2 = 4(2)[(12k - 12) - 1] = 8(12k - 13) = 96k - 104

.

Substitute

a = 12k - 12

and

b = 2

into

4b(a - 1)

.

4

Set the resulting expression equal to 88 and solve for

k

.

96k - 104 = 88 \implies 96k = 192 \implies k = 2

.

Linear algebraic equation solving yields the value of

k

.

Anahtar Kavram

Custom Binary Operations and Nested Function Evaluation

Soru 1642Soru →

A line with a slope of $-2$ passes through the point $(1, 7)$ and contains the point $P(a, 3)$ . What is the distance between point $P$ and the point $Q(11, -3)$ ?

2\sqrt{10}

6\sqrt{5}

10

100

14

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

The distance between point

P

and point

Q

is

10

.

Using the slope definition

m = \frac{\Delta y}{\Delta x}

, we set up

\frac{3 - 7}{a - 1} = -2

, which solves to

a = 3

, giving point

P(3, 3)

. Applying the distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

between

P(3, 3)

and

Q(11, -3)

yields

\sqrt{(11 - 3)^2 + (-3 - 3)^2} = \sqrt{64 + 36} = 10

.

Adım Adım Çözüm

1

Use the slope formula to find the missing coordinate

a

of point

P(a, 3)

.

The slope

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 7}{a - 1} = -2

, which simplifies to

\frac{-4}{a - 1} = -2

, yielding

a - 1 = 2

, so

a = 3

.

The slope between any two points on a straight line must equal the given slope of

-2

.

2

Identify the coordinates of point

P

.

Point

P

has coordinates

(3, 3)

.

Substituting

a = 3

into

P(a, 3)

gives the exact position of

P

.

3

Apply the distance formula between

P(3, 3)

and

Q(11, -3)

.

d = \sqrt{(11 - 3)^2 + (-3 - 3)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10

.

The distance formula calculates the Euclidean distance between two points in the coordinate plane.

Anahtar Kavram

Line slope equation and coordinate distance formula

Alternatif Yöntem

Find the line equation directly in slope-intercept form:

y - 7 = -2(x - 1) \implies y = -2x + 9

. Substitute

y = 3

to get

3 = -2a + 9 \implies a = 3

. Then compute the distance between

(3, 3)

and

(11, -3)

using the standard distance formula.

Tahmini Süre:1m 30s

Soru 1643Soru →

In triangle $PQR$ , point $S$ lies on segment $QR$ such that segment $PS$ is perpendicular to $QR$ . The length of altitude $PS$ is $8$ units. If the area of triangle $PQR$ is $56$ square units and the ratio of the area of triangle $PQS$ to the area of triangle $PSR$ is $3 : 4$ , what is the length of side $PQ$ ?

10

4\sqrt{13}

\sqrt{174.25}

4\sqrt{5}

14

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

10

The area of triangle

PQR

is given as 56 square units and altitude

PS = 8

. Using

\text{Area} = \frac{1}{2} \times QR \times 8

, we find

QR = 14

. Because triangles

PQS

and

PSR

share altitude

PS

, the ratio of their areas equals the ratio of their bases

QS : SR = 3 : 4

. Dividing

QR = 14

into 7 equal parts yields

QS = 6

. In right-angled triangle

PQS

, legs are

PS = 8

and

QS = 6

, giving hypotenuse

PQ = \sqrt{8^2 + 6^2} = 10

.

Adım Adım Çözüm

1

Calculate the total length of base

QR

using the area of triangle

PQR

.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \implies 56 = \frac{1}{2} \times QR \times 8 \implies 56 = 4 \times QR \implies QR = 14

.

The area formula for any triangle relates base, height, and total area.

2

Determine the length of segment

QS

using the given area ratio.

Since triangles

PQS

and

PSR

share the same altitude

PS

, their areas are proportional to their base lengths

QS

and

SR

. Therefore,

QS : SR = 3 : 4

. The total parts are

3 + 4 = 7

. Thus,

QS = 14 \times \frac{3}{7} = 6

.

Triangles sharing a common altitude have areas proportional to their respective bases.

3

Apply the Pythagorean theorem in right triangle

PQS

to find hypotenuse

PQ

.

PQ^2 = PS^2 + QS^2 = 8^2 + 6^2 = 64 + 36 = 100 \implies PQ = 10

.

Segment

PS

is perpendicular to

QR

, forming right-angled triangle

PQS

with legs of length 8 and 6.

Anahtar Kavram

Area of triangles, common altitude area ratio, and the Pythagorean theorem.

Soru 1644Soru →

In trapezoid $ABCD$ , side $AB$ is perpendicular to parallel bases $AD$ and $BC$ . Diagonal $AC$ is perpendicular to side $CD$ . If the measure of angle $CAD$ is $30^\circ$ and the length of $BC$ is $9$ , what is the length of side $CD$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

Because base

BC

and base

AD

are parallel, transversal

AC

creates equal alternate interior angles, making

\angle ACB = \angle CAD = 30^\circ

. Triangle

ABC

is a right triangle with

\angle B = 90^\circ

, so

BC

is adjacent to

30^\circ

. The ratio of the side adjacent to

30^\circ

to the hypotenuse

AC

is

\frac{\sqrt{3}}{2}

, which yields

AC = \frac{9}{\sqrt{3}/2} = 6\sqrt{3}

. Triangle

ACD

is also a

30^\circ-60^\circ-90^\circ

right triangle with right angle at

C

. Side

CD

is opposite the

30^\circ

angle and side

AC

is opposite the

60^\circ

angle. The ratio of the short leg to the long leg is

\frac{1}{\sqrt{3}}

, giving

CD = \frac{6\sqrt{3}}{\sqrt{3}} = 6

.

Adım Adım Çözüm

1

Determine angle measures in right triangle ABC using parallel line properties

Angle ACB = 30 degrees and angle BAC = 60 degrees

Since base BC is parallel to base AD, alternate interior angles formed by transversal AC are equal: angle ACB = angle CAD = 30 degrees.

