Tüm alıştırma soruları

Soru 2001Soru →

Consider the following sentence fragment: 'Despite the Lead Curator's insistence that the restoration of the Renaissance mural was executed with strictly ________ adherence to historic conservation protocols, independent analysts discovered unapproved synthetic materials.' Which of the following classifications correctly pairs each candidate vocabulary term or pair with its distractor role in sentence equivalence analysis?

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct pairings match 'scrupulous / meticulous' to the true contextual synonym pair, 'dogmatic / doctrinaire' to the false synonym trap pair, 'artistic' to the topically related distractor, and 'diligent' to the unpaired distractor.

The pair 'scrupulous / meticulous' correctly completes the sentence by conveying thoroughness and precise adherence to standards. 'Dogmatic / doctrinaire' forms a false synonym trap because its tone of rigid ideology does not match the sentence. 'Artistic' is a topically related distractor related to art but lacking a pair or contextual fit, and 'diligent' is an unpaired distractor.

Adım Adım Çözüm

1

Analyze the sentence context and structural clues.

The blank modifies 'adherence to historic conservation protocols'. The contrast signal 'Despite... discovered unapproved synthetic materials' indicates that the curator claimed high standards of exactness and carefulness.

Establishing the precise contextual meaning required for the blank dictates which word pair is appropriate.

2

Evaluate synonym pairs for contextual fit vs. tone traps.

'Scrupulous' and 'meticulous' form a true synonym pair meaning careful and exact. 'Dogmatic' and 'doctrinaire' form a synonym pair, but mean opinionated and inflexible, which introduces an unjustified negative judgment.

Sentence Equivalence distractors often include valid synonym pairs whose connotation or specific definition fails to match the sentence.

3

Identify topically related and unpaired single distractors.

'Artistic' is relevant to mural restoration but illogical when describing 'adherence to protocols'. 'Diligent' is logically plausible but lacks a matching synonym option.

Eliminating options without a true equivalent pair prevents falling for isolated vocabulary words that seem topically plausible.

Anahtar Kavram

Eliminating False Synonyms and Topically Related Distractors

Soru 2002Soru →

A jar contains $4$ red marbles, $6$ blue marbles, and $5$ green marbles. If two marbles are selected at random one after another without replacement, what is the probability that the first marble selected is red and the second marble selected is blue?

\frac{4}{35}

\frac{8}{75}

\frac{73}{105}

\frac{12}{55}

\frac{10}{29}

Cevabı ve açıklamayı göster

Cevap: $\frac{4}{35}$

Cevap

The probability that the first marble selected is red and the second marble selected is blue is

\frac{4}{35}

.

To find the joint probability of two dependent sequential events, multiply the probability of the first event by the conditional probability of the second event. The probability of drawing a red marble first is

\frac{4}{15}

. Because the selection is made without replacement, there are

14

marbles remaining in the jar,

6

of which are blue. The probability of drawing a blue marble second is

\frac{6}{14}

. Multiplying these together yields

\frac{4}{15} \times \frac{6}{14} = \frac{24}{210} = \frac{4}{35}

.

Adım Adım Çözüm

1

Calculate the total number of marbles in the jar initially.

Total marbles =

4 + 6 + 5 = 15

.

The sample space size for the first draw is the sum of all marbles.

2

Determine the probability of selecting a red marble on the first draw.

P(\text{Red}_1) = \frac{4}{15}

.

There are

4

favorable outcomes out of

15

total outcomes.

3

Determine the probability of selecting a blue marble on the second draw given that one red marble was removed without replacement.

P(\text{Blue}_2 \mid \text{Red}_1) = \frac{6}{14} = \frac{3}{7}

.

After removing one red marble, there are still

6

blue marbles, but only

14

total marbles remaining.

4

Apply the multiplication rule for dependent events.

P(\text{Red}_1 \text{ and } \text{Blue}_2) = \frac{4}{15} \times \frac{6}{14} = \frac{24}{210} = \frac{4}{35}

.

