Tüm alıştırma soruları

Soru 81Soru →

In urban ecosystems, green roofs offer a sustainable method to mitigate the heat island effect by cooling densely populated areas. By replacing conventional dark asphalt shingles with living vegetation, city planners can successfully lower ambient air temperatures, decrease stormwater runoff, and ________ solar radiation absorption. These ecological installations not only lower energy costs for individual buildings but also enhance local biodiversity by providing habitats for native pollinators.

Which choice completes the text so that it conforms to the conventions of Standard English?

minimize

minimizing

minimizes

to minimize

Cevabı ve açıklamayı göster

Cevap: minimize

Cevap

minimize

The correct option is 'minimize'. The sentence presents a list of actions that city planners 'can' perform: 'lower ambient air temperatures,' 'decrease stormwater runoff,' and the blank. Because 'lower' and 'decrease' are base verbs (bare infinitives) following the helper verb 'can', the final element must also be a base verb ('minimize') to maintain parallel structure.

Adım Adım Çözüm

1

Identify the list of items in the sentence and determine the grammatical structure of the established items.

The sentence presents a list of three actions that city planners 'can' take: 'lower ambient air temperatures,' 'decrease stormwater runoff,' and the blank. Both 'lower' and 'decrease' are base verbs (bare infinitives) following the auxiliary verb 'can'.

Parallel structure requires that items in a list or comparison share the same grammatical form.

2

Select the option that matches the established grammatical structure of base verb forms.

The base verb 'minimize' completes the list in a parallel manner ('can lower... decrease... and minimize').

This maintains the grammatical consistency of the list.

Anahtar Kavram

Parallel structure in lists requires all items to have the same grammatical form.

Soru 82Soru →

While preparing a presentation on urban sociology, a student takes the following notes:
* In 2024, sociologist Dr. Marcus Vance studied community cohesion across various city neighborhoods.
* Vance measured community cohesion by tracking the annual frequency of neighborhood-organized events.
* Neighborhoods with at least two public parks averaged twelve neighborhood-organized events per year.
* Neighborhoods with fewer than two public parks averaged only four neighborhood-organized events per year.
* Vance concluded that access to public green spaces directly fosters community engagement.

The student wants to support Dr. Vance's conclusion using information from the notes. Which choice most effectively uses information from the given notes to accomplish this goal?

In 2024, sociologist Dr. Marcus Vance conducted a study that tracked the annual frequency of neighborhood-organized events in city neighborhoods.

Neighborhoods with at least two public parks averaged twelve events per year compared to just four in neighborhoods with fewer parks, supporting Vance's conclusion that green spaces foster community engagement.

Because public parks offer residents free areas to exercise and socialize, neighborhoods with multiple parks have much higher levels of community engagement.

Vance concluded that public green spaces foster engagement after finding that neighborhoods with fewer than two public parks averaged twelve events per year.

Cevabı ve açıklamayı göster

Cevap: Neighborhoods with at least two public parks averaged twelve events per year compared to just four in neighborhoods with fewer parks, supporting Vance's conclusion that green spaces foster community engagement.

Cevap

The statement comparing the average number of events in neighborhoods with at least two parks to those with fewer parks, linking this data to Vance's conclusion that green spaces foster community engagement.

The correct option effectively uses the data from the notes (the comparison of twelve events in neighborhoods with at least two parks versus four events in neighborhoods with fewer parks) to support Vance's conclusion that green spaces foster community engagement.

Adım Adım Çözüm

1

Identify the goal of the student's presentation.

The goal is to support Dr. Marcus Vance's conclusion that access to public green spaces directly fosters community engagement.

This is the specific claim the prompt requires the student to support.

2

Examine the notes to find evidence that supports this conclusion.

The notes show a contrast between neighborhoods with at least two parks (averaging twelve events per year) and those with fewer than two parks (averaging only four events per year).

This comparative data provides the empirical basis for Vance's conclusion.

3

Evaluate the options to find the one that combines this specific evidence with the conclusion.

The correct choice highlights the comparison in event frequency and directly connects it to Vance's conclusion about green spaces and community engagement.

It fulfills the rhetorical goal using only the provided facts.

Anahtar Kavram

Rhetorical Synthesis: Supporting Claims

Soru 83Soru →

\begin{aligned} y &= x^2 - 8x + c \\ y &= 2x - 5 \end{aligned}

In the system of equations above, $c$ is a constant. If the system has two real solutions, $(x_1, y_1)$ and $(x_2, y_2)$ , such that the product of the $y$ -coordinates of the solutions, $y_1 y_2$ , is equal to $5$ , what is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The value of the constant

c

is

15

.

Substituting the expression for

y

from the linear equation into the quadratic equation yields the single variable quadratic equation

x^2 - 10x + (c+5) = 0

. Using Vieta's formulas, the sum of the roots is

x_1 + x_2 = 10

and the product of the roots is

x_1 x_2 = c+5

. Substituting these relationships into the expanded product of the

y

-coordinates,

y_1 y_2 = (2x_1 - 5)(2x_2 - 5) = 4x_1 x_2 - 10(x_1 + x_2) + 25

, allows us to set up the equation

5 = 4(c+5) - 100 + 25

. Solving for

c

yields

15

. Checking the discriminant of the quadratic equation at

c=15

gives

100 - 80 = 20

, which is positive, confirming the existence of two distinct real solutions.