2

Calculate the length of diagonal AC using special right triangle ratios

AC = 6*sqrt(3)

In 30-60-90 right triangle ABC, the ratio of the side adjacent to 30 degrees (BC) to the hypotenuse (AC) is sqrt(3)/2. Therefore, 9 / AC = sqrt(3)/2, which gives AC = 18 / sqrt(3) = 6*sqrt(3).

3

Calculate the length of side CD using right triangle ACD

CD = 6

In 30-60-90 right triangle ACD, angle ACD = 90 degrees and angle CAD = 30 degrees. The ratio of the short leg opposite 30 degrees (CD) to the long leg opposite 60 degrees (AC) is 1/sqrt(3). Therefore, CD = AC / sqrt(3) = (6*sqrt(3)) / sqrt(3) = 6.

Anahtar Kavram

Side ratios of 30-60-90 special right triangles and alternate interior angles

Soru 1645Soru →

A technology firm surveyed $100$ software developers regarding their proficiency in three programming languages: Python ( $P$ ), Java ( $J$ ), and C++ ( $C$ ). The survey revealed the following results:
- $55$ developers are proficient in Python.
- $50$ developers are proficient in Java.
- $35$ developers are proficient in C++.
- $25$ developers are proficient in both Python and Java.
- $20$ developers are proficient in both Java and C++.
- $15$ developers are proficient in both Python and C++.
- $10$ developers are proficient in none of these three languages.

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

Geçerli olan tümünü seçin

Exactly

10

developers are proficient in all three programming languages.

Exactly

50

developers are proficient in at least two of these programming languages.

The number of developers proficient in Python only is

25

.

The total number of developers proficient in exactly one of these languages is

50

.

The percentage of Python-proficient developers who are proficient in Python only is

25\%

.

Cevabı ve açıklamayı göster

Cevap: Exactly $10$ developers are proficient in all three programming languages.; The number of developers proficient in Python only is $25$ .; The total number of developers proficient in exactly one of these languages is $50$ .

Cevap

The correct statements are those asserting that exactly 10 developers know all three languages, that 25 developers know Python only, and that 50 developers know exactly one of these languages.

The statements confirming that 10 developers know all three languages, 25 know Python only, and 50 know exactly one language are correct based on standard 3-set Venn diagram calculations. Using Inclusion-Exclusion,

|P \cup J \cup C| = 90

, which gives

|P \cap J \cap C| = 10

. Subtracting overlapping regions yields

25

for Python only,

15

for Java only,

10

for C++ only, summing to

50

for exactly one language.

Adım Adım Çözüm

1

Calculate the total number of developers proficient in at least one language.

N_{\text{at least one}} = 100 - 10 = 90

.

Subtracting the developers proficient in none of the languages from the total survey population yields the union

|P \cup J \cup C|

.

2

Apply the Principle of Inclusion-Exclusion for 3 sets to find the triple intersection

|P \cap J \cap C|

.

90 = 55 + 50 + 35 - (25 + 20 + 15) + |P \cap J \cap C| \implies 90 = 80 + |P \cap J \cap C| \implies |P \cap J \cap C| = 10

.

The formula

|P \cup J \cup C| = |P| + |J| + |C| - (|P \cap J| + |J \cap C| + |P \cap C|) + |P \cap J \cap C|

links all given set quantities.

3

Calculate the count of developers proficient in exactly two languages for each pair.

Python and Java only =

25 - 10 = 15

; Java and C++ only =

20 - 10 = 10

; Python and C++ only =

15 - 10 = 5

.

Subtracting the triple intersection count (

10

) from each pairwise intersection yields the exclusive two-set regions.

4

Calculate the single-language proficiency counts (exactly one language).

Python only =

55 - (15 + 5 + 10) = 25

; Java only =

50 - (15 + 10 + 10) = 15

; C++ only =

35 - (5 + 10 + 10) = 10

. Total exactly one =

25 + 15 + 10 = 50

.

Subtracting all overlapping regions containing each language from its total count yields the single-language region size.

5

Evaluate the statement options based on computed regional values.

Statements stating 10 all three, 25 Python only, and 50 exactly one language are true.

Comparing calculated values (

10

for all three,

25

for Python only,

50

for exactly one language) against each option validates the true choices.

Anahtar Kavram

Three-set inclusion-exclusion principle and Venn diagram region decomposition

Soru 1646Soru →

A technology institute surveyed a cohort of $300$ software engineers regarding their proficiency in three programming paradigms: Functional ( $F$ ), Object-Oriented ( $O$ ), and Concurrent ( $C$ ). Every surveyed engineer is proficient in at least one of these three paradigms. The ratio of the total number of engineers proficient in $F$ , $O$ , and $C$ is $5 : 6 : 4$ , respectively. Furthermore, exactly $20\%$ of the engineers proficient in $F$ are proficient in all three paradigms. If exactly $54$ engineers are proficient in both $F$ and $O$ , $48$ are proficient in both $O$ and $C$ , and $30$ are proficient in both $F$ and $C$ , how many engineers in the cohort are proficient in exactly one programming paradigm?

96

141

222

273

78

Cevabı ve açıklamayı göster

Cevap: 222

Cevap

222 engineers are proficient in exactly one programming paradigm.

Using the Principle of Inclusion-Exclusion for three sets,

|F \cup O \cup C| = |F| + |O| + |C| - (|F \cap O| + |O \cap C| + |F \cap C|) + |F \cap O \cap C|

. Substituting

|F|=5k

,

|O|=6k

,

|C|=4k

,

|F \cap O \cap C|=k

, and the total cohort

300

gives

300 = 16k - 132

, yielding

k = 27

. Consequently, the triple intersection is

27

. Decomposing into disjoint regions:

F

only

= 78

,

O

only

= 87

, and

C

only

= 57

. Summing these gives

78 + 87 + 57 = 222

.

Adım Adım Çözüm

1

Set up algebraic representations for the set sizes using the given ratio.

Let

|F| = 5k

,

|O| = 6k

, and

|C| = 4k

for some positive constant

k

.

The total proficiencies follow the ratio

5:6:4

.

2

Express the triple intersection

|F \cap O \cap C|

in terms of

k

.

|F \cap O \cap C| = 0.20 \times |F| = 0.20 \times 5k = k

.

Exactly 20% of engineers proficient in

F

are proficient in all three paradigms.