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

Anahtar Kavram

Conditional Probability and Dependent Events

Tahmini Süre:1m 30s

Soru 2003Soru →

In rhombus $ABCD$ , the perimeter is $52$ and the length of diagonal $AC$ is $10$ . What is the area of rhombus $ABCD$ ?

60

96

120

169

240

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

The area of rhombus

ABCD

is 120.

Because all four sides of a rhombus are equal in length, a perimeter of 52 implies each side measures 13. The diagonals of a rhombus intersect at right angles and bisect each other. Given diagonal AC has a length of 10, half of AC is 5. Using the Pythagorean theorem on one of the right triangles formed by the intersecting diagonals gives a half-diagonal length of sqrt(13^2 - 5^2) = 12 for BD. Thus, the total length of diagonal BD is 24. The area of the rhombus is (1/2) * 10 * 24 = 120.

Adım Adım Çözüm

1

Find the side length of rhombus

ABCD

.

Side length

s = 52 / 4 = 13

.

A rhombus has four equal sides, so its perimeter divided by 4 gives the length of one side.

2

Use the properties of rhombus diagonals to find half of diagonal

BD

.

Half of

AC = 5

. Half of

BD = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12

.

The diagonals of a rhombus bisect each other at right angles, forming four congruent right triangles whose hypotenuse is the side length (

13

) and legs are the half-diagonals.

3

Calculate the full length of diagonal

BD

.

Length of diagonal

BD = 2 \times 12 = 24

.

The full diagonal length is twice the length of its half.

4

Calculate the area of rhombus

ABCD

.

Area

= \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 10 \times 24 = 120

.

The area of any rhombus is equal to half the product of its diagonal lengths.

Anahtar Kavram

Properties of a Rhombus: Perpendicular Bisecting Diagonals and Area Formula

Tahmini Süre:1m 30s

Soru 2004Soru →

In her critique of mid-twentieth-century urban planning, Jane Jacobs did not merely challenge prevailing architectural orthodoxy; she aimed to puncture the balloon of professional hubris that dominated city development agencies. Planning theorists of the era had envisioned vast, orderly superblocks, arguing that traditional, dense neighborhoods were archaic relics that stifled modern progress. Jacobs demonstrated that these sprawling modernist developments, despite their pristine aesthetic promises, frequently eroded social cohesion and economic vitality. In describing Jacobs's polemic, contemporary historians note that her intent was neither to condemn structural engineering nor to demand an absolute return to pre-industrial municipal governance. Rather, by deploying incisive empirical observations of neighborhood life, Jacobs sought to deflate the unexamined assumptions of omniscient expertise that shielded bureaucratic planners from public accountability.

Based on the passage, which of the following statements accurately capture the contextual meaning and rhetorical function of the phrase "puncture the balloon of professional hubris"? Select all that apply.

Geçerli olan tümünü seçin

Expose and diminish the unwarranted self-importance and unexamined assumptions of urban planning experts

Challenge the inflated confidence and sense of superiority held by city development theorists of the era

Physically dismantle or damage high-altitude aerial survey equipment utilized in municipal mapping

Reject all forms of technical expertise and specialized knowledge across every domain of civic administration

Demand the immediate demolition of all physical modernist superblocks constructed during the mid-twentieth century

Cevabı ve açıklamayı göster

Cevap: Expose and diminish the unwarranted self-importance and unexamined assumptions of urban planning experts; Challenge the inflated confidence and sense of superiority held by city development theorists of the era

Cevap

The correct options are those stating that the phrase means to expose and diminish the unwarranted self-importance and unexamined assumptions of urban planning experts, and to challenge the inflated confidence and sense of superiority held by city development theorists.

The phrase 'puncture the balloon of professional hubris' is an extended metaphor. In context, 'hubris' refers to the excessive pride of planning theorists, and 'puncturing the balloon' means deflating or exposing that arrogance. The passage later restates this concept directly, explaining that Jacobs aimed to 'deflate the unexamined assumptions of omniscient expertise.' Therefore, options highlighting the exposure of unwarranted self-importance and the challenging of inflated confidence are contextually accurate.

