Tüm alıştırma soruları

Soru 41Soru →

A vertical flagpole stands on flat, horizontal ground. An observer at point $S$ on the ground measures the angle of elevation to the top of the flagpole to be $60^\circ$ . A second observer at point $P$ on the ground, located $30$ feet further from the base of the flagpole along the same line extending from the base through $S$ , measures the angle of elevation to the top of the flagpole to be $30^\circ$ . What is the height of the flagpole, in feet?

15\sqrt{3}

30\sqrt{3}

45

30

15

Cevabı ve açıklamayı göster

Cevap: $15\sqrt{3}$

Cevap

The height of the flagpole is

15\sqrt{3}

feet.

Let

B

be the base and

T

be the top of the flagpole. The right triangle

TBS

has angles

30^\circ

,

60^\circ

, and

90^\circ

at

B

. If

BS = x

, then the height

TB = x\sqrt{3}

. In right triangle

TBP

, the angle at

P

is

30^\circ

, so the base

BP = TB\sqrt{3} = (x\sqrt{3})\sqrt{3} = 3x

. Since

BP = BS + 30

, we set up

3x = x + 30

, yielding

x = 15

. Therefore, the height

TB = 15\sqrt{3}

feet.

Adım Adım Çözüm

1

Model the scenario using right triangles.

Let

B

be the base of the flagpole and

T

be the top. Triangle

TBS

is a

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

right triangle at

B

, and triangle

TBP

is also a

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

right triangle at

B

.

The flagpole is perpendicular to the horizontal ground.

2

Express side lengths in terms of base distance

x = BS

.

In

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

triangle

TBS

,

TB = x\sqrt{3}

and

ST = 2x

.

The side opposite the

60^\circ

angle is

\sqrt{3}

times the side opposite the

30^\circ

angle.

3

Set up the equation for triangle

TBP

.

In

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

triangle

TBP

, the side opposite the

30^\circ

angle is

TB = x\sqrt{3}

, so the adjacent side

BP = (x\sqrt{3})\sqrt{3} = 3x

.

The side opposite the

60^\circ

angle (

BP

) is

\sqrt{3}

times the side opposite the

30^\circ

angle (

TB

).

4

Solve for

x

using the given distance

SP = 30

.

Since

BP = BS + SP

, we have

3x = x + 30

, which simplifies to

2x = 30

, so

x = 15

.

The total horizontal distance

BP

is the sum of segments

BS

and

SP

.

5

Calculate the height

TB

.

TB = 15\sqrt{3}

.

Substitute

x = 15

into

TB = x\sqrt{3}

.

Anahtar Kavram

Special Right Triangle Ratios (

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

)

Tahmini Süre:1m 30s

Soru 42Soru →

Read the sentence below concerning astrobiological research. Based on the explicit continuation structural signal in the sentence, enter a word or phrase that logically completes the blank.

Aşağıdaki boşlukları doldurun

The discovery of intact organic molecules within subterranean basalt deposits did not merely suggest the possibility of microbial life in extreme environments; , by confirming active metabolic pathways miles beneath the seafloor, the study forced astrobiologists to broaden their criteria for planetary habitability.

Cevabı ve açıklamayı göster

Cevap

indeed (or an equivalent intensifying continuation transition such as 'in fact', 'furthermore', or 'moreover')

The sentence follows an escalating continuation pattern initiated by 'did not merely'. The second independent clause enhances the scope of the discovery from a mere possibility to a confirmed finding that alters scientific criteria. An intensifying continuation word such as 'indeed' or 'furthermore' correctly bridges the two clauses while maintaining logical coherence.

Adım Adım Çözüm

1

Analyze the structural construction of the sentence.

Identified the correlative escalating framework 'did not merely [X]; [blank], [Y]'.

The phrase 'did not merely suggest' sets up an intensifying continuation where the second clause provides stronger evidence or a more conclusive assertion.

2

Determine the required directional signal for the blank.

The blank requires a continuation/intensification transition.

The second clause reinforces and expands upon the first clause rather than contrasting with it or expressing concession.

3

Select a fitting transition word.

Words such as 'indeed', 'in fact', 'furthermore', or 'moreover' logically complete the causal continuation.

These terms correctly signal that the second statement builds upon and confirms the claim made in the first statement.

Anahtar Kavram

Interpreting Continuation and Causal Structural Signals

Tahmini Süre:1m 0s

Soru 43Soru →

A beverage mixture is prepared by combining apple juice, orange juice, and grape juice in a ratio of $3 : 4 : 5$ by volume. If a pitcher contains a total of $48$ fluid ounces of this mixture, how many fluid ounces of orange juice are in the pitcher?

12

fluid ounces

16

fluid ounces

20

fluid ounces

24

fluid ounces

36

fluid ounces

Cevabı ve açıklamayı göster

Cevap: $16$ fluid ounces

Cevap

16

fluid ounces

The ratio

3 : 4 : 5

indicates that the total mixture is divided into

3 + 4 + 5 = 12

equal parts. Orange juice accounts for

4

of these

12

parts, representing

\frac{4}{12} = \frac{1}{3}

of the total volume. Taking

\frac{1}{3}

of

48

fluid ounces gives

16

fluid ounces.

Adım Adım Çözüm

1

Calculate the total number of ratio parts.

Total parts =

3 + 4 + 5 = 12

.