3

Apply the Principle of Inclusion-Exclusion for three sets to solve for

k

.

|F \cup O \cup C| = |F| + |O| + |C| - (|F \cap O| + |O \cap C| + |F \cap C|) + |F \cap O \cap C| \implies 300 = 5k + 6k + 4k - (54 + 48 + 30) + k \implies 300 = 16k - 132 \implies 16k = 432 \implies k = 27

.

Every engineer is proficient in at least one paradigm, so

|F \cup O \cap C| = 300

.

4

Calculate the total size of each set and each exclusive intersection region.

|F| = 135

,

|O| = 162

,

|C| = 108

, and

|F \cap O \cap C| = 27

.
Exclusively

F \cap O = 54 - 27 = 27

.
Exclusively

O \cap C = 48 - 27 = 21

.
Exclusively

F \cap C = 30 - 27 = 3

.

Subtracting the triple intersection from pairwise intersections yields the two-set-only regions.

5

Determine the number of engineers proficient in exactly one paradigm.

Only

F = 135 - (27 + 3 + 27) = 78

.
Only

O = 162 - (27 + 21 + 27) = 87

.
Only

C = 108 - (3 + 21 + 27) = 57

.
Total exactly one =

78 + 87 + 57 = 222

.

Subtracting all overlapping regions from each total set size gives the single-category populations.

Anahtar Kavram

Three-set Principle of Inclusion-Exclusion and Venn Diagram region decomposition.

Soru 1647Soru →

A solar power facility operates two types of solar panel arrays: Array Alpha and Array Beta. When operational, Array Beta produces electricity at a constant hourly rate that is $25\%$ greater than the constant hourly rate of Array Alpha. On a clear day, Array Alpha operated for $8$ hours and Array Beta operated for $6$ hours, together generating a total of $3,100$ kilowatt-hours (kWh) of electricity. What was the hourly production rate of Array Alpha, in kWh per hour?

160

193.75

200

248

250

Cevabı ve açıklamayı göster

Cevap: 200

Cevap

200 kWh per hour

Let the hourly rate of Array Alpha be

r

kWh per hour. Since Array Beta produces at a rate

25\%

greater, its hourly rate is

1.25r

kWh per hour. Multiply each rate by the respective number of hours operated: Array Alpha produced

8r

kWh and Array Beta produced

6 \times 1.25r = 7.5r

kWh. Combining these gives

8r + 7.5r = 15.5r = 3,100

. Solving for

r

yields

r = 200

kWh per hour.

Adım Adım Çözüm

1

Define variables for the hourly rates of Array Alpha and Array Beta.

Let

r

be the hourly rate of Array Alpha in kWh per hour. Since Array Beta's rate is

25\%

greater, Array Beta's rate is

r + 0.25r = 1.25r

kWh per hour.

Establishing the linear relationship between the two unknown rates.

2

Set up the total electricity output equation using rate times time for each array.

\text{Total Energy} = (8 \text{ hours} \times r) + (6 \text{ hours} \times 1.25r) = 3,100

Total production is the sum of production from Array Alpha and Array Beta.

3

Simplify the algebraic equation and solve for

r

.

8r + 7.5r = 3,100 \implies 15.5r = 3,100 \implies r = \frac{3,100}{15.5} = 200

Isolating

r

gives the hourly rate of Array Alpha.

Anahtar Kavram

Linear Algebraic Modeling of Combined Rates and Percentages

Tahmini Süre:1m 30s

Soru 1648Soru →

In convex quadrilateral $ABCD$ , diagonals $AC$ and $BD$ intersect at point $E$ . The area of $\triangle ABE$ is $4$ , the area of $\triangle BCE$ is $8$ , and the area of $\triangle CDE$ is $16$ . Which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

The area of

\triangle DAE

is

8

.

Quadrilateral

ABCD

is a trapezoid with

AB

parallel to

CD

.

The length of segment

CD

is twice the length of segment

AB

.

The total area of quadrilateral

ABCD

is

32

.

The midpoint of diagonal

AC

lies on segment

DE

.

Cevabı ve açıklamayı göster

Cevap: The area of $\triangle DAE$ is $8$ .; Quadrilateral $ABCD$ is a trapezoid with $AB$ parallel to $CD$ .; The length of segment $CD$ is twice the length of segment $AB$ .

Cevap

The correct statements are that the area of triangle DAE is 8, quadrilateral ABCD is a trapezoid with AB parallel to CD, and the length of segment CD is twice the length of segment AB.

The statements confirming that the area of triangle DAE is 8, that ABCD is a trapezoid with AB parallel to CD, and that CD is twice length AB are all derived using area ratio properties of intersecting diagonals and triangle similarity.

Adım Adım Çözüm

1

Determine the area of triangle DAE using diagonal segment ratios.

Area of triangle DAE = 8.

Triangles sharing a base line have areas proportional to the segments into which the intersecting line divides that base line: AE/EC = Area(ABE)/Area(BCE) = 4/8 = 1/2, so Area(DAE) = (1/2) * 16 = 8.

2

Check parallelism of opposite sides AB and CD.

AB is parallel to CD, making ABCD a trapezoid.

Area(ABC) = 4 + 8 = 12 and Area(ABD) = 4 + 8 = 12. Triangles with equal areas on the common base AB must have equal heights, implying line CD is parallel to line AB.

3

Calculate side ratio CD / AB using similar triangles.

CD = 2 * AB.

Since AB || CD, triangle ABE is similar to triangle CDE. The ratio of their areas is 16/4 = 4, so the side length ratio CD/AB = sqrt(4) = 2.

4

Compute total area of quadrilateral ABCD and evaluate diagonal midpoint position.

Total area is 36 (not 32), and the midpoint of AC is at (1/2)AC from A, which is distinct from E at (1/3)AC.

Total area = 4 + 8 + 16 + 8 = 36. Since E divides AC in a 1:2 ratio, E is not the midpoint of AC.

Anahtar Kavram

Properties of convex quadrilaterals, area decomposition via diagonal ratios, trapezoid parallelism criteria, and triangle similarity.