Adım Adım Çözüm

1

Substitute the expression for

y

from the second equation into the first equation.

2x - 5 = x^2 - 8x + c \implies x^2 - 10x + (c+5) = 0

This substitution reduces the system to a single quadratic equation whose roots,

x_1

and

x_2

, represent the

x

-coordinates of the intersection points.

2

Apply Vieta's formulas to the resulting quadratic equation.

x_1 + x_2 = 10

and

x_1 x_2 = c+5

Vieta's formulas state that for a quadratic equation

ax^2 + bx + d = 0

, the sum of the roots is

-\frac{b}{a}

and the product of the roots is \frac{d}{a}.

3

Express the product of the

y

-coordinates,

y_1 y_2

, in terms of

x_1

and

x_2

using the linear relationship.

y_1 y_2 = (2x_1 - 5)(2x_2 - 5) = 4x_1 x_2 - 10(x_1 + x_2) + 25

Since both intersection points lie on the line

y = 2x - 5

, we can substitute

y_1 = 2x_1 - 5

and

y_2 = 2x_2 - 5

and expand.

4

Substitute the Vieta's formulas relations into the product equation and solve for

c

.

5 = 4(c + 5) - 10(10) + 25 \implies 5 = 4c + 20 - 100 + 25 \implies 5 = 4c - 55 \implies 60 = 4c \implies c = 15

By substituting the known values of

(x_1 + x_2)

and

(x_1 x_2)

and setting the product

y_1 y_2

to

5

, we obtain a linear equation in terms of

c

that we can solve directly.

Anahtar Kavram

Solving systems of linear-quadratic equations using algebraic substitution and Vieta's formulas.

Soru 84Soru →

A landscaping company uses the equation $12x + 18y = 240$ to model the total cost, in dollars, of renting a wood chipper for $x$ hours and a soil aerator for $y$ hours. If the wood chipper was rented for 5 more hours than the soil aerator, and the company spent a total of $240$ on these rentals, for how many hours was the soil aerator rented?

5

6

10

11

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The soil aerator was rented for 6 hours.

The correct answer is 6. By writing the relationship between the rental hours as

x = y + 5

and substituting this expression into the cost equation, we obtain

12(y + 5) + 18y = 240

. Distributing the 12 results in

12y + 60 + 18y = 240

. Combining the like terms gives

30y + 60 = 240

. Subtracting 60 from both sides of the equation yields

30y = 180

. Dividing both sides by 30 results in

y = 6

. Therefore, the soil aerator was rented for 6 hours.

Adım Adım Çözüm

1

Express the relationship between the rental hours of the wood chipper (

x

) and the soil aerator (

y

) as an equation.

x = y + 5

The wood chipper was rented for 5 more hours than the soil aerator.

2

Substitute the expression for

x

into the cost equation

12x + 18y = 240

.

12(y + 5) + 18y = 240

Substitution reduces the equation to a single variable, allowing us to solve for

y

.

3

Distribute the 12 and combine like terms.

12y + 60 + 18y = 240 \implies 30y + 60 = 240

Simplification isolates the variable terms on one side.

4

Isolate

y

by subtracting 60 from both sides and then dividing by 30.

30y = 180 \implies y = 6

This calculation yields the rental hours for the soil aerator.

Anahtar Kavram

Solving a system of linear equations in two variables where one equation represents a budget constraint and the other relates the quantities of the two variables.

Soru 85Soru →

Many astronomers believe that the presence of liquid water is the single most important factor in determining a planet's habitability. However, researcher Dr. Elena Rostova argues that a planet’s magnetic field is equally critical. Without a robust magnetic field, solar winds can strip away a planet’s atmosphere, preventing liquid water from ever accumulating on the surface. Thus, Rostova’s team focuses their search on exoplanets that exhibit signs of strong magnetospheres.

Which choice best describes the function of the underlined sentence in the text as a whole?

It summarizes the main finding that Rostova's team reached at the conclusion of their research.

It introduces a potential flaw in the methods Rostova's team uses to locate exoplanets.

It explains the process by which a weak magnetic field prevents a planet from maintaining liquid water.

It describes a rare scenario in which liquid water is not necessary for a planet to support life.

Cevabı ve açıklamayı göster

Cevap: It explains the process by which a weak magnetic field prevents a planet from maintaining liquid water.

Cevap

It explains the process by which a weak magnetic field prevents a planet from maintaining liquid water.

The correct option correctly identifies that the underlined sentence describes the mechanism (atmosphere stripping by solar winds) that prevents liquid water accumulation when a magnetic field is absent. This explanation supports Rostova's argument that a magnetic field is critical to habitability.

Adım Adım Çözüm

1

Analyze the role of the underlined sentence in context.

The underlined sentence states that without a magnetic field, solar winds strip the atmosphere and prevent water from accumulating.