To find the fraction of each ingredient in the mixture, we must sum all parts of the ratio.

2

Determine the fraction of the total volume that represents orange juice.

Orange juice fraction =

\frac{4}{12} = \frac{1}{3}

.

Orange juice corresponds to

4

parts out of the

12

total parts.

3

Multiply the orange juice fraction by the total volume.

\frac{1}{3} \times 48 = 16

fluid ounces.

Multiplying the part-to-whole fraction by the total volume yields the exact volume of orange juice.

Anahtar Kavram

Part-to-Whole Ratio Relationships

Tahmini Süre:45s

Soru 44Soru →

If $x$ and $y$ are real numbers such that $|2x - 5| \le 9$ and $|y + 3| \le 4$ , what is the maximum possible value of $|x - 2y + 1|$ ?

22

20

14

13

6

Cevabı ve açıklamayı göster

Cevap: $22$

Cevap

The maximum possible value of

|x - 2y + 1|

is 22.

The correct answer is 22 because solving the absolute value inequalities gives

-2 \le x \le 7

and

-7 \le y \le 1

. To maximize

x - 2y + 1

, we take the largest possible value of

x

(

7

), the smallest possible value of

y

(

-7

, making

-2y = 14

), and add

1

, obtaining

7 + 14 + 1 = 22

. Since the lower bound of

x - 2y + 1

is

-3

, the maximum absolute value is

|22| = 22

.

Adım Adım Çözüm

1

Find the range of possible values for

x

from the inequality

|2x - 5| \le 9

.

-9 \le 2x - 5 \le 9 \implies -4 \le 2x \le 14 \implies -2 \le x \le 7

.

An absolute value inequality

|A| \le B

unwraps to

-B \le A \le B

.

2

Find the range of possible values for

y

from the inequality

|y + 3| \le 4

.

-4 \le y + 3 \le 4 \implies -7 \le y \le 1

.

Unwrapping the absolute value inequality gives the upper and lower bounds for

y

on the real number line.

3

Determine the range of possible values for

-2y

.

Multiplying

-7 \le y \le 1

by

-2

and reversing the inequality signs yields

-2(1) \le -2y \le -2(-7) \implies -2 \le -2y \le 14

.

Multiplying an inequality by a negative number reverses the direction of the inequality signs.

4

Combine the ranges to find the minimum and maximum bounds for

z = x - 2y + 1

.

Minimum

z = (-2) + (-2) + 1 = -3

; Maximum

z = 7 + 14 + 1 = 22

. Thus,

-3 \le x - 2y + 1 \le 22

.

Adding the individual minimums gives the absolute minimum, and adding the individual maximums gives the absolute maximum.

5

Calculate the maximum value of the absolute value

|x - 2y + 1|

.

\\max(|x - 2y + 1|) = \\max(|-3|, |22|) = 22

.

The absolute value of a quantity ranging from

-3

to

22

reaches its maximum distance from zero at

22

.

Anahtar Kavram

Properties of Real Numbers, Number Line Inequalities, and Absolute Value Operations

Tahmini Süre:2m 30s

Soru 45Soru →

In analyzing the ecological impact of rapid industrial development, environmental scientists note that expanding infrastructure continues to exact a heavy toll on local biodiversity, demanding sacrifices from surrounding ecosystems that cannot easily be recovered.

Which of the following words is closest in meaning to the word 'exact' as it is used in the passage above?

impose

precise

measure

lessen

isolate

Cevabı ve açıklamayı göster

Cevap: impose

Cevap

impose

In the context of the passage, 'exact' is used as a verb meaning to demand, inflict, or levy (e.g., 'to exact a toll'). Substituting 'impose' preserves the intended meaning that infrastructure expansion forces or inflicts a severe cost on biodiversity.

Adım Adım Çözüm

1

Identify the target word and its grammatical role in the sentence

The word 'exact' follows the infinitive marker 'to' ('to exact a heavy toll'), indicating it functions as a verb, not an adjective.

Determining the part of speech filters out primary adjective definitions that do not fit the syntactical structure.

2

Analyze surrounding contextual clues

The clause 'demanding sacrifices from surrounding ecosystems' directly clarifies what 'exact a heavy toll' means.

The surrounding text provides an explicit restatement of the secondary verb meaning.

3

Substitute options to find the semantic match

Replacing 'exact' with 'impose' yields 'continues to impose a heavy toll', maintaining the exact meaning of demanding or inflicting a cost.

'Impose' correctly captures the secondary verb definition of requiring or inflicting something burdensome.

Anahtar Kavram

Secondary verb meanings of polysemous words in academic contexts

Soru 46Soru →

Although early biographers depicted the diplomat’s private journal as a purely (i)________ account designed to settle old scores, subsequent historical analysis reveals that the entries were surprisingly (ii)________, offering an extraordinarily balanced and objective record of the negotiations.

Which of the following pairs of terms best completes the passage in a logically coherent manner?

(i) rancorous, (ii) dispassionate

(i) rancorous, (ii) tendentious

(i) meticulous, (ii) dispassionate

(i) scrupulous, (ii) tendentious

(i) rancorous, (ii) superficial

Cevabı ve açıklamayı göster

Cevap: (i) rancorous, (ii) dispassionate

Cevap

The pair '(i) rancorous, (ii) dispassionate' correctly completes the text by establishing the required contrast between a bitter early portrayal and a balanced historical reality.