Adım Adım Çözüm

1

Analyze the figurative expression within its surrounding text.

The target phrase 'puncture the balloon of professional hubris' uses a metaphor of deflating a balloon to describe attacking 'hubris' (excessive pride or arrogance).

Contextual clues following the phrase elaborate that Jacobs sought to 'deflate the unexamined assumptions of omniscient expertise that shielded bureaucratic planners.'

2

Evaluate candidate interpretations against passage evidence.

The options describing the exposure of unwarranted self-importance and the challenging of inflated confidence directly align with 'deflating unexamined assumptions of omniscient expertise.'

Both correct choices capture the metaphorical meaning of curbing arrogance without introducing literal or overly extreme claims.

3

Eliminate incorrect options based on literal readings, extreme tone, or scope errors.

Discard literal interpretations regarding aerial survey equipment, extreme claims rejecting all civic expertise, and physical demolition of buildings.

The text explicitly qualifies Jacobs's position, indicating her critique targeted bureaucratic assumptions rather than engineering or physical destruction.

Anahtar Kavram

Interpreting Idiomatic Usage and Figurative Language in Context

Tahmini Süre:1m 30s

Soru 2005Soru →

If $a$ and $b$ are distinct real numbers, what is the simplified form of the algebraic expression $\frac{a^3 - b^3 - a(a^2 - b^2) + b(a - b)^2}{a - b}$ ?

ab + 2b^2

-ab

ab

a^2 + b^2

a(a + b)

Cevabı ve açıklamayı göster

Cevap: $ab$

Cevap

ab

Factoring the common binomial term

(a - b)

from each term in the numerator yields

(a - b)[(a^2 + ab + b^2) - a(a + b) + b(a - b)]

. Expanding inside the brackets gives

a^2 + ab + b^2 - a^2 - ab + ab - b^2

, which reduces completely to

ab

. Dividing

(a - b)(ab)

by

(a - b)

leaves the simplified expression

ab

.

Adım Adım Çözüm

1

Apply standard algebraic factoring identities to each component of the numerator.

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

,

-a(a^2 - b^2) = -a(a - b)(a + b)

, and

b(a - b)^2 = b(a - b)(a - b)

.

Rewriting each grouping with the common binomial factor

(a - b)

allows for structured factoring of the numerator.

2

Factor out

(a - b)

from the entire numerator.

Numerator

= (a - b) \left[ (a^2 + ab + b^2) - a(a + b) + b(a - b) \right]

.

Extracting the common factor isolates the remaining polynomial terms inside brackets.

3

Expand and collect like terms within the bracketed expression.

(a^2 + ab + b^2) - (a^2 + ab) + (ab - b^2) = a^2 - a^2 + ab - ab + ab + b^2 - b^2 = ab

.

Canceling opposite terms (

a^2 - a^2 = 0

,

ab - ab = 0

,

b^2 - b^2 = 0

) leaves only

ab

inside the brackets.

4

Divide the factored numerator by the denominator

(a - b)

.

\frac{(a - b)(ab)}{a - b} = ab

.

Since

a

and

b

are distinct (

a \neq b

),

a - b \neq 0

, allowing non-zero cancellation.

Anahtar Kavram

Simplifying complex algebraic expressions by factoring common binomial terms and applying algebraic identities

Soru 2006Soru →

The positive integer $N$ is divisible by $60$ . If the greatest common divisor of $N$ and $500$ is $100$ , which of the following statements MUST be true? Select all such statements.

Geçerli olan tümünü seçin

N

is a multiple of

300

.

N

is divisible by

125

.

The prime factorization of

N

contains at least two factors of

2

.

N

is divisible by

9

.

The greatest common divisor of

N

and

250

is

50

.