This helps identify the logical relationship between the underlined sentence and the surrounding sentences.

2

Connect the sentence to the main claim of the passage.

Dr. Rostova claims that a magnetic field is critical; the underlined sentence explains why by showing the consequence of not having one.

This establishes the rhetorical function of the sentence as supporting the researcher's argument.

3

Evaluate the choices to find the one that matches this function.

The option stating that it explains how a weak magnetic field prevents water retention matches the analysis, whereas other options mischaracterize the text or speculate beyond it.

This confirms the correct option.

Anahtar Kavram

Identifying the rhetorical function of a specific sentence within a passage, particularly how it supports a main claim.

Soru 86Soru →

A quadratic function $g$ is defined by $g(x) = 3(x - 5)^2 - 4$ . If the graph of $y = g(x)$ in the $xy$ -plane is translated $6$ units up to produce the graph of $y = f(x)$ , what is the $y$ -coordinate of the vertex of the graph of $y = f(x)$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct answer is 2.

The original function

g(x) = 3(x - 5)^2 - 4

is in vertex form

y = a(x - h)^2 + k

, where the vertex is

(h, k)

. Therefore, the vertex of the graph of

g

is

(5, -4)

. Translating a graph upward by

6

units is represented by adding

6

to the function, so

f(x) = g(x) + 6

. This transformation shifts the vertex from

(5, -4)

to

(5, -4 + 6)

, which is

(5, 2)

. The

y

-coordinate of this new vertex is

2

.

Adım Adım Çözüm

1

Identify the vertex of the original quadratic function.

The vertex of the graph of

g(x) = 3(x - 5)^2 - 4

is

(5, -4)

.

A quadratic function written in vertex form

y = a(x - h)^2 + k

has its vertex at

(h, k)

.

2

Determine the vertex of the translated function.

The vertex of the graph of

y = f(x)

is

(5, 2)

.

Translating a graph

6

units up increases the

y

-coordinate of all points, including the vertex, by

6

.

Anahtar Kavram

Vertex form of a quadratic function and vertical translation of graphs.

Tahmini Süre:45s

Soru 87Soru →

A commercial agricultural irrigation system distributes liquid fertilizer from a storage tank. The volume of fertilizer remaining in the tank, $V$ , in liters, $m$ minutes after the system is turned on is modeled by the equation:

V = 1,800 - 15(m - 3c)

where $c$ is the number of times the system's nozzles are cleaned during the irrigation process. During each cleaning cycle, the flow of fertilizer is completely paused, and no fertilizer is distributed. Based on the model, what is the duration, in minutes, of a single cleaning cycle?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The duration of a single cleaning cycle is 3 minutes.

In the model

V = 1,800 - 15(m - 3c)

, the term

(m - 3c)

represents the total time, in minutes, that the irrigation system is actively distributing fertilizer. Since the total elapsed time is

m

minutes and the system pauses during cleaning cycles, the term

3c

represents the total paused time for

c

cleanings. Therefore, the duration of a single cleaning cycle is

\frac{3c}{c} = 3

minutes.

Adım Adım Çözüm

1

Analyze the structure of the equation to identify the meaning of each term.

The coefficient 15 is the active flow rate in liters per minute, and

(m - 3c)

is the active distribution time in minutes.

To understand how the time variables affect the volume of fertilizer remaining.

2

Relate the total elapsed time to the active time and the paused time.

The total elapsed time is

m

minutes, and the active time is

m - 3c

minutes, meaning the system is paused for a total of

3c

minutes.

To find the expression for the total duration of all cleaning cycles.

3

Determine the duration of a single cleaning cycle.

Since

c

cleaning cycles result in a total pause of

3c

minutes, each cycle lasts

\frac{3c}{c} = 3

minutes.

To calculate the duration of one individual cleaning cycle.

Anahtar Kavram

Interpreting Linear Relationships in Context

Soru 88Soru →

In the $xy$ -plane, the graph of the quadratic function $f(x) = ax^2 + bx + c$ , where $a > 0$ , has vertex $V$ . The graph intersects the $x$ -axis at points $A$ and $B$ . If triangle $VAB$ is an equilateral triangle with an area of $12\sqrt{3}$ , what is the value of $a$ ?

\frac{1}{8}

\frac{1}{2}

1

2

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{2}$

Cevap

The correct answer is the value 1/2.

The value 1/2 is correct because the area of the equilateral triangle determines its side length to be

4\sqrt{3}

and its height to be

6

. Since the parabola opens upwards and its base lies on the

x

-axis, the vertex is

V(h, -6)

. The two

x

-intercepts are symmetric about the axis of symmetry

x = h

, meaning they are located at

h \pm 2\sqrt{3}

. Substituting one of these points into the vertex form

f(x) = a(x-h)^2 - 6

yields

0 = 12a - 6

, which gives

a = 1/2

.

Adım Adım Çözüm

1

Determine the side length and height of the equilateral triangle from the given area.

The side length

s

is

4\sqrt{3}

, and the height of the triangle is

6

.