The sentence relies on a concessive transition ('Although') paired with internal elaboration clues. The descriptive phrase 'designed to settle old scores' points to Blank (i) needing a word meaning resentful or bitter ('rancorous'). In contrast, the second clause describes the entries as an 'extraordinarily balanced and objective record', requiring Blank (ii) to mean impartial or unbiased ('dispassionate'). Only the pair featuring 'rancorous' for Blank (i) and 'dispassionate' for Blank (ii) satisfies both semantic requirements concurrently.

Adım Adım Çözüm

1

Analyze the clue for Blank (ii)

The phrase 'offering an extraordinarily balanced and objective record' directly describes the second blank.

Blank (ii) must be a synonym or positive attribute corresponding to 'balanced' and 'objective', pointing to 'dispassionate'.

2

Identify structural transition and clue for Blank (i)

The initial contrast word 'Although' indicates that early biographers held a view opposite to the 'balanced and objective' reality, while 'designed to settle old scores' provides a specific negative clue for Blank (i).

Blank (i) requires a term indicating bitterness or hostility, pointing to 'rancorous'.

3

Verify inter-blank dependency and overall sentence coherence

Combining 'rancorous' and 'dispassionate' completes the contrast smoothly: early biographers saw it as a bitter account, but later analysis revealed it to be surprisingly objective.

Both blanks must work simultaneously to satisfy both the internal clause clues and the overarching concession structure ('Although...').

Anahtar Kavram

Multi-Blank Dependency Tracking

Soru 47Soru →

A laboratory mixture is created by combining two solutions containing a specific tracer compound. Solution X has a volume of $2.5 \times 10^3$ milliliters and a tracer concentration of $4.8 \times 10^{-7}$ grams per milliliter. Solution Y has a volume of $7.5 \times 10^3$ milliliters and a tracer concentration of $1.6 \times 10^{-6}$ grams per milliliter. What is the total mass, in grams, of the tracer compound present in the combined mixture?

1.32 \times 10^{-2}

2.4 \times 10^{-3}

1.32 \times 10^{-5}

2.4 \times 10^{-2}

1.32 \times 10^{-1}

Cevabı ve açıklamayı göster

Cevap: $1.32 \times 10^{-2}$

Cevap

1.32 \times 10^{-2}

grams

The mass in Solution X is

(2.5 \times 10^3) \times (4.8 \times 10^{-7}) = 1.2 \times 10^{-3}

grams, and the mass in Solution Y is

(7.5 \times 10^3) \times (1.6 \times 10^{-6}) = 1.2 \times 10^{-2}

grams. Rewriting

1.2 \times 10^{-3}

as

0.12 \times 10^{-2}

allows direct addition:

(0.12 + 1.2) \times 10^{-2} = 1.32 \times 10^{-2}

grams.

Adım Adım Çözüm

1

Calculate the mass of the tracer in Solution X

M_X = (2.5 \times 10^3) \times (4.8 \times 10^{-7}) = 12.0 \times 10^{-4} = 1.2 \times 10^{-3}

grams

Mass equals volume multiplied by concentration.

2

Calculate the mass of the tracer in Solution Y

M_Y = (7.5 \times 10^3) \times (1.6 \times 10^{-6}) = 12.0 \times 10^{-3} = 1.2 \times 10^{-2}

grams

Mass equals volume multiplied by concentration.

3

Sum the tracer masses and adjust to scientific notation

M_{\text{total}} = 1.2 \times 10^{-3} + 1.2 \times 10^{-2} = 0.12 \times 10^{-2} + 1.2 \times 10^{-2} = 1.32 \times 10^{-2}

grams

To add terms in scientific notation, convert them to have matching exponents before adding coefficients.

Anahtar Kavram

Arithmetic operations with Scientific Notation and Decimals

Tahmini Süre:1m 30s

Soru 48Soru →

If $x + y = 8$ and $2x - y = 7$ , what is the value of $x$ ?

5

3

1

15

11

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

Adding the two equations yields

(x + y) + (2x - y) = 8 + 7

, which simplifies to

3x = 15

. Dividing both sides by 3 gives the correct value,

x = 5

.

Adım Adım Çözüm

1

Add the two equations together to eliminate the variable

y

.

(x + y) + (2x - y) = 8 + 7 \implies 3x = 15

Since the coefficients of

y

are

+1

and

-1

, adding the equations eliminates

y

directly.

2

Solve for

x

by dividing both sides of the equation by 3.

x = \frac{15}{3} = 5

Isolating

x

gives the required solution.

Anahtar Kavram

Elimination Method in Systems of Linear Equations

Soru 49Soru →

If $\frac{2x - 3}{4} - \frac{x - 5}{3} = \frac{x + 15}{12}$ , what is the value of $3x - 2$ ?

10

4

37

55

130

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The value of

3x - 2

is

10

.

Multiplying the given linear equation by the least common denominator 12 yields

3(2x - 3) - 4(x - 5) = x + 15

. Expanding both sides gives

6x - 9 - 4x + 20 = x + 15

, which simplifies to

2x + 11 = x + 15

, leading to

x = 4

. Substituting

x = 4

into

3x - 2

gives

3(4) - 2 = 10

.

Adım Adım Çözüm

1

Clear the denominators by multiplying both sides of the equation by the least common multiple of 4, 3, and 12, which is 12.