Cevabı ve açıklamayı göster

Cevap: $N$ is a multiple of $300$ .; The prime factorization of $N$ contains at least two factors of $2$ .; The greatest common divisor of $N$ and $250$ is $50$ .

Cevap

The statements asserting that

N

is a multiple of

300

, that the prime factorization of

N

contains at least two factors of

2

, and that the greatest common divisor of

N

and

250

is

50

must all be true.

Analyzing the prime factorizations:

60 = 2^2 \cdot 3^1 \cdot 5^1

and

500 = 2^2 \cdot 5^3

. Since

N

is divisible by

60

, the prime factorization of

N

must have exponents

a \ge 2

for prime

2

,

b \ge 1

for prime

3

, and

c \ge 1

for prime

5

. Furthermore,

\gcd(N, 500) = 100 = 2^2 \cdot 5^2

. The GCD rule requires taking the minimum exponent for each prime factor:

\min(a, 2) = 2 \implies a \ge 2

, and

\min(c, 3) = 2 \implies c = 2

. Therefore,

N = 2^{a} \cdot 3^{b} \cdot 5^{2}

where

a \ge 2

and

b \ge 1

. This means

N

is divisible by

2^2 \cdot 3^1 \cdot 5^2 = 300

, contains at least two prime factors of

2

, and has

\gcd(N, 250) = \gcd(2^a \cdot 3^b \cdot 5^2, 2^1 \cdot 5^3) = 2^1 \cdot 5^2 = 50

.

Adım Adım Çözüm

1

Express

60

and

500

in their prime factorizations.

60 = 2^2 \cdot 3^1 \cdot 5^1

and

500 = 2^2 \cdot 5^3

.

Prime factorization allows us to analyze divisibility and GCD conditions in terms of prime exponents.

2

Apply the condition that

N

is divisible by

60

.

The exponent of

2

in

N

is at least

2

, the exponent of

3

is at least

1

, and the exponent of

5

is at least

1

.

For

N

to be divisible by an integer,

N

must contain at least as many of each prime factor as that integer.

3

Apply the GCD condition

\gcd(N, 500) = 100 = 2^2 \cdot 5^2

.

The exponent of

5

in

N

must be exactly

2

.

Since

500

has

5^3

and the GCD has

5^2

, taking the minimum exponent of

5

between

N

and

500

yields

2

. Thus,

N

has

5^2

and not

5^3

or higher.

4

Evaluate each statement against the established exponent bounds for

N

.

N

has prime factor powers

2^{\ge 2} \cdot 3^{\ge 1} \cdot 5^2

. This guarantees

N

is a multiple of

2^2 \cdot 3^1 \cdot 5^2 = 300

, contains at least two factors of

2

, and yields

\gcd(N, 250) = 2^1 \cdot 5^2 = 50

.

Comparing the prime factor requirements of each choice confirms which statements MUST be true.

Anahtar Kavram

Prime factorization rules for divisibility and greatest common divisor (GCD)

Soru 2007Soru →

Far from being (i)________ by the discovery of irregular orbital oscillations in the outer solar system, the astrophysicist's unified theory of planetary resonance was actually (ii)________; the seemingly anomalous data provided the exact empirical validation needed to confirm the model’s underlying predictive structure.

Which of the following choices best completes the blanks in the passage?

(i) discredited, (ii) vindicated

(i) discredited, (ii) undermined

(i) bolstered, (ii) vindicated

(i) bolstered, (ii) undermined

(i) obfuscated, (ii) ignored

Cevabı ve açıklamayı göster

Cevap: (i) discredited, (ii) vindicated

Cevap

The pair '(i) discredited, (ii) vindicated' logically completes both blanks by aligning the contrast pivot with the concluding evidence.

The combination of '(i) discredited' and '(ii) vindicated' correctly fulfills all structural signals in the sentence. The phrase 'Far from being (i)________' anticipates a negative expectation (discredited) which is then contrasted by 'was actually (ii)________' with a positive resolution (vindicated), matching the semicolon clue that the data provided empirical validation.