The area of an equilateral triangle with side length

s

is given by the formula

\text{Area} = \frac{s^2\sqrt{3}}{4}

. Setting this equal to

12\sqrt{3}

yields

s^2 = 48

, which means

s = \sqrt{48} = 4\sqrt{3}

. The height of the triangle is given by

h_{\text{tri}} = \frac{s\sqrt{3}}{2} = \frac{4\sqrt{3} \cdot \sqrt{3}}{2} = 6

.

2

Relate the height and side length of the triangle to the vertex and

x

-intercepts of the parabola.

The vertex of the parabola is

V(h, -6)

and the

x

-intercepts are at

x = h \pm 2\sqrt{3}

.

Because

a > 0

, the parabola opens upwards, meaning the vertex

V(h, k)

lies below the

x

-axis. Since the base of the triangle lies on the

x

-axis, the vertical distance from the vertex to the

x

-axis is the height of the triangle, so

k = -6

. The axis of symmetry

x = h

bisects the base

AB

, so the

x

-intercepts are located at a distance of half the side length,

\frac{s}{2} = 2\sqrt{3}

, to the left and right of

h

.

3

Write the quadratic function in vertex form and substitute one of the

x

-intercepts to solve for

a

.

a = \frac{1}{2}

The vertex form of the quadratic function is

f(x) = a(x - h)^2 - 6

. Substituting the

x

-intercept

(h + 2\sqrt{3}, 0)

into this equation gives

0 = a(h + 2\sqrt{3} - h)^2 - 6

, which simplifies to

0 = a(2\sqrt{3})^2 - 6 \implies 12a = 6 \implies a = \frac{1}{2}

.

Anahtar Kavram

Using the symmetry properties and vertex form of a quadratic function to relate the geometric features of its graph to its algebraic coefficients.

Alternatif Yöntem

Alternatively, you can use the relationship between the roots and coefficients. For any quadratic equation

ax^2 + bx + c = 0

with real roots, the distance between the roots is

\frac{\sqrt{b^2 - 4ac}}{|a|}

. Since the triangle is equilateral, the height is

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

times this distance. The vertex is at

(-\frac{b}{2a}, -\frac{b^2-4ac}{4a})

. Setting the absolute value of the vertex

y

-coordinate equal to the height allows you to solve for

a

directly when combined with the area formula.

Tahmini Süre:3m 0s

Soru 89Soru →

If $\frac{8}{x - 3} = 2$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

To solve the equation

\frac{8}{x - 3} = 2

, we clear the denominator by multiplying both sides of the equation by

x - 3

. This gives

8 = 2(x - 3)

. Distributing the

2

results in

8 = 2x - 6

. Adding

6

to both sides yields

14 = 2x

. Dividing both sides by

2

gives the solution

x = 7

.

Adım Adım Çözüm

1

Multiply both sides of the equation by

x - 3

to eliminate the fraction.

8 = 2(x - 3)

To clear the denominator and simplify the equation.

2

Distribute the 2 to the terms inside the parentheses.

8 = 2x - 6

To expand the right-hand side of the equation.

3

Add 6 to both sides of the equation.

14 = 2x

To isolate the term with the variable

x

.

4

Divide both sides by 2 to solve for

x

.

x = 7

To find the final value of the variable.

Anahtar Kavram

Solving a rational equation that simplifies to a linear equation by multiplying by the common denominator.

Soru 90Soru →

A sample of a radioactive isotope decays such that its mass, in grams, is modeled by the function $M(t) = A \cdot d^{\frac{t}{12}}$ , where $t$ is the time in hours since the measurement began, and $A$ and $d$ are positive constants. The mass of the sample decreases by $75\%$ every $24$ hours. If the mass of the sample is $15$ grams when $t = 36$ , what was the initial mass, in grams, of the sample?

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

The initial mass of the sample was 120 grams.

The correct answer is 120. A decrease of

75\%

over 24 hours means the mass at

t+24

is

0.25

times the mass at

t

. According to the function,

M(t+24) = A \cdot d^{(t+24)/12} = A \cdot d^{t/12} \cdot d^2 = M(t) \cdot d^2

. Equating the two yields

d^2 = 0.25

, so

d = 0.5

. Substituting

t = 36

and

M(36) = 15

gives

15 = A \cdot (0.5)^{36/12} = A \cdot (0.5)^3 = 0.125A

. Solving for

A

gives

A = 120

.

Adım Adım Çözüm

1

Relate the 24-hour decay rate to the exponent in the function to set up an equation for

d

.

d^2 = 0.25

Every 24 hours (

t

increases by 24), the mass decreases by

75\%

, so it becomes

25\%

(

0.25

) of its previous value. The exponent increases by

24/12 = 2

, multiplying the mass by

d^2

.

2

Solve for the decay base

d

.

d = 0.5

Since

d

is a positive constant, taking the square root of

0.25

gives

0.5

.

3

Substitute the given mass at

t = 36

into the model to solve for the initial mass

A

.

A = 120

Plugging

t = 36

and

M(36) = 15

into

M(t) = A \cdot (0.5)^{t/12}

gives

15 = A \cdot (0.5)^3

, which simplifies to

15 = 0.125A

.