12 \cdot \left(\frac{2x - 3}{4}\right) - 12 \cdot \left(\frac{x - 5}{3}\right) = 12 \cdot \left(\frac{x + 15}{12}\right) \implies 3(2x - 3) - 4(x - 5) = x + 15

Eliminating fractions simplifies the algebraic manipulation.

2

Distribute the coefficients through the terms in parentheses, paying careful attention to the negative sign on the second term.

6x - 9 - 4x + 20 = x + 15

Distributing

-4

across

(x - 5)

yields

-4x + 20

.

3

Combine like terms on the left side of the equation.

2x + 11 = x + 15

Grouping variable terms

(6x - 4x = 2x)

and constant terms

(-9 + 20 = 11)

.

4

Isolate the variable

x

by subtracting

x

and

11

from both sides.

x = 4

Solves the linear equation for

x

.

5

Substitute

x = 4

into the target expression

3x - 2

.

3(4) - 2 = 12 - 2 = 10

Calculates the final answer requested in the stem.

Anahtar Kavram

Linear Equations with Fractional Coefficients

Soru 50Soru →

The processing times for a specific type of database query on a server cluster are normally distributed with a mean of $180$ milliseconds and a standard deviation of $25$ milliseconds. Which of the following statements must be true? Indicate all such statements.

Geçerli olan tümünü seçin

Approximately

68\%

of the query processing times are between

155

milliseconds and

205

milliseconds.

A query processing time of

230

milliseconds corresponds to a

z

-score of

+2.0

.

A query processing time of

205

milliseconds corresponds to approximately the

84\text{th}

percentile rank.

A query processing time of

155

milliseconds corresponds to approximately the

32\text{nd}

percentile rank.

A

z

-score of

+1.0

indicates that the query processing time is

25\%

greater than the mean processing time.

Cevabı ve açıklamayı göster

Cevap: Approximately $68\%$ of the query processing times are between $155$ milliseconds and $205$ milliseconds.; A query processing time of $230$ milliseconds corresponds to a $z$ -score of $+2.0$ .; A query processing time of $205$ milliseconds corresponds to approximately the $84\text{th}$ percentile rank.

Cevap

The correct statements are those asserting that approximately 68% of processing times lie between 155 ms and 205 ms, that a processing time of 230 ms corresponds to a z-score of +2.0, and that 205 ms corresponds to approximately the 84th percentile rank.

For a normal distribution with mean

\mu = 180

and standard deviation

\sigma = 25

: (1) The interval

\mu \pm 1\sigma = [155, 205]

contains approximately

68\%

of all observations. (2) A score of

230

ms has a

z

-score of

\frac{230-180}{25} = +2.0

. (3) A score of

205

ms corresponds to

z = +1.0

, which accumulates

50\%

(area below mean)

+ 34\%

(area between mean and

+1\sigma

) =

84\%

percentile rank.

Adım Adım Çözüm

1

Evaluate the 1 standard deviation interval

\mu \pm 1\sigma = 180 \pm 25 = [155, 205]

ms

The 68-95-99.7 empirical rule states that approximately 68% of observations in a normal distribution fall within 1 standard deviation of the mean.

2

Calculate the z-score for a query processing time of 230 ms

z = \frac{230 - 180}{25} = \frac{50}{25} = +2.0

The standard score formula

z = \frac{x - \mu}{\sigma}

measures how many standard deviations an observation is above or below the mean.

3

Determine the percentile rank for 205 ms and 155 ms

For 205 ms (

z = +1

), percentile =

50\% + 34\% = 84\%

. For 155 ms (

z = -1

), percentile =

50\% - 34\% = 16\%

.

50% of the distribution lies below the mean. For

z = +1

, add the 34% between mean and

z = +1

. For

z = -1

, subtract the 34% from 50%.

4

Evaluate relative percentage change versus standard deviation

A 1 standard deviation increase (

25

ms) above the mean (

180

ms) equals an increase of

\frac{25}{180} \approx 13.89\%

.

Standard deviation measures absolute dispersion in the units of the variable, not a percentage of the mean.

Anahtar Kavram

Empirical Rule, z-Score Calculation, and Percentile Ranks in Normal Distributions

Soru 51Soru →

A sequence $a_1, a_2, a_3, \dots$ is defined by $a_n = \frac{1}{\sqrt{n} + \sqrt{n+2}}$ for all positive integers $n$ . If $S = \sum_{n=1}^{98} a_n$ , which of the following is closest to the value of $S$ when rounded to the nearest tenth?

8.8

4.5

9.0

9.9

17.5

Cevabı ve açıklamayı göster

Cevap: 8.8

Cevap

The value of

S

rounded to the nearest tenth is 8.8.

The term

a_n

simplifies to

\frac{\sqrt{n+2} - \sqrt{n}}{2}

upon rationalizing the denominator. Summing from

n=1

to

98

causes all middle terms to cancel out, leaving

\frac{\sqrt{99} + \sqrt{100} - \sqrt{1} - \sqrt{2}}{2}

. Substituting

\sqrt{100} = 10

,

\sqrt{1} = 1

,

\sqrt{99} \approx 9.95

, and

\sqrt{2} \approx 1.41

gives approximately

8.77

, which rounds to 8.8.