Adım Adım Çözüm

1

Identify structural pivots and sentence direction.

The opening phrase 'Far from being (i)________' indicates that blank (i) requires a negative condition that one might expect anomalous data to cause, while 'was actually (ii)________' signals a reversal to a positive outcome.

'Far from being X, Y was actually Z' sets up an antonymic relationship between the expected negative impact X and the actual positive impact Z.

2

Analyze the semicolon context clue for inter-blank dependency.

The clause following the semicolon states that the anomalous data provided 'the exact empirical validation needed to confirm the model's underlying predictive structure.'

This confirms that the actual outcome in blank (ii) must mean supported or proven true ('vindicated').

3

Verify cross-blank semantic alignment.

If the theory was actually 'vindicated' (ii), then far from being 'discredited' (i), the anomalies actually proved its strength.

The pair '(i) discredited, (ii) vindicated' maintains complete thematic and structural coherence.

Anahtar Kavram

Multi-Blank Dependency Tracking & Contrast Pivot Alignment

Soru 2008Soru →

In Year 1, a retail company's total revenue was generated by two divisions: Division X, which accounted for $40\%$ of total revenue, and Division Y, which accounted for the remaining $60\%$ . From Year 1 to Year 2, revenue from Division X increased by $20\%$ , while revenue from Division Y decreased by $10\%$ . From Year 2 to Year 3, revenue from Division X decreased by $10\%$ , while revenue from Division Y increased by $20\%$ . Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

Total company revenue in Year 3 was

8\%

greater than total company revenue in Year 1.

From Year 1 to Year 3, revenue from Division X increased by

8\%

.

In Year 2, Division X accounted for more than

50\%

of total company revenue.

Total company revenue in Year 2 was

10\%

greater than total company revenue in Year 1.

From Year 1 to Year 3, revenue from Division Y increased by

8\%

.

Cevabı ve açıklamayı göster

Cevap: Total company revenue in Year 3 was $8\%$ greater than total company revenue in Year 1.; From Year 1 to Year 3, revenue from Division X increased by $8\%$ .; From Year 1 to Year 3, revenue from Division Y increased by $8\%$ .

Cevap

The correct statements are: total company revenue in Year 3 was 8% greater than in Year 1; revenue from Division X increased by 8% from Year 1 to Year 3; and revenue from Division Y increased by 8% from Year 1 to Year 3.

The statements confirming an 8% overall increase for total company revenue, Division X revenue, and Division Y revenue from Year 1 to Year 3 are all correct. A 20% increase followed by a 10% decrease (and vice versa) multiplies the starting amount by

1.20 \times 0.90 = 1.08

, which represents a net 8% increase on each respective division's initial base.

Adım Adım Çözüm

1

Assign a variable to initial total revenue and calculate division revenues for Year 1.

Let total revenue in Year 1 be

R

. Revenue from Division X is

0.40 R

and revenue from Division Y is

0.60 R

.

Establishing initial quantities provides the base for computing percentage changes across subsequent years.

2

Calculate division revenues and total revenue for Year 2.

Division X revenue in Year 2 =

0.40 R \times 1.20 = 0.48 R

. Division Y revenue in Year 2 =

0.60 R \times 0.90 = 0.54 R

. Total Year 2 revenue =

0.48 R + 0.54 R = 1.02 R

.

Applying the respective 20% increase and 10% decrease to Year 1 base values yields Year 2 amounts.

3

Calculate division revenues and total revenue for Year 3.

Division X revenue in Year 3 =

0.48 R \times 0.90 = 0.432 R

. Division Y revenue in Year 3 =

0.54 R \times 1.20 = 0.648 R

. Total Year 3 revenue =

0.432 R + 0.648 R = 1.08 R

.

Applying the respective 10% decrease and 20% increase to Year 2 values yields Year 3 amounts.

4

Evaluate the percent changes from Year 1 to Year 3 for each division and the total company.