Anahtar Kavram

Interpreting and solving exponential decay functions with fractional exponents

Alternatif Yöntem

Instead of solving for

d

first, recognize that

36

hours is exactly

1.5

intervals of

24

hours. Since the mass is multiplied by

0.25

(or

\frac{1}{4}

) every

24

hours, after

36

hours it will be multiplied by

(\frac{1}{4})^{1.5} = (\frac{1}{4})^{3/2} = \frac{1}{8}

of its initial value. Therefore,

15 = A \cdot \frac{1}{8} \implies A = 120

.

Tahmini Süre:3m 0s

Soru 91Soru →

If the expression $\frac{3x^2 - bx + 10}{x - 2}$ is equivalent to $3x - c - \frac{12}{x - 2}$ for all $x \neq 2$ , where $b$ and $c$ are positive constants, what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

17

Multiplying both sides of the equivalence by

x - 2

results in

3x^2 - bx + 10 = (3x - c)(x - 2) - 12

. Expanding the right side gives

3x^2 - (6 + c)x + (2c - 12)

. By equating the coefficients of corresponding terms, we get

2c - 12 = 10

, which solves to

c = 11

. Substituting this into the

x

-coefficient equivalence

-b = -(6 + c)

gives

b = 6 + 11 = 17

.

Adım Adım Çözüm

1

Multiply both sides of the expression by the denominator

x - 2

to equate the numerators.

3x^2 - bx + 10 = (3x - c)(x - 2) - 12

Since the rational expressions are equivalent for all

x \neq 2

, their numerators must be equal when written over a common denominator.

2

Expand and simplify the right side of the equation.

3x^2 - bx + 10 = 3x^2 - (6 + c)x + (2c - 12)

Expanding

(3x - c)(x - 2)

yields

3x^2 - 6x - cx + 2c

, and subtracting

12

gives the simplified polynomial expression.

3

Equate the constant terms to solve for

c

.

10 = 2c - 12 \Rightarrow c = 11

For the polynomials to be equivalent, their constant terms must be equal.

4

Equate the coefficients of the

x

terms to solve for

b

.

b = 6 + c \Rightarrow b = 6 + 11 = 17

Equating the coefficients of

x

gives

-b = -(6 + c)

, which simplifies to

b = 6 + c

.

Anahtar Kavram

Equivalence of rational and polynomial expressions via coefficient comparison

Alternatif Yöntem

Alternatively, you can evaluate the equivalence at a convenient value of

x

. For instance, substituting

x = 0

into the expression gives

-5 = -c + 6

, yielding

c = 11

. Then, evaluating the equation at another convenient value such as

x = 1

allows you to solve for

b

directly using the now-known value of

c

.

Tahmini Süre:1m 30s

Soru 92Soru →

A survey of 140 students in the 9th and 10th grades asked about their preferred after-school activity. The results are summarized in the table below.

Grade	Sports	Arts	Total
9th Grade	40	20	60
10th Grade	30	50	80
Total	70	70	140

If a student is selected at random from the 9th-grade students, what is the probability that the selected student preferred sports?

\frac{2}{7}

\frac{1}{3}

\frac{2}{3}

\frac{4}{7}

Cevabı ve açıklamayı göster

Cevap: $\frac{2}{3}$

Cevap

\frac{2}{3}

The correct answer is

\frac{2}{3}

. The question asks for the probability that a student preferred sports, given that the student is in the 9th grade. This means we only consider the 60 students in the 9th grade. Of those 60 students, 40 preferred sports. Therefore, the probability is

\frac{40}{60}

, which simplifies to

\frac{2}{3}

.

Adım Adım Çözüm

1

Identify the group of interest from which the student is selected.

The selection is made from the 9th-grade students, so the denominator for the probability is the total number of 9th-grade students, which is

60

.

The problem states: 'If a student is selected at random from the 9th-grade students,' establishing this as a conditional probability where the sample space is restricted to 9th graders.

2

Identify the number of students within this group who satisfy the target condition.

The number of 9th-grade students who preferred sports is

40

.

We look at the intersection of the '9th Grade' row and the 'Sports' column in the table.

3

Calculate the probability by dividing the number of target outcomes by the total outcomes in the conditional sample space, and simplify the fraction.

The probability is

\frac{40}{60}

, which simplifies to

\frac{2}{3}

.

Dividing both the numerator and the denominator by their greatest common divisor, 20, gives the simplified probability.

Anahtar Kavram

Calculating conditional probability from a two-way frequency table by restricting the sample space to a specific row or column.

Soru 93Soru →

What is the set of all real solutions to the equation below?