Soru 1649Soru →

An investor allocates a total of $\$24,000$ among three accounts: Account A, which earns $3\%$ annual simple interest; Account B, which earns $5\%$ annual simple interest; and Account C, which earns $7\%$ annual simple interest. The total annual interest earned from all three accounts combined at the end of one year is $\$1,260$ . If the amount invested in Account C is $\$2,000$ more than twice the amount invested in Account A, what is the amount invested in Account B?

\$1,000

\$4,000

\$11,000

\$15,000

\$19,000

Cevabı ve açıklamayı göster

Cevap: $\$19,000$

Cevap

The amount invested in Account B is

\$19,000

.

The system of linear equations representing the total investment, interest earned, and relative account values yields

A = \$1,000

,

C = \$4,000

, and

B = \$19,000

. Thus, the amount invested in Account B is

\$19,000

.

Adım Adım Çözüm

1

Define variables and write the system of three linear equations based on total investment, total annual interest, and account relationship.

Let

A

,

B

, and

C

represent the dollars invested in Accounts A, B, and C respectively.
Equation (1):

A + B + C = 24,000

Equation (2):

0.03A + 0.05B + 0.07C = 1,260

, which simplifies to

3A + 5B + 7C = 126,000

Equation (3):

C = 2A + 2,000

, or

2A - C = -2,000

Translate the verbal conditions into an explicit

3 \times 3

system of linear equations.

2

Substitute

C = 2A + 2,000

into Equations (1) and (2) to reduce the system to two variables (

A

and

B

).

From Equation (1):

A + B + (2A + 2,000) = 24,000 \implies 3A + B = 22,000 \implies B = 22,000 - 3A

.
From Equation (2):

3A + 5B + 7(2A + 2,000) = 126,000 \implies 17A + 5B + 14,000 = 126,000 \implies 17A + 5B = 112,000

.

Eliminate variable

C

to simplify solving the linear system.

3

Substitute

B = 22,000 - 3A

into

17A + 5B = 112,000

to solve for

A

.

17A + 5(22,000 - 3A) = 112,000 \implies 17A + 110,000 - 15A = 112,000 \implies 2A = 2,000 \implies A = 1,000

.

Solve for the single variable

A

.

4

Determine the values of

C

and

B

.

C = 2(1,000) + 2,000 = 4,000

.

B = 22,000 - 3(1,000) = 19,000

.

Substitute

A = 1,000

back into the expressions for

C

and

B

to find the targeted investment amount.

Anahtar Kavram

Setting up and solving a system of three linear equations in three variables by substitution and elimination.

Tahmini Süre:2m 0s

Soru 1650Soru →

A survey of $300$ urban commuters evaluated their usage of three transit services: the Bus ( $B$ ), the Commuter Rail ( $R$ ), and the Express Ferry ( $F$ ). The survey revealed the following information:

- $160$ commuters use the Bus.
- $140$ commuters use the Commuter Rail.
- $110$ commuters use the Express Ferry.
- $60$ commuters use both the Bus and the Commuter Rail.
- $45$ commuters use both the Commuter Rail and the Express Ferry.
- $50$ commuters use both the Bus and the Express Ferry.
- $25$ commuters use all three transit services.

How many of the surveyed commuters use none of these three transit services?

20

45

70

280

145

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20 commuters use none of the three transit services.

Using the inclusion-exclusion principle for three overlapping sets, the total number of commuters using at least one of the transit services is

160 + 140 + 110 - (60 + 45 + 50) + 25 = 280

. Subtracting this value from the total surveyed group of

300

commuters gives

300 - 280 = 20

commuters who use none of the three services.

Adım Adım Çözüm

1

Apply the Principle of Inclusion-Exclusion for three sets to find the total number of commuters who use at least one transit service,

|B \cup R \cup F|

.

|B \cup R \cup F| = |B| + |R| + |F| - (|B \cap R| + |R \cap F| + |F \cap B|) + |B \cap R \cap F|

Simply adding set sizes double-counts elements in pairwise intersections and triple-counts elements in all three sets.

2

Substitute the given numerical values into the inclusion-exclusion formula.

|B \cup R \cup F| = 160 + 140 + 110 - (60 + 45 + 50) + 25 = 410 - 155 + 25 = 280

Combining the sums and differences yields the exact count of commuters using at least one mode of transit.

3

Subtract the number of commuters using at least one service from the total surveyed population to find those using none.

\text{None} = 300 - 280 = 20

The universe of surveyed commuters consists of those using at least one service plus those using none.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets

Tahmini Süre:1m 30s

Soru 1651Soru →

In $\triangle ABC$ , angle $B$ is a right angle, and line segment $BD$ is an altitude drawn to side $AC$ with point $D$ lying on $AC$ . If $AD = 4$ units and $DC = 16$ units, what is the area, in square units, of $\triangle ABC$ ?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

The area of triangle ABC is 80 square units.

In right triangle

ABC

with right angle at

B

, altitude

BD

drawn to hypotenuse

AC

divides the hypotenuse into segments

AD

and

DC

. By the geometric mean theorem,

BD^2 = AD \times DC = 4 \times 16 = 64

, which gives

BD = 8

units. The length of hypotenuse

AC

is

AD + DC = 4 + 16 = 20

units. The area of triangle

ABC

is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 20 \times 8 = 80

square units.

Adım Adım Çözüm

1

Calculate the height (altitude)

BD

of the triangle.

BD = 8

units.

In a right triangle, the altitude to the hypotenuse divides the hypotenuse into two segments such that

BD^2 = AD \times DC

. Substituting the given values yields

BD^2 = 4 \times 16 = 64

, so

BD = \sqrt{64} = 8

.

2

Calculate the total length of hypotenuse

AC

.

AC = 20

units.

Since point

D

lies on segment

AC

, the total length is the sum of its parts:

AC = AD + DC = 4 + 16 = 20

.

3

Calculate the area of

\triangle ABC

.

Area = 80 square units.

Using the triangle area formula

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

, with base

AC = 20

and altitude

BD = 8

, we get

\text{Area} = \frac{1}{2} \times 20 \times 8 = 80

.

Anahtar Kavram

Geometric mean theorem for right triangle altitude and area of a triangle

Soru 1652Soru →

Two industrial pumps, Pump A and Pump B, are used to fill a storage tank. Working alone at its constant standard operating rate, Pump A can fill the empty tank in $12$ hours. Working alone at its constant standard operating rate, Pump B can fill the empty tank in $18$ hours.