Division X change:

\frac{0.432 R - 0.40 R}{0.40 R} = 8\%

. Division Y change:

\frac{0.648 R - 0.60 R}{0.60 R} = 8\%

. Total revenue change:

\frac{1.08 R - 1.00 R}{1.00 R} = 8\%

. Also, Division X proportion in Year 2 was

\frac{0.48 R}{1.02 R} \approx 47.06\%

.

Determines which of the given statements hold true based on accurate relative changes.

Anahtar Kavram

Weighted and Successive Percent Change

Tahmini Süre:2m 0s

Soru 2009Soru →

Let $p$ , $q$ , and $r$ be non-zero integers satisfying the following three conditions:

I. $p^3 q r^2 < 0$
II. $(-p)^q < 0$
III. $(-1)^{p + r} = 1$

Which of the following expressions MUST be negative?

q \cdot p^r

p \cdot (-q)^r

(-p)^r \cdot q

(p - q)^r

q \cdot (r - p)^q

Cevabı ve açıklamayı göster

Cevap: $q \cdot p^r$

Cevap

The expression

q \cdot p^r

MUST be negative.

Condition II establishes that

p > 0

and

q

is odd. Condition I establishes that

q < 0

. Since

p > 0

, raising

p

to any integer exponent

r

results in a strictly positive value (

p^r > 0

). Multiplying this positive value by negative

q

guarantees a negative outcome regardless of the value or sign of

r

.

Adım Adım Çözüm

1

Analyze Condition II:

(-p)^q < 0

Base

-p < 0 \implies p > 0

, and exponent

q

must be an odd integer.

A negative number raised to an integer power is negative if and only if the exponent is odd.

2

Analyze Condition I:

p^3 q r^2 < 0

q < 0

(negative odd integer).

Since

p > 0

,

p^3 > 0

. Also

r \neq 0 \implies r^2 > 0

. For the overall product

p^3 q r^2

to be negative,

q

must be negative.

3

Analyze Condition III:

(-1)^{p + r} = 1

p + r

is an even integer, so

p

and

r

have the same parity.

(-1)^k = 1

requires

k

to be an even integer.

4

Evaluate the sign of

q \cdot p^r

q \cdot p^r < 0

for all valid values.

Since

p > 0

, any integer power

p^r > 0

. Multiplying positive

p^r

by negative

q

yields a negative result.

Anahtar Kavram

Even-odd exponent rules and negative base sign determination

Soru 2010Soru →

In paleoclimatology, ice core isotopic signatures are frequently cited as unambiguous records of historical temperature fluctuations; nevertheless, recent methodological reevaluations suggest that post-depositional atmospheric interactions can significantly discount the accuracy of these proxy measurements. Rather than confirming atmospheric stability over millennia, local sublimation processes alter isotope ratios prior to compaction. Consequently, researchers who treat raw isotopic data as unblemished historical records risk misinterpreting transient surface anomalies as systemic climatic shifts.

As used in the passage, which of the following choices best expresses the meaning of the word 'discount'?

diminish the validity or reliability of

reduce the financial cost or monetary price of

substantiate or confirm the precision of

completely erase through physical degradation

exaggerate the systemic magnitude of

Cevabı ve açıklamayı göster

Cevap: diminish the validity or reliability of

Cevap

The word 'discount' as used in this context means to diminish the validity or reliability of.

The contrast signal 'nevertheless' shifts the focus from ice cores being 'unambiguous records' to a factor that undermines their accuracy. Furthermore, the subsequent sentence clarifies that these processes 'alter isotope ratios' '[r]ather than confirming atmospheric stability'. Thus, in this context, 'discount' means to lessen, diminish, or undermine the reliability or validity of the measurements.

Adım Adım Çözüm

1

Identify the target word and locate structural transition signals in the sentence.

The target word is 'discount', preceded by the contrast transition 'nevertheless'.

Contrast signals indicate a shift from the preceding claim that ice core signatures are 'unambiguous records'.

2

Analyze the surrounding elaboration and continuation clues.