\sqrt{3x+10} = x+2

\{-3, 2\}

\{2\}

\{-3\}

\{-2, 3\}

Cevabı ve açıklamayı göster

Cevap: $\{2\}$

Cevap

The set containing only 2

The correct answer is the set containing only 2. Squaring both sides of the equation yields

3x + 10 = x^2 + 4x + 4

. Rearranging the terms to set the equation to zero gives

x^2 + x - 6 = 0

. Factoring the quadratic expression gives

(x + 3)(x - 2) = 0

, indicating potential solutions of

x = -3

and

x = 2

. Substituting

x = 2

back into the original equation gives

\sqrt{16} = 4

, which is valid. Substituting

x = -3

back into the original equation gives

\sqrt{1} = -1

, which is invalid because the principal square root is non-negative. Thus,

x = -3

is extraneous and must be discarded.

Adım Adım Çözüm

1

Square both sides of the equation to eliminate the radical.

3x + 10 = (x + 2)^2

, which expands to

3x + 10 = x^2 + 4x + 4

.

Squaring both sides is the standard algebraic method to isolate the term under a square root.

2

Move all terms to one side to form a quadratic equation equal to zero.

x^2 + x - 6 = 0

.

Setting the quadratic expression to zero allows us to solve it by factoring.

3

Factor the quadratic equation.

(x + 3)(x - 2) = 0

, which gives the potential solutions

x = -3

and

x = 2

.

Finding the factors helps us determine the roots of the equation.

4

Substitute both potential solutions back into the original equation to check for extraneous solutions.

For

x = 2

:

\sqrt{3(2)+10} = \sqrt{16} = 4

and

2+2 = 4

, which is true. For

x = -3

:

\sqrt{3(-3)+10} = \sqrt{1} = 1

and

-3+2 = -1

, which is false.

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original equation.

Anahtar Kavram

Identifying extraneous solutions in radical equations

Tahmini Süre:1m 30s

Soru 94Soru →

If $x > 2$ and satisfies the equation

x - \frac{8}{x^2 - 2x} = 5 - \frac{4}{x - 2}

what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct answer is 4.

To solve the rational equation, we first determine the domain restrictions:

x \neq 0

and

x \neq 2

. We then clear the denominators by multiplying the entire equation by the least common denominator

x(x - 2)

, which simplifies the equation to the cubic form

x^3 - 7x^2 + 14x - 8 = 0

. Factoring the cubic polynomial yields

(x - 1)(x - 2)(x - 4) = 0

. The potential solutions are

x = 1

,

x = 2

, and

x = 4

. Since

x = 2

is an extraneous solution and the constraint requires

x > 2

,

x = 4

is the only valid solution.

Adım Adım Çözüm

1

Determine the domain restrictions of the rational terms in the equation.

x \neq 0

and

x \neq 2

Division by zero is undefined, so the denominators

x(x-2)

and

x-2

cannot be zero.

2

Multiply the entire equation by the least common denominator

x(x - 2)

to eliminate the fractions.

x^2(x - 2) - 8 = 5x(x - 2) - 4x

Multiplying by the common denominator converts the rational equation into an equivalent polynomial equation within the defined domain.

3

Expand the terms and collect them on one side to form a standard polynomial equation.

x^3 - 7x^2 + 14x - 8 = 0

Simplifying the expressions allows us to find the roots of the polynomial.

4

Factor the cubic polynomial using the rational root theorem or synthetic division.

(x - 1)(x - 2)(x - 4) = 0

Since

x = 1

makes the polynomial zero, we factor out

(x-1)

to get the remaining quadratic factor

x^2 - 6x + 8

, which factors into

(x - 2)(x - 4)

.

5

Identify the valid solution based on the domain restriction and the given inequality constraint.

x = 4

The value

x = 2

is extraneous because it makes the original equation undefined. The value

x = 1

is rejected because the problem specifies

x > 2

.

Anahtar Kavram

Solving rational equations by clearing denominators, factoring polynomials, identifying extraneous solutions, and applying inequality constraints.

Soru 95Soru →

What is the value of the real solution to the equation below?

\sqrt{3x - 2} - \sqrt{x + 3} = 1

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The only real solution to the equation is 6.

Substituting

x = 6

into the original equation yields

\sqrt{3(6) - 2} - \sqrt{6 + 3} = \sqrt{16} - \sqrt{9} = 4 - 3 = 1

, which is a true statement. Therefore, the only real solution is 6.

Adım Adım Çözüm

1

Isolate one of the radical terms

\sqrt{3x - 2} = \sqrt{x + 3} + 1

Preparing the equation to square both sides.

2

Square both sides of the equation

3x - 2 = x + 4 + 2\sqrt{x + 3}

Eliminating one of the square root radicals.

3

Isolate the remaining radical term and simplify

x - 3 = \sqrt{x + 3}

Simplifying the equation by isolating the second radical and dividing both sides by 2.

4

Square both sides again to eliminate the second radical

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

Converting the radical equation into a polynomial equation.

5

Write in standard quadratic form and factor

(x - 6)(x - 1) = 0

Setting the quadratic equation to zero and factoring to find potential solutions.

6

Verify solutions in the original equation to check for extraneous solutions

x = 6

is valid;

x = 1

is extraneous

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original equation.

Anahtar Kavram

Solving equations containing radical expressions and identifying extraneous solutions.