To fill the tank, both pumps begin operating simultaneously at their standard rates. After $4$ hours of joint operation, Pump A undergoes maintenance that reduces its operating rate by $25\%$ , while Pump B continues operating at its standard rate. Exactly $2$ hours after Pump A's rate is reduced, Pump B's operating rate is increased by $50\%$ above its standard rate due to a valve adjustment. If both pumps continue operating at these adjusted rates until the tank is full, how many total hours from the initial start does it take to completely fill the tank?

6\frac{2}{3}

hours

7\frac{1}{7}

hours

7\frac{3}{7}

hours

8\frac{1}{7}

hours

8\frac{2}{3}

hours

Cevabı ve açıklamayı göster

Cevap: $7\frac{3}{7}$ hours

Cevap

The total time required to completely fill the tank from the initial start is

7\frac{3}{7}

hours.

The solution proceeds in three stages. In the first 4 hours, both pumps at standard rates completed 5/9 of the job. In the next 2 hours, Pump A operated at 1/16 tank/hr and Pump B at 1/18 tank/hr, completing an additional 17/72 of the job, bringing total completed work to 19/24 of the tank. For the final 5/24 of the tank, Pump A (1/16 tank/hr) and Pump B (1/12 tank/hr) worked at a combined rate of 7/48 tank/hr, taking 10/7 (or 1 3/7) hours. The total time is 4 + 2 + 1 3/7 = 7 3/7 hours.

Adım Adım Çözüm

1

Determine individual standard rates and combined initial rate.

Pump A rate

r_A = \frac{1}{12}

tank/hr; Pump B rate

r_B = \frac{1}{18}

tank/hr. Initial combined rate

r_1 = \frac{1}{12} + \frac{1}{18} = \frac{5}{36}

tank/hr.

Work rate is the reciprocal of the total time taken to complete one full job.

2

Calculate work completed in Phase 1 (first 4 hours).

Work completed

= 4 \times \frac{5}{36} = \frac{20}{36} = \frac{5}{9}

of the tank. Remaining work

= 1 - \frac{5}{9} = \frac{4}{9}

of the tank.

Both pumps work at standard combined rate for 4 hours.

3

Adjust rates for Phase 2 (hours 4 to 6) and calculate work done.

Pump A reduced rate

= 0.75 \times \frac{1}{12} = \frac{1}{16}

tank/hr. Combined rate

r_2 = \frac{1}{16} + \frac{1}{18} = \frac{17}{144}

tank/hr. Work done in 2 hours

= 2 \times \frac{17}{144} = \frac{17}{72}

. Cumulative work

= \frac{5}{9} + \frac{17}{72} = \frac{57}{72} = \frac{19}{24}

of the tank.

Phase 2 lasts 2 hours with Pump A operating at

75\%

efficiency and Pump B at standard rate.

4

Adjust rates for Phase 3 (from hour 6 onwards) and calculate remaining time.

Pump B increased rate

= 1.50 \times \frac{1}{18} = \frac{1}{12}

tank/hr. Combined rate

r_3 = \frac{1}{16} + \frac{1}{12} = \frac{7}{48}

tank/hr. Remaining work

= 1 - \frac{19}{24} = \frac{5}{24}

. Additional time required

= \frac{5/24}{7/48} = \frac{5}{24} \times \frac{48}{7} = \frac{10}{7} = 1\frac{3}{7}

hours.

Divide remaining fractional work by the new combined rate.

5

Calculate total elapsed time.

Total time

= 4 + 2 + 1\frac{3}{7} = 7\frac{3}{7}

hours.

Sum the durations of all three phases.

Anahtar Kavram

Multi-stage work-rate problems with variable individual rates and fractional job completion tracking.

Alternatif Yöntem

Define the tank capacity as 144 units (the LCM of 12, 18, 16, 48). Pump A standard rate = 12 units/hr; Pump B standard rate = 8 units/hr. Phase 1 (4 hrs): combined rate = 20 units/hr, work done = 80 units. Phase 2 (2 hrs): Pump A rate = 9 units/hr, Pump B rate = 8 units/hr, combined rate = 17 units/hr, work done = 34 units. Total work done in 6 hrs = 114 units. Remaining work = 30 units. Phase 3: Pump A rate = 9 units/hr, Pump B rate = 12 units/hr, combined rate = 21 units/hr. Additional time = 30/21 = 10/7 hrs. Total time = 6 + 10/7 = 7 3/7 hrs.

Tahmini Süre:2m 30s

Soru 1653Soru →

An environmental auditing agency surveyed $250$ manufacturing plants regarding their compliance with three environmental standards: Air Quality ( $A$ ), Water Discharge ( $W$ ), and Waste Management ( $M$ ). The survey yielded the following data:

- $130$ plants meet Air Quality standards ( $A$ ).
- $140$ plants meet Water Discharge standards ( $W$ ).
- $120$ plants meet Waste Management standards ( $M$ ).
- $40$ plants meet all three standards.
- $20$ plants meet none of the three standards.
- The number of plants meeting both Air Quality and Water Discharge standards is equal to the number of plants meeting both Water Discharge and Waste Management standards.
- The number of plants meeting both Air Quality and Waste Management standards is $10$ fewer than the number meeting both Air Quality and Water Discharge standards.

How many of the surveyed plants meet exactly one of the three environmental standards?

Cevabı ve açıklamayı göster

Cevap: 110

Cevap

110

By setting up the 3-set inclusion-exclusion equation, the unknown pairwise intersections are found to be 70, 70, and 60. Subtracting the 40 plants that meet all three standards gives the exclusive double-overlap regions (30, 30, and 20). Subtracting these along with the central intersection from each single set yields 40 plants meeting only Air Quality, 40 meeting only Water Discharge, and 30 meeting only Waste Management, totaling 110 plants.

Adım Adım Çözüm

1

Find total number of plants meeting at least one standard

|A ∪ W ∪ M| = 250 - 20 = 230

Subtracting plants that meet no standards from the total surveyed gives the union of all three sets.