Adım Adım Çözüm

1

Rationalize the general term

a_n

a_n = \frac{1}{\sqrt{n} + \sqrt{n+2}} \cdot \frac{\sqrt{n+2} - \sqrt{n}}{\sqrt{n+2} - \sqrt{n}} = \frac{\sqrt{n+2} - \sqrt{n}}{(n+2) - n} = \frac{\sqrt{n+2} - \sqrt{n}}{2}

Eliminating radicals from the denominator reveals the underlying telescoping structure of the sequence.

2

Expand the summation

S = \sum_{n=1}^{98} a_n

S = \frac{1}{2} \left[ (\sqrt{3} - \sqrt{1}) + (\sqrt{4} - \sqrt{2}) + (\sqrt{5} - \sqrt{3}) + \dots + (\sqrt{99} - \sqrt{97}) + (\sqrt{100} - \sqrt{98}) \right]

Writing out initial and final terms demonstrates which terms cancel.

3

Simplify the telescoping sum

S = \frac{\sqrt{99} + \sqrt{100} - \sqrt{1} - \sqrt{2}}{2} = \frac{\sqrt{99} + 10 - 1 - \sqrt{2}}{2} = \frac{9 + \sqrt{99} - \sqrt{2}}{2}

All intermediate terms cancel out, leaving two positive boundary terms and two negative boundary terms.

4

Estimate square root values and perform rounding

Since

\sqrt{99} \approx 9.94987

and

\sqrt{2} \approx 1.41421

,

S \approx \frac{9 + 9.94987 - 1.41421}{2} = \frac{17.53566}{2} = 8.76783 \approx 8.8

Evaluating the radicals to two decimal places allows accurate rounding to the nearest tenth.

Anahtar Kavram

Telescoping Series Summation and Square Root Estimation

Soru 52Soru →

If $x > 1$ and $\sqrt{x^{\sqrt{x}}} = x^2$ , what is the value of $x$ ?

2

4

8

16

64

Cevabı ve açıklamayı göster

Cevap: $16$

Cevap

16

By rewriting the square root as an exponent of

1/2

, the left side becomes

x^{\frac{\sqrt{x}}{2}}

. Since the base

x

is greater than 1, set the exponents equal to each other:

\frac{\sqrt{x}}{2} = 2

, which gives

\sqrt{x} = 4

. Squaring both sides yields

x = 16

.

Adım Adım Çözüm

1

Express the radical on the left side of the equation as a fractional exponent.

\sqrt{x^{\sqrt{x}}} = (x^{\sqrt{x}})^{\frac{1}{2}} = x^{\frac{\sqrt{x}}{2}}

The square root rule states that

\sqrt[k]{a} = a^{\frac{1}{k}}

.

2

Set the exponent of the left side equal to the exponent of the right side.

\frac{\sqrt{x}}{2} = 2

Since the bases are equal (

x > 1

), their corresponding exponents must be equal.

3

Solve for

\sqrt{x}

by multiplying both sides by 2.

\sqrt{x} = 4

Isolating the radical term allows determination of the root value.

4

Square both sides to solve for

x

.

x = 4^2 = 16

Squaring a principal square root yields the underlying radicand.

Anahtar Kavram

Combining Fractional Exponents and Radical Expressions

Soru 53Soru →

A renewable energy facility distributes stored electricity among three battery banks: Bank 1, Bank 2, and Bank 3. Initially, the ratio of the energy stored in Bank 1 to Bank 2 to Bank 3 is $3 : 4 : 5$ .

To balance the system, two sequential transfers of energy are performed without any energy loss:
1. Energy is transferred from Bank 3 to Bank 1 such that the ratio of the energy in Bank 1 to Bank 2 becomes $5 : 4$ .
2. Energy is then transferred from Bank 2 to Bank 3 such that the ratio of the energy in Bank 2 to Bank 3 becomes $1 : 4$ .

If Bank 3 contains $60\text{ MWh}$ more energy after the second transfer than it did initially, what was the total amount of energy, in MWh, stored across all three battery banks?

1,200\text{ MWh}

1,500\text{ MWh}

900\text{ MWh}

720\text{ MWh}

500\text{ MWh}

Cevabı ve açıklamayı göster

Cevap: $1,200\text{ MWh}$

Cevap

The total energy stored across all three battery banks is

1,200\text{ MWh}

.

The correct answer is

1,200\text{ MWh}

. By setting up algebraic expressions for the energy in each bank relative to total energy

T

, we track how each transfer modifies individual bank totals while conserving total energy. The intermediate transfer reduces Bank 3 from

\frac{25}{60}T

to

\frac{15}{60}T

, and the subsequent transfer increases it to

\frac{28}{60}T

. The resulting difference of

\frac{3}{60}T = \frac{1}{20}T

equals

60\text{ MWh}

, solving directly to

T = 1,200\text{ MWh}

.

Adım Adım Çözüm

1

Express initial energy quantities in terms of total energy

T

Bank 1 =

\frac{3}{12}T

, Bank 2 =

\frac{4}{12}T

, Bank 3 =

\frac{5}{12}T = \frac{25}{60}T

The ratio

3 : 4 : 5

sums to

12

total parts.

2

Calculate energy amounts after the first transfer (Bank 3 to Bank 1)

Bank 2 remains

\frac{4}{12}T

; Bank 1 becomes

\frac{5}{12}T

; Bank 3 becomes

T - \frac{5}{12}T - \frac{4}{12}T = \frac{3}{12}T

The new ratio of Bank 1 to Bank 2 is

5 : 4

, and Bank 2's energy did not change.