Tahmini Süre:2m 30s

Soru 96Soru →

A transit agency surveyed a sample of 160 commuters about their primary mode of transportation and whether they travel during peak commuting hours. The results are summarized in the table below.

Commute Mode	Peak Hours	Off-Peak Hours	Total
Bus	35	45	80
Train	55	25	80
Total	90	70	160

If a commuter who travels during peak hours is selected at random from those surveyed, what is the probability that the commuter's primary mode of transportation is the train?

\frac{11}{32}

\frac{11}{18}

\frac{11}{16}

\frac{7}{18}

Cevabı ve açıklamayı göster

Cevap: $\frac{11}{18}$

Cevap

The probability that the commuter's primary mode of transportation is the train, given they travel during peak hours, is

\frac{11}{18}

.

The correct answer is

\frac{11}{18}

. To find the probability that a selected commuter's primary mode of transportation is the train, given that they travel during peak hours, we must look only at the column representing peak-hour commuters. The total number of peak-hour commuters is

35 + 55 = 90

. Among these commuters, the number whose primary mode of transportation is the train is

55

. Therefore, the probability is

\frac{55}{90}

, which simplifies to

\frac{11}{18}

.

Adım Adım Çözüm

1

Identify the total number of commuters who meet the condition of traveling during peak hours.

The total number of peak-hour commuters is

90

.

The question asks for the probability given that the commuter travels during peak hours, restricting our sample space to this column total.

2

Identify the number of commuters in the peak hours group whose primary mode of transportation is the train.

There are

55

commuters who travel by train during peak hours.

This represents the favorable outcomes within our restricted sample space.

3

Calculate the conditional probability by dividing the number of favorable outcomes by the size of the restricted sample space.

\frac{55}{90} = \frac{11}{18}

Probability is the ratio of favorable outcomes to the total possible outcomes under the given condition.

Anahtar Kavram

Conditional Probability from a Two-Way Table

Soru 97Soru →

In 2024, a city recycled a certain number of tons of plastic. In 2025, the number of tons of plastic recycled was $15\%$ greater than the number of tons recycled in 2024. In 2026, the number of tons of plastic recycled was $20\%$ less than the number of tons recycled in 2025. If the city recycled $5,520$ tons of plastic in 2026, how many tons of plastic did the city recycle in 2024?

Cevabı ve açıklamayı göster

Cevap: 6000

Cevap

The city recycled

6,000

tons of plastic in 2024.

To find the number of tons of plastic recycled in 2024, we must work backward from the 2026 value of 5,520 tons. Let

x

represent the tons recycled in 2025. Since the 2026 value represents a

20\%

decrease from 2025, we have

0.80x = 5,520

, which simplifies to

x = 6,900

tons. Next, let

y

represent the tons recycled in 2024. Since the 2025 value represents a

15\%

increase from 2024, we have

1.15y = 6,900

. Solving for

y

yields

y = \frac{6,900}{1.15} = 6,000

tons.

Adım Adım Çözüm

1

Set up an equation to find the amount of plastic recycled in 2025 using the 2026 amount of 5,520 tons.

The amount of plastic recycled in 2025 was

6,900

tons.

The 2026 amount is a

20\%

decrease from the 2025 amount, so

5,520

is equal to

80\%

of the 2025 amount. Solving

0.80x = 5,520

gives

x = 6,900

.

2

Set up an equation to find the amount of plastic recycled in 2024 using the 2025 amount of 6,900 tons.

The amount of plastic recycled in 2024 was

6,000

tons.

The 2025 amount is a

15\%

increase from the 2024 amount, so

6,900

is equal to

115\%

of the 2024 amount. Solving

1.15y = 6,900

gives

y = 6,000

.

Anahtar Kavram

Working backward through multiple sequential percentage changes.

Soru 98Soru →

The graph of the function $y = f(x)$ in the $xy$ -plane passes through the point $(3, -4)$ . If the function $g$ is defined by $g(x) = f(x + 2) - 1$ , which of the following points must lie on the graph of $y = g(x)$ ?

(1, -5)

(5, -5)

(1, -3)

(5, -3)

Cevabı ve açıklamayı göster

Cevap: $(1, -5)$

Cevap

(1, -5)

The correct answer is the point

(1, -5)

. Since the point

(3, -4)

lies on the graph of

f

, we know that

f(3) = -4

. For the function

g(x) = f(x + 2) - 1

, substituting

x = 1

gives

g(1) = f(1 + 2) - 1 = f(3) - 1

. Substituting

-4

for

f(3)

yields

g(1) = -4 - 1 = -5

. Thus, the point

(1, -5)

must lie on the graph of

g

.

Adım Adım Çözüm

1

Identify the given function value from the point on the graph of

f

.

Since the graph of

y = f(x)

passes through

(3, -4)

, we have

f(3) = -4

.

Any point

(x, y)

on the graph of a function satisfies the equation

y = f(x)

.

2

Determine the input variable

x

for the function

g(x) = f(x + 2) - 1

that corresponds to the known input of

3

for

f

.

Set the argument of

f

in

g(x)

equal to

3

:

x + 2 = 3

, which simplifies to

x = 1

.