The text explains that processes 'alter isotope ratios' 'rather than confirming atmospheric stability' and lead to 'misinterpreting' data.

These contextual clues demonstrate that post-depositional interactions detract from or lessen the accuracy of the measurements.

3

Evaluate the definitions against structural clues to select the contextually appropriate meaning.

'Diminish the validity or reliability of' perfectly matches the required negative impact on measurement accuracy.

Secondary definitions of polysemous words must be selected based on structural signals rather than primary commercial definitions.

Anahtar Kavram

Deciphering Vocabulary in Context via Contrast Signals

Tahmini Süre:1m 30s

Soru 2011Soru →

On the real number line, the distance between $x$ and $3$ is equal to twice the distance between $x$ and $-9$ . What is the sum of all possible real values of $x$ ?

-26

-21

-14

-5

22

Cevabı ve açıklamayı göster

Cevap: $-26$

Cevap

The sum of all possible real values of

x

is

-26

.

The distance between

x

and

a

on the real number line is expressed as

|x - a|

. Thus, the condition translates to

|x - 3| = 2|x - (-9)|

, which simplifies to

|x - 3| = 2|x + 9|

. Setting up the two algebraic cases gives

x - 3 = 2(x + 9)

, leading to

x = -21

, and

x - 3 = -2(x + 9)

, leading to

x = -5

. Adding these two solutions yields

(-21) + (-5) = -26

.

Adım Adım Çözüm

1

Translate the geometric statement on the number line into an absolute value equation.

The distance between

x

and

3

is

|x - 3|

, and the distance between

x

and

-9

is

|x - (-9)| = |x + 9|

. Therefore,

|x - 3| = 2|x + 9|

.

Distance between two points

a

and

b

on the number line is given by

|a - b|

.

2

Solve Case 1 where the expressions inside the absolute values have the same sign.

x - 3 = 2(x + 9) \implies x - 3 = 2x + 18 \implies x = -21

.

Removing absolute values with identical signs gives a linear equation in

x

.

3

Solve Case 2 where the expressions inside the absolute values have opposite signs.

x - 3 = -2(x + 9) \implies x - 3 = -2x - 18 \implies 3x = -15 \implies x = -5

.

Removing absolute values with opposite signs accounts for the second possible geometric location.

4

Sum the valid real solutions.

(-21) + (-5) = -26

.

The question asks for the sum of all possible real values of

x

.

Anahtar Kavram

Absolute Value as Distance on the Real Number Line

Soru 2012Soru →

A bakery produces sourdough loaves using a mixture of two types of flour, Flour X and Flour Y. A standard batch requires a total of $84$ kilograms of flour. The amount of Flour X used is $12$ kilograms more than twice the amount of Flour Y used. If $y$ represents the amount of Flour Y, in kilograms, used in one standard batch, which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

y

is a positive integer multiple of

8

.

The ratio of the amount of Flour X to the amount of Flour Y used in one batch is

5:2

.

Flour X accounts for

75\%

of the total flour used in one batch.

The amount of Flour X used in one batch exceeds the amount of Flour Y used by

36

kilograms.

The combined amount of Flour Y used in

4

standard batches is

100

kilograms.

Cevabı ve açıklamayı göster

Cevap: $y$ is a positive integer multiple of $8$ .; The ratio of the amount of Flour X to the amount of Flour Y used in one batch is $5:2$ .; The amount of Flour X used in one batch exceeds the amount of Flour Y used by $36$ kilograms.

Cevap

The correct statements are that

y

is a positive integer multiple of

8

, the ratio of Flour X to Flour Y used in one batch is

5:2

, and the amount of Flour X used in one batch exceeds the amount of Flour Y used by

36

kilograms.

Solving

y + (2y + 12) = 84

yields

y = 24

kilograms of Flour Y and

60

kilograms of Flour X. Since

24

is divisible by

8

, the statement asserting

y

is a multiple of

8

is true. The ratio of Flour X to Flour Y is

60:24 = 5:2

, making the ratio statement true. Finally,

60 - 24 = 36

, confirming that Flour X exceeds Flour Y by

36

kilograms.