3

Calculate energy amounts after the second transfer (Bank 2 to Bank 3)

Combined energy in Banks 2 and 3 =

\frac{4}{12}T + \frac{3}{12}T = \frac{7}{12}T

. Final Bank 3 =

\frac{4}{5} \times \frac{7}{12}T = \frac{28}{60}T

Bank 1 remains unchanged at

\frac{5}{12}T

. The remaining energy is divided between Bank 2 and Bank 3 in a

1 : 4

ratio.

4

Determine the net change in Bank 3 and solve for total energy

T

Net change =

\frac{28}{60}T - \frac{25}{60}T = \frac{3}{60}T = \frac{1}{20}T

. Since

\frac{1}{20}T = 60\text{ MWh}

,

T = 1,200\text{ MWh}

Bank 3 ended with

60\text{ MWh}

more than its initial amount.

Anahtar Kavram

Multi-stage ratio rebalancing and conservation of total quantity

Soru 54Soru →

For all real numbers

x

such that

x \neq 3

and

x \neq -5

, the algebraic expression

\frac{(x^2 - 9)^2 - 4(x - 3)^2}{(x - 3)(x + 5)}

can be simplified to the equivalent polynomial expression

x^2 + ax + b

, where

a

and

b

are constants. What is the value of

a + b

?

Cevabı ve açıklamayı göster

Cevap: -5

Cevap

The value of

a + b

is

-5

.

Factoring the numerator yields

(x - 3)^2(x + 1)(x + 5)

. Canceling the common factors

(x - 3)

and

(x + 5)

with the denominator simplifies the expression to

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

. Comparing this to

x^2 + ax + b

identifies

a = -2

and

b = -3

, giving a sum of

a + b = -5

.

Adım Adım Çözüm

1

Factor the numerator by recognizing

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

(x^2 - 9)^2 - 4(x - 3)^2 = [(x - 3)(x + 3)]^2 - 4(x - 3)^2 = (x - 3)^2 (x + 3)^2 - 4(x - 3)^2

Applying the difference of squares identity inside the squared term allows factoring out common factors.

2

Factor out

(x - 3)^2

from the numerator

(x - 3)^2 \left[ (x + 3)^2 - 4 \right]

Extracting the greatest common algebraic factor simplifies the remaining expression.

3

Apply difference of squares to

(x + 3)^2 - 4

(x + 3)^2 - 2^2 = ((x + 3) - 2)((x + 3) + 2) = (x + 1)(x + 5)

Recognizing

(x + 3)^2 - 2^2

as

A^2 - B^2

yields factored linear terms directly.

4

Substitute the fully factored numerator back into the rational expression and simplify

\frac{(x - 3)^2 (x + 1)(x + 5)}{(x - 3)(x + 5)} = (x - 3)(x + 1)

Canceling non-zero common factors

(x - 3)

and

(x + 5)

simplifies the rational function.

5

Expand

(x - 3)(x + 1)

and determine

a + b

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

, so

a = -2

and

b = -3

. Therefore,

a + b = -2 + (-3) = -5

.

Matching coefficients with

x^2 + ax + b

gives

a = -2

and

b = -3

.

Anahtar Kavram

Simplifying complex rational expressions through nested difference of squares factoring.

Soru 55Soru →

Let $p$ and $q$ be integers such that $p < 0 < q$ , $(-1)^p + (-1)^q = 0$ , and $p^2 q + p$ is an even integer. Which of the following expressions must be negative?

(-p)^q

(-q)^p

q^p

p^q

(q - p)^p

Cevabı ve açıklamayı göster

Cevap: $p^q$

Cevap

p^q

From

(-1)^p + (-1)^q = 0

,

p

and

q

must have opposite parities. Factoring

p^2 q + p

into

p(pq + 1)

shows that if

p

were odd,

q

would be even, leading to an odd product

\text{odd} \times \text{odd} = \text{odd}

. Because

p^2 q + p

is given as even,

p

must be an even integer and

q

must be an odd integer. Given

p < 0

,

p

is a negative even integer, and

q

is a positive odd integer. The expression

p^q

represents a negative base raised to an odd exponent, which is guaranteed to be negative.

Adım Adım Çözüm

1

Determine the parities of

p

and

q

using the equation

(-1)^p + (-1)^q = 0

.

One of

p

or

q

is even and the other is odd.

For the sum

(-1)^p + (-1)^q

to equal

0

, one term must equal

1

and the other must equal

-1

, meaning

p

and

q

have opposite parities.

2

Analyze the parity of

p^2 q + p = p(pq + 1)

to determine which variable is even.

p

must be even, and

q

must be odd.

If

p

were odd,

q

would be even, making

pq

even,

pq+1

odd, and

p(pq+1)

odd. Since

p^2 q + p

is even,

p

must be even and

q

must be odd.

3

Evaluate the sign of

p^q

using sign rules for exponents.

p^q < 0

p

is negative (

p < 0

) and

q

is a positive odd integer. A negative number raised to an odd power is always negative.

Anahtar Kavram

Parity and sign rules for negative bases and integer exponents

Soru 56Soru →

Consider the system of linear equations in two variables

x

and

y

:

\begin{aligned} 2x - 3y &= a \\ 4x - 6y &= b \end{aligned}

where

a

and

b

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

Geçerli olan tümünü seçin

If

b = 2a

, the system has infinitely many solutions.