This allows us to substitute the known value

f(3)

into the expression for

g(x)

.

3

Evaluate

g(1)

using the value of

f(3)

.

g(1) = f(1 + 2) - 1 = f(3) - 1 = -4 - 1 = -5

.

Substituting

f(3) = -4

into the simplified expression gives the output value of

g

at

x = 1

.

4

State the resulting point on the graph of

y = g(x)

.

The point is

(1, -5)

.

An input of

x = 1

yields an output of

y = -5

for the function

g

.

Anahtar Kavram

Function transformations and their effects on individual coordinate points.

Soru 99Soru →

A pharmaceutical company produces two strengths of a certain medication, Strength P and Strength Q, by mixing an active ingredient with a liquid base. In Strength P, the ratio of the volume of the active ingredient to the volume of the liquid base is $2$ to $5$ . In Strength Q, the ratio of the volume of the active ingredient to the volume of the liquid base is $3$ to $8$ . A technician prepares a batch of Strength P and a batch of Strength Q such that the volume of the liquid base used in the batch of Strength P is equal to the volume of the liquid base used in the batch of Strength Q. If the total volume of the active ingredient used in both batches combined is $62$ liters, what is the total volume of the liquid base, in liters, used in both batches combined?

Cevabı ve açıklamayı göster

Cevap: 160

Cevap

The total volume of the liquid base used in both batches combined is 160 liters.

The total volume of the liquid base used in both batches combined is 160 liters. To find this, let the volume of the liquid base in each batch be

L

liters. The volume of the active ingredient in Strength P is

\frac{2}{5}L

liters, and the volume of the active ingredient in Strength Q is

\frac{3}{8}L

liters. The sum of the active ingredients is

\frac{2}{5}L + \frac{3}{8}L = 62

liters. Finding a common denominator gives

\frac{16}{40}L + \frac{15}{40}L = 62

, which simplifies to

\frac{31}{40}L = 62

. Solving for

L

yields

L = 62 \times \frac{40}{31} = 80

liters. Since both batches contain equal volumes of liquid base, the combined total volume of the liquid base is

2L = 2 \times 80 = 160

liters.

Adım Adım Çözüm

1

Represent the volumes of the ingredients in terms of a single variable.

Let the volume of the liquid base used in each batch be

40x

liters, where 40 is chosen as the least common multiple of 5 and 8. The volume of the active ingredient in Strength P is

\frac{2}{5} \times 40x = 16x

liters, and the volume of the active ingredient in Strength Q is

\frac{3}{8} \times 40x = 15x

liters.

This establishes a common algebraic scale for both mixtures, utilizing the fact that the volume of the liquid base is identical in both batches.

2

Formulate an equation using the total volume of the active ingredient.

16x + 15x = 62

, which simplifies to

31x = 62

.

The problem states that the combined volume of the active ingredient from both batches is 62 liters.

3

Solve for the variable

x

.

x = 2

Dividing both sides of the equation by 31 yields the scaling factor.

4

Calculate the total volume of the liquid base used in both batches combined.

Total liquid base =

40x + 40x = 80x = 80(2) = 160

liters.

Since each batch uses

40x

liters of liquid base, the total volume for both batches is the sum of their individual base volumes, which is

80x

liters.

Anahtar Kavram

Solving ratio and proportion problems with a shared quantity

Tahmini Süre:2m 30s

Soru 100Soru →

In right triangle $PQR$ , the measure of angle $Q$ is $90^\circ$ . If $\tan(P) = \frac{4}{3}$ , what is the value of $\sin(P)$ ?

\frac{4}{5}

\frac{3}{5}

\frac{3}{4}

\frac{5}{4}

Cevabı ve açıklamayı göster

Cevap: $\frac{4}{5}$

Cevap

\frac{4}{5}

The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Given that

\tan(P) = \frac{4}{3}

, we can set the length of the side opposite to angle P as

4

and the adjacent side as

3

. Using the Pythagorean theorem, the hypotenuse is

\sqrt{3^2 + 4^2} = 5

. Since the sine of an angle is the ratio of the opposite side to the hypotenuse,

\sin(P) = \frac{4}{5}

.

Adım Adım Çözüm

1

Identify the relationship between the tangent ratio and the sides of the right triangle.

The tangent of angle P is defined as the opposite side divided by the adjacent side. Since

\tan(P) = \frac{4}{3}

, we can set the opposite side to

4

and the adjacent side to

3

.

This allows us to establish the relative lengths of the sides of the right triangle.

2

Calculate the length of the hypotenuse using the Pythagorean theorem.

The hypotenuse is

\sqrt{3^2 + 4^2} = \sqrt{25} = 5

.

The hypotenuse is required to calculate the sine ratio.

3

Determine the sine of angle P.

The sine of angle P is defined as the opposite side divided by the hypotenuse, which is

\frac{4}{5}

.

This yields the final requested trigonometric ratio.

Anahtar Kavram

Using basic trigonometric ratios (SOH CAH TOA) and the Pythagorean theorem to evaluate trigonometric ratios in a right triangle.

Tahmini Süre:45s

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