Adım Adım Çözüm

1

Set up the linear equation in terms of

y

.

Amount of Flour Y =

y

, Amount of Flour X =

2y + 12

. Total amount =

y + (2y + 12) = 84

.

The problem states Flour X is

12

kg more than twice Flour Y, and their sum equals

84

kg.

2

Solve the linear equation for

y

.

3y + 12 = 84 \implies 3y = 72 \implies y = 24

.

Subtract

12

from both sides and divide by

3

to isolate the variable

y

.

3

Calculate the amount of Flour X used.

Amount of Flour X =

2(24) + 12 = 48 + 12 = 60

kilograms.

Substitute

y = 24

back into the expression for Flour X.

4

Evaluate each given statement.

1)

y = 24

, which is a multiple of

8

(

24 = 8 \times 3

). (True)
2) Ratio of Flour X to Flour Y is

60 : 24 = 5 : 2

. (True)
3) Percentage of Flour X is

\frac{60}{84} = \frac{5}{7} \approx 71.43\% \neq 75\%

. (False)
4) Difference is

60 - 24 = 36

kg. (True)
5)

4

batches of Flour Y equal

4 \times 24 = 96

kg

\neq 100

kg. (False)

Direct calculation confirms which statements match the derived values.

Anahtar Kavram

Formulating and solving a linear equation in one variable from a word problem scenario, followed by logical evaluation of derived quantities.

Soru 2013Soru →

A quality control manager records the durability ratings, on a scale from $1$ to $50$ , for $8$ randomly selected component batches. The ratings are listed below:

34, 18, 42, 27, 18, 39, 45, 25

Two additional component batches with identical ratings, $x$ and $y$ (where $x = y$ ), are included in the dataset. If the median rating of the updated dataset of $10$ batches is equal to the median rating of the original dataset of $8$ batches, what is the value of $x$ ?

22.5

26.0

30.5

31.5

36.5

Cevabı ve açıklamayı göster

Cevap: $30.5$

Cevap

The value of

x

is

30.5

.

To find the median of the original dataset, the numbers must first be ordered from smallest to largest:

18, 18, 25, 27, 34, 39, 42, 45

. Since there are

8

numbers, the median is the average of the 4th and 5th numbers:

(27 + 34)/2 = 30.5

. When two equal values

x

are added to form a

10

-element dataset, having

x = 30.5

places both new values directly between

27

and

34

. The 5th and 6th terms of the updated set are both

30.5

, making the new median

(30.5 + 30.5)/2 = 30.5

, which matches the original median.

Adım Adım Çözüm

1

Sort the original dataset of 8 durability ratings in ascending order.

The sorted dataset is

18, 18, 25, 27, 34, 39, 42, 45

.

Finding the median of a numerical dataset requires arranging the numbers in order from least to greatest.

2

Calculate the median of the original 8 ratings.

The 4th value is

27

and the 5th value is

34

. The median is

\frac{27 + 34}{2} = 30.5

.

For a dataset with an even number of elements (

n = 8

), the median is the average of the two middle elements (the 4th and 5th terms).

3

Analyze how adding two identical values

x

and

y

(

x = y

) affects the median of the 10-element dataset.

Inserting two values equal to

30.5

places them as the 5th and 6th elements in the 10-element sorted array:

18, 18, 25, 27, 30.5, 30.5, 34, 39, 42, 45

. The new median is

\frac{30.5 + 30.5}{2} = 30.5

.

To keep the median unchanged at

30.5

when adding two identical values, the added values must fall between

27

and

34

and specifically equal

30.5

so that the middle two terms of the 10 values average to

30.5

.

Anahtar Kavram

Median of a Dataset

Tahmini Süre:1m 30s

Soru 2014Soru →

Fill in the blanks in the passage below to complete the sentence logically based on inter-blank structural clues.