2

Set up algebraic expressions for pairwise intersections

|A ∩ M| = k, |A ∩ W| = k + 10, |W ∩ M| = k + 10

Define the smallest pairwise intersection as k and express the other two based on the given relationships.

3

Apply the Principle of Inclusion-Exclusion (PIE) for three sets to solve for k

230 = 130 + 140 + 120 - (k + 10 + k + 10 + k) + 40 => k = 60

Substitute set sizes and the triple intersection into the 3-set inclusion-exclusion formula.

4

Calculate the number of plants in each exclusive region

Only (A ∩ W) = 30, Only (W ∩ M) = 30, Only (A ∩ M) = 20

Subtract the triple intersection (40) from each pairwise intersection.

5

Calculate plants meeting exactly one standard and sum them

Only A = 40, Only W = 40, Only M = 30; Total = 40 + 40 + 30 = 110

Subtract all double-overlap and triple-overlap regions from each individual set total.

Anahtar Kavram

Three-Set Principle of Inclusion-Exclusion and Venn Diagram Region Decomposition

Soru 1654Soru →

A university surveyed a cohort of $150$ freshmen regarding their membership in three student organizations: the Art Club ( $A$ ), the Music Society ( $M$ ), and the Theater Guild ( $T$ ). The survey revealed the following data:

- $68$ students belong to the Art Club.
- $62$ students belong to the Music Society.
- $54$ students belong to the Theater Guild.
- $22$ students belong to both the Art Club and the Music Society.
- $18$ students belong to both the Music Society and the Theater Guild.
- $15$ students belong to both the Art Club and the Theater Guild.
- $8$ students belong to all three organizations.

How many of the surveyed students belong to exactly one of these three organizations?

Cevabı ve açıklamayı göster

Cevap: 98

Cevap

98 students belong to exactly one of the three organizations.

To find the number of students belonging to exactly one organization, analyze the regions of a 3-set Venn diagram starting from the innermost region (all three clubs =

8

). Subtracting

8

from each pairwise intersection gives the students in exactly two clubs: Art & Music only (

14

), Music & Theater only (

10

), and Art & Theater only (

7

). Next, subtract the overlapping regions from each club total: Art only is

68 - (14 + 7 + 8) = 39

; Music only is

62 - (14 + 10 + 8) = 30

; Theater only is

54 - (7 + 10 + 8) = 29

. Summing these single-club regions gives

39 + 30 + 29 = 98

.

Adım Adım Çözüm

1

Find the number of students belonging strictly to each pair of organizations (two-set intersections only).

Art and Music only =

14

; Music and Theater only =

10

; Art and Theater only =

7

.

The given pairwise totals include students who belong to all three organizations (

8

), so subtracting

8

isolates those in exactly two groups.

2

Determine the number of students belonging to each individual organization exclusively.

Art only =

39

; Music only =

30

; Theater only =

29

.

Subtract all shared membership regions (both two-group only and three-group) from each total organization membership.

3

Add the counts of students belonging to exactly one group.

39 + 30 + 29 = 98

.

The question requests the sum of all students in the non-overlapping single-set regions.

Anahtar Kavram

Three-Set Venn Diagram Region Partitioning

Soru 1655Soru →

For any real number $x$ , the custom unary operation $\triangle$ is defined by $\triangle(x) = 2x - x^2$ . For all real numbers $u$ and $v$ , the custom binary operation $\odot$ is defined by $u \odot v = \triangle(u) + \triangle(v) + uv$ . The function $f$ is defined for all real numbers $x$ by $f(x) = x \odot (3 - x)$ . What is the maximum value of $f(x)$ ?

Cevabı ve açıklamayı göster

Cevap: 3.75

Cevap

The maximum value of

f(x)

is

3.75

(or

\frac{15}{4}

).

Expanding

f(x) = x \odot (3 - x)

yields

\triangle(x) + \triangle(3 - x) + x(3 - x)

. Substituting

\triangle(t) = 2t - t^2

gives

(2x - x^2) + (6 - 2x - (9 - 6x + x^2)) + (3x - x^2) = -3x^2 + 9x - 3

. The vertex of this quadratic function occurs at

x = \frac{9}{6} = 1.5

, where

f(1.5) = -3(2.25) + 13.5 - 3 = 3.75

.

Adım Adım Çözüm

1

Evaluate the custom unary operation

\triangle

for each argument

\triangle(x) = 2x - x^2

and

\triangle(3-x) = 2(3-x) - (3-x)^2 = -x^2 + 4x - 3

Substitute

x

and

3-x

into the definition

\triangle(t) = 2t - t^2

and expand carefully.

2

Compute the product term

uv

x(3 - x) = 3x - x^2

The binary definition

u \odot v

includes an additive product term

uv

.

3

Sum all components to construct the explicit quadratic expression for

f(x)

f(x) = (2x - x^2) + (-x^2 + 4x - 3) + (3x - x^2) = -3x^2 + 9x - 3

Combine like terms for

x^2

,

x

, and the constant.

4

Determine the vertex of the downward-opening parabola

f(x) = -3x^2 + 9x - 3

x = \frac{3}{2} = 1.5

, yielding

f(1.5) = 3.75$

Since the coefficient of

x^2

is negative (

-3 < 0

), the maximum occurs at

x = -\frac{b}{2a} = \frac{9}{6} = 1.5

.

Anahtar Kavram

Custom operations combined with quadratic function optimization

Tahmini Süre:2m 0s

Soru 1656Soru →

At a technology conference attended by $80$ software engineers, $50$ engineers write code in Python, $40$ write code in Java, and $15$ write code in neither Python nor Java. How many of the software engineers write code in both Python and Java?

10

15

25

30

35

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The number of software engineers who write code in both Python and Java is 25.

Out of

80

engineers,

15

write neither language, which means

80 - 15 = 65

engineers write Python, Java, or both. Using the inclusion-exclusion principle,

|P \cup J| = |P| + |J| - |P \cap J|

, substituting the values gives

65 = 50 + 40 - |P \cap J|

. Solving for the overlap gives

|P \cap J| = 90 - 65 = 25

. Thus, 25 engineers write code in both languages.