If

b \neq 2a

, the system has no solution.

If

a = 0

and

b = 0

, the system has a unique solution at

(0, 0)

.

There exist real values of

a

and

b

for which the system has exactly one solution.

If

b = 2a + 1

, the lines represented by the equations in the

xy

-plane are parallel and distinct.

Cevabı ve açıklamayı göster

Cevap: If $b = 2a$ , the system has infinitely many solutions.; If $b \neq 2a$ , the system has no solution.; If $b = 2a + 1$ , the lines represented by the equations in the $xy$ -plane are parallel and distinct.

Cevap

The correct statements are: 'If

b = 2a

, the system has infinitely many solutions.', 'If

b \neq 2a

, the system has no solution.', and 'If

b = 2a + 1

, the lines represented by the equations in the

xy

-plane are parallel and distinct.'

Statements asserting that the system has infinitely many solutions when

b = 2a

, no solution when

b \neq 2a

, and parallel distinct lines when

b = 2a + 1

are all mathematically sound. Multiplying the first equation by 2 reveals that the left sides are identical (

4x - 6y

). Equality of the right sides (

b = 2a

) makes the lines identical, whereas inequality (

b \neq 2a

) makes them parallel and distinct.

Adım Adım Çözüm

1

Analyze the coefficients of the system of equations.

The coefficients of

x

and

y

in the second equation (

4

and

-6

) are exactly twice the coefficients of

x

and

y

in the first equation (

2

and

-3

).

Comparing coefficient ratios determines line relationships (slopes).

2

Multiply the first equation by 2.

2(2x - 3y) = 2(a) \implies 4x - 6y = 2a

.

This puts the left side of the first equation in exact alignment with the second equation (

4x - 6y = b

).

3

Evaluate the condition for infinitely many solutions.

If

b = 2a

, the two equations become

4x - 6y = 2a

and

4x - 6y = 2a

, which describe the exact same line, giving infinitely many solutions.

Coincident lines intersect at every point along the line.

4

Evaluate the condition for no solution.

If

b \neq 2a

, subtracting the equations gives

0 = b - 2a \neq 0

, a contradiction. Hence, the lines are parallel and distinct, meaning no solution exists.

Distinct parallel lines never intersect.

5

Examine specific cases such as

b = 2a + 1

and

a=0, b=0

.

For

b = 2a + 1

, since

2a + 1 \neq 2a

,

b \neq 2a

, confirming parallel distinct lines. For

a=0, b=0

,

b = 2(0) = 0

, which yields infinitely many solutions rather than a unique solution.

Verifies specific option claims.

Anahtar Kavram

Systems of Linear Equations (Solvability and Geometric Interpretation)

Soru 57Soru →

If $x$ is a real number such that $\sqrt[4]{16^{x+3}} = 8^{x-1}$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

x

is

3

.

To solve

\sqrt[4]{16^{x+3}} = 8^{x-1}

, express both sides with the base 2. The left side simplifies to

\sqrt[4]{(2^4)^{x+3}} = 2^{\frac{4(x+3)}{4}} = 2^{x+3}

. The right side simplifies to

(2^3)^{x-1} = 2^{3(x-1)} = 2^{3x-3}

. Equating the exponents yields

x + 3 = 3x - 3

, which solves to

2x = 6

, giving

x = 3

.

Adım Adım Çözüm

1

Rewrite 16 and 8 using prime base 2

16 = 2^4

and

8 = 2^3

Converting terms to a common base allows direct comparison of exponents.

2

Simplify the left-hand side radical expression

\sqrt[4]{16^{x+3}} = \sqrt[4]{(2^4)^{x+3}} = 2^{\frac{4(x+3)}{4}} = 2^{x+3}

The

n

-th root

\sqrt[n]{a^m}

is equivalent to

a^{m/n}

.

3

Simplify the right-hand side exponential expression

8^{x-1} = (2^3)^{x-1} = 2^{3(x-1)} = 2^{3x-3}

Applying the exponent power rule

(a^m)^n = a^{m \cdot n}

requires multiplying

3

by

(x - 1)

.

4

Equate the exponents and solve for

x

x + 3 = 3x - 3 \implies 2x = 6 \implies x = 3

When

a^u = a^v

for

a > 0

and

a \neq 1

, it follows that

u = v

.

Anahtar Kavram

Solving exponential equations using prime base factorization and radical conversion rules

Soru 58Soru →

If the pair $(x, y)$ satisfies the system of linear equations below, what is the value of $x - y$ ?

\begin{aligned} 3x - 5y &= 14 \\ 2x + 7y &= -1 \end{aligned}

-1

2

3

4

5

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x - y

is

4

.

Solving the system of linear equations by elimination gives

x = 3

and

y = -1

. Substituting these values into the target expression

x - y

yields

3 - (-1) = 3 + 1 = 4

.

Adım Adım Çözüm

1

Eliminate one variable using the elimination method.

Multiply the first equation by

2

and the second equation by

3

:

6x - 10y = 28

6x + 21y = -3

Creating matching coefficients for

x

allows elimination of

x

by subtraction.

2

Subtract the first modified equation from the second modified equation to solve for

y

.