Adım Adım Çözüm

1

Find the number of engineers who write code in at least one of the two languages (Python or Java).

|P \cup J| = 80 - 15 = 65

Subtracting the engineers who write neither language from the total gives the union of the two sets.

2

Apply the Principle of Inclusion-Exclusion formula for two sets.

|P \cup J| = |P| + |J| - |P \cap J|

The total number of engineers in the union is equal to the sum of the individual sets minus their intersection.

3

Substitute the known values into the equation to solve for the intersection

|P \cap J|

.

65 = 50 + 40 - |P \cap J| \implies 65 = 90 - |P \cap J| \implies |P \cap J| = 25

Solving the linear equation yields the number of engineers writing both languages.

Anahtar Kavram

Principle of Inclusion-Exclusion for Two Sets

Tahmini Süre:1m 0s

Soru 1657Soru →

An environmental protection agency audited $240$ coastal wetland sites to evaluate contamination by three specific pollutants: Microplastics ( $M$ ), Heavy metals ( $H$ ), and Agricultural runoff ( $A$ ). The audit revealed the following findings:

- Exactly $30$ of the audited wetlands showed no contamination from any of the three pollutants.
- The total number of wetlands containing Microplastics, Heavy metals, and Agricultural runoff were $130$ , $110$ , and $100$ , respectively.
- The number of wetlands containing Microplastics and Heavy metals but NOT Agricultural runoff was $25$ .
- The number of wetlands containing Heavy metals and Agricultural runoff but NOT Microplastics was $35$ .
- The number of wetlands containing Microplastics and Agricultural runoff but NOT Heavy metals was $20$ .

Based on the audit data, which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

Exactly

25

wetlands contain all three pollutants.

The number of wetlands containing only Microplastics is

60

.

The total number of wetlands containing exactly one pollutant is

85

.

The percentage of all surveyed wetlands that contain at least two pollutants is

50\%

.

The ratio of wetlands containing only Heavy metals to wetlands containing only Agricultural runoff is

5

to

4

.

Cevabı ve açıklamayı göster

Cevap: Exactly $25$ wetlands contain all three pollutants.; The number of wetlands containing only Microplastics is $60$ .; The ratio of wetlands containing only Heavy metals to wetlands containing only Agricultural runoff is $5$ to $4$ .

Cevap

The statements confirming that exactly 25 wetlands contain all three pollutants, that 60 wetlands contain only Microplastics, and that the ratio of wetlands containing only Heavy metals to only Agricultural runoff is 5 to 4 are all correct.

The correct options are those stating that 25 wetlands contain all three pollutants, that 60 wetlands contain only Microplastics, and that the ratio of Heavy metals only to Agricultural runoff only is 5 to 4. Each of these follows directly from setting up the inclusion-exclusion equation for three sets and determining all eight mutually exclusive regions of the Venn diagram.

Adım Adım Çözüm

1

Determine the number of wetlands containing at least one pollutant.

Total in union

|M \cup H \cup A| = 240 - 30 = 210

.

The total audited set is 240, and 30 sites have no pollutants.

2

Apply the Principle of Inclusion-Exclusion for three sets.

|M \cup H \cup A| = |M| + |H| + |A| - (|M \cap H| + |H \cap A| + |A \cap M|) + |M \cap H \cap A|

, which simplifies to

210 = 130 + 110 + 100 - S_2 + x \implies S_2 - x = 130

, where

S_2

is the sum of pairwise intersections and

x

is the triple intersection.

This relates the total union to the individual set sizes and intersection regions.

3

Express

S_2

in terms of the given 'exactly two' regions and solve for

x

.

The number of wetlands with exactly two pollutants is

25 + 35 + 20 = 80

. Since

S_2 - 3x = 80

, substituting

S_2 = 130 + x

gives

(130 + x) - 3x = 80 \implies 2x = 50 \implies x = 25

.

The sum of pairwise overlaps counts the triple intersection three times, so subtracting

3x

yields the 'exactly two' region.

4

Calculate single-category region sizes.

Microplastics only =

130 - (25 + 20 + 25) = 60

; Heavy metals only =

110 - (25 + 35 + 25) = 25

; Agricultural runoff only =

100 - (20 + 35 + 25) = 20

. Total single-category =

60 + 25 + 20 = 105

.

Subtracting all dual and triple overlap counts from each set total isolates the exclusive membership.

5

Verify each choice statement against the calculated region counts.

Triple intersection is 25 (True). Microplastics only is 60 (True). Exactly one pollutant total is 105, not 85 (False). Percentage of total sample with at least two pollutants is

105 / 240 = 43.75\%

, not

50\%

(False). Ratio of Heavy metals only to Agricultural runoff only is

25:20 = 5:4

(True).

Validates exact values against option assertions.

Anahtar Kavram

Three-set Principle of Inclusion-Exclusion and Venn diagram region decomposition

Soru 1658Soru →

In the $xy$ -plane, line $\ell_1$ passes through the points $(2, k)$ and $(k, 14)$ , where $k$ is a constant. Line $\ell_2$ passes through the point $(k, 14)$ and has a $y$ -intercept at $(0, 22)$ . If line $\ell_1$ is perpendicular to line $\ell_2$ , what is the sum of all possible values of $k$ ?

-22

-14

-6

6

8

Cevabı ve açıklamayı göster

Cevap: -6

Cevap

The sum of all possible values of

k

is -6.

The correct answer is -6 because the slope of line

\ell_1

is

m_1 = \frac{14-k}{k-2}

and the slope of line

\ell_2

is

m_2 = -\frac{8}{k}

. Since the lines are perpendicular, their slopes multiply to

-1

, yielding

\frac{8(14-k)}{k(k-2)} = 1

. Solving the resulting quadratic equation

k^2 + 6k - 112 = 0

yields

k = 8

and

k = -14

. The sum of these values is

8 + (-14) = -6

.

Adım Adım Çözüm

1

Calculate the slope of line

\ell_1

in terms of

k

.

m_1 = \frac{14 - k}{k - 2}

for

k \neq 2

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Calculate the slope of line

\ell_2

using

(k, 14)

and the

y

-intercept

(0, 22)

.

m_2 = \frac{14 - 22}{k - 0} = -\frac{8}{k}

for

k \neq 0

.

The

y

-intercept

(0, 22)

provides a second point on line

\ell_2

to find its slope.

3

Set up the perpendicularity condition

m_1 \cdot m_2 = -1

.

\left(\frac{14 - k}{k - 2}\right)\left(-\frac{8}{k}\right) = -1 \implies \frac{8(14 - k)}{k(k - 2)} = 1$.

Two non-vertical lines are perpendicular if and only if the product of their slopes equals

-1

.

4

Expand and rearrange the equation into standard quadratic form.

112 - 8k = k^2 - 2k \implies k^2 + 6k - 112 = 0.

Multiplying both sides by

k(k - 2)

clears the denominator to form a quadratic equation.

5

Factor the quadratic equation to find all possible values of

k

.

(k + 14)(k - 8) = 0 \implies k = -14 \text{ or } k = 8.

The quadratic expression factors neatly, giving two valid non-zero values for

k

.

6

Calculate the sum of all possible values of

k

.

(-14) + 8 = -6.

Summing the two solutions gives the required final numerical value.

Anahtar Kavram

Perpendicular Slopes and Quadratic Line Equations

Tahmini Süre:2m 30s

Soru 1659Soru →

A parabola defined by the quadratic function $f(x) = ax^2 + bx + c$ , where $a, b,$ and $c$ are real constants with $a > 0$ , has its vertex at a minimum value of $-16$ . The distance between the two $x$ -intercepts of the parabola is $8$ . If $f(1) = -7$ and the $x$ -coordinate of the vertex is positive, what is the value of $f(-2)$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The value of

f(-2)

is 20.

By converting the parabola into vertex form

f(x) = a(x - h)^2 - 16

, the

x

-intercepts are found at

h \pm \frac{4}{\sqrt{a}}

. Equating their difference to

8

yields

a = 1

. Substituting

f(1) = -7

gives

(1 - h)^2 = 9

, which yields

h = 4

under the condition

h > 0

. Evaluating

f(-2) = (-2 - 4)^2 - 16

produces

20

.

Adım Adım Çözüm

1

Express the quadratic function in vertex form using the minimum value

f(x) = a(x - h)^2 - 16

, where

(h, -16)

is the vertex.

Since

a > 0

, the parabola opens upwards and its minimum value occurs at the vertex

y

-coordinate,

k = -16

.

2

Determine the leading coefficient

a

using the distance between

x

-intercepts

a = 1

Setting

f(x) = 0

yields

a(x - h)^2 - 16 = 0 \implies (x - h)^2 = \frac{16}{a} \implies x = h \pm \frac{4}{\sqrt{a}}

. The distance between roots is

\frac{8}{\sqrt{a}} = 8

, which gives

\sqrt{a} = 1 \implies a = 1

.

3

Determine the vertex

x

-coordinate

h

using the point

f(1) = -7

h = 4

Substituting

a = 1

and

x = 1

into the vertex form gives

(1 - h)^2 - 16 = -7 \implies (1 - h)^2 = 9

. Taking square roots gives

1 - h = 3 \implies h = -2

or

1 - h = -3 \implies h = 4

. Since

h > 0

, we select

h = 4

.

4

Evaluate

f(-2)

using the fully specified function

f(-2) = 20

With

f(x) = (x - 4)^2 - 16

, substituting

x = -2

yields

f(-2) = (-2 - 4)^2 - 16 = (-6)^2 - 16 = 36 - 16 = 20

.

Anahtar Kavram

Quadratic Vertex Form, Root Separation, and Evaluation

Soru 1660Soru →

For all real numbers $x$ and $y$ , the custom binary operation $\odot$ is defined by $x \odot y = x + y - xy$ . Which of the following statements must be true for all real numbers $a$ , $b$ , and $c$ ? Select all such statements.

Geçerli olan tümünü seçin

a \odot b = b \odot a

a \odot 1 = 1

a \odot 0 = 0

(a \odot b) \odot c = a \odot (b \odot c)

a \odot a = a^2

Cevabı ve açıklamayı göster

Cevap: $a \odot b = b \odot a$ ; $a \odot 1 = 1$ ; $(a \odot b) \odot c = a \odot (b \odot c)$

Cevap

The statements

a \odot b = b \odot a

,

a \odot 1 = 1

, and

(a \odot b) \odot c = a \odot (b \odot c)

must be true for all real numbers

a

,

b

, and

c

.

The custom operation is commutative (

a \odot b = b \odot a

), maps any real number operated with

1

to

1

(

a \odot 1 = 1

), and is associative (

(a \odot b) \odot c = a \odot (b \odot c)

). These three properties hold universally for all real numbers.

Adım Adım Çözüm

1

Evaluate

a \odot b

and

b \odot a

to test commutativity.

a \odot b = a + b - ab

and

b \odot a = b + a - ba

. Since addition and multiplication of real numbers are commutative,

a + b - ab = b + a - ba

.

Verify if order of operands alters the result.

2

Evaluate

a \odot 1

using the operation definition.

a \odot 1 = a + 1 - a(1) = a + 1 - a = 1

.

Test the behavior of operating with the constant

1

.

3

Evaluate

a \odot 0

to verify the zero property statement.

a \odot 0 = a + 0 - a(0) = a

. This equals

a

, not

0

for general values of

a

.

Check if operating with

0

results in

0

.

4

Evaluate both sides of

(a \odot b) \odot c = a \odot (b \odot c)

to test associativity.

Left side:

(a \odot b) \odot c = (a + b - ab) \odot c = (a + b - ab) + c - (a + b - ab)c = a + b + c - ab - ac - bc + abc

. Right side:

a \odot (b \odot c) = a \odot (b + c - bc) = a + (b + c - bc) - a(b + c - bc) = a + b + c - bc - ab - ac + abc

. Both sides are identical.

Verify whether grouping alters the result.

5

Evaluate

a \odot a

.

a \odot a = a + a - a(a) = 2a - a^2

. This is not equal to

a^2

except when

2a - a^2 = a^2 \implies 2a^2 - 2a = 0 \implies a=0

or

a=1

.

Determine if the self-operation yields

a^2

for all real numbers.

Anahtar Kavram

Evaluating algebraic properties (commutativity, identity, associativity) of custom binary operations.

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