(6x + 21y) - (6x - 10y) = -3 - 28

31y = -31 \implies y = -1

Subtracting eliminates

x

, isolating

y

.

3

Substitute

y = -1

back into one of the original equations to solve for

x

.

3x - 5(-1) = 14 \implies 3x + 5 = 14 \implies 3x = 9 \implies x = 3

Plugging in

y

allows direct solution for

x

.

4

Evaluate the requested expression

x - y

.

x - y = 3 - (-1) = 3 + 1 = 4

Subtracting a negative value is equivalent to adding its positive value.

Anahtar Kavram

Solving systems of linear equations using elimination and evaluating algebraic expressions.

Tahmini Süre:1m 30s

Soru 59Soru →

The quadratic equation $x^2 - kx + 36 = 0$ , where $k$ is a positive constant, has two distinct real roots $r_1$ and $r_2$ such that $r_2 - r_1 = 5$ . What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

The value of

k

is 13.

According to Vieta's formulas, for the quadratic equation

x^2 - kx + 36 = 0

, the sum of the roots is

r_1 + r_2 = k

and the product of the roots is

r_1 r_2 = 36

. Using the identity

(r_2 - r_1)^2 = (r_1 + r_2)^2 - 4r_1 r_2

, we substitute the known values

r_2 - r_1 = 5

,

r_1 + r_2 = k

, and

r_1 r_2 = 36

. This gives

5^2 = k^2 - 4(36)

, which simplifies to

25 = k^2 - 144

. Solving for

k^2

gives

k^2 = 169

. Since

k

is specified as a positive constant,

k = 13

.

Adım Adım Çözüm

1

Apply Vieta's formulas to the given quadratic equation

The sum of the roots is

r_1 + r_2 = k

and the product of the roots is

r_1 r_2 = 36

.

For any quadratic equation

ax^2 + bx + c = 0

, the sum of roots is

-b/a

and the product of roots is

c/a

.

2

Relate the difference of the roots to their sum and product

(r_2 - r_1)^2 = (r_1 + r_2)^2 - 4r_1 r_2

Expanding both sides shows that

r_2^2 - 2r_1 r_2 + r_1^2 = r_1^2 + 2r_1 r_2 + r_2^2 - 4r_1 r_2

, which is an algebraic identity.

3

Substitute the known values into the identity

5^2 = k^2 - 4(36) \implies 25 = k^2 - 144

We are given that

r_2 - r_1 = 5

,

r_1 r_2 = 36

, and

r_1 + r_2 = k

.

4

Solve for the positive constant

k

k^2 = 169 \implies k = 13

Adding 144 to both sides gives

k^2 = 169

. Taking the positive square root because

k > 0

yields

k = 13

.

Anahtar Kavram

Vieta's Formulas and Root Difference Identity

Soru 60Soru →

Read the passage below:

While early twentieth-century art conservators viewed the micro-cracking in Renaissance oil paintings as an alarming sign of (i)________, subsequent micro-analytical studies revealed that these fissures actually release internal structural tension. Far from presaging imminent structural failure, the phenomenon is largely (ii)________, serving to stabilize the paint layers against environmental fluctuations over centuries. Consequently, aggressive interventionist attempts to artificially seal these micro-cracks often prove not merely redundant but outright (iii)________, inducing synthetic stresses that accelerate physical flaking.

Which of the following combinations of words best completes the blanks in the passage to maintain logical and contextual coherence?

(i) degradation, (ii) benign, (iii) deleterious

(i) maturation, (ii) benign, (iii) advantageous

(i) degradation, (ii) catastrophic, (iii) ineffective

(i) renovation, (ii) indispensable, (iii) deleterious

(i) degradation, (ii) benign, (iii) restorative

Cevabı ve açıklamayı göster

Cevap: (i) degradation, (ii) benign, (iii) deleterious

Cevap

(i) degradation, (ii) benign, (iii) deleterious

The combination of 'degradation', 'benign', and 'deleterious' satisfies all local syntax requirements and maintains overall thematic coherence across the three sentences. 'Degradation' fits the alarm of conservators, 'benign' reflects the harmless nature demonstrated by scientific analysis, and 'deleterious' matches the destructive consequences of intervention.

Adım Adım Çözüm

1

Analyze Blank (i) using sentence clues.

Blank (i) describes what early conservators feared micro-cracking represented. The phrase 'alarming sign of' points to a negative physical process, making 'degradation' the logical choice.

The sentence sets up an initial misconception of damage before the second clause introduces micro-analytical evidence.

2

Analyze Blank (ii) using cross-sentence structural signals.

The phrase 'Far from presaging imminent structural failure' establishes a contrast. Since it is not indicative of failure and actually serves to 'stabilize', the phenomenon is 'benign'.

The contrast signal 'Far from...' requires a term that opposes 'imminent structural failure'.

3

Analyze Blank (iii) based on consequence and degree.

The sentence begins with 'Consequently' and uses the intensifying frame 'not merely redundant but outright...'. Because interventions 'induce synthetic stresses' and 'accelerate physical flaking', the outcome is harmful, fitting 'deleterious'.

The discourse marker 'not merely X but outright Y' requires Y to be a stronger, more severe escalation of negative impact than simple redundancy.

Anahtar Kavram

Cross-Sentence Contextual Coherence in Multi-Blank Text Completion

Tahmini Süre:1m 30s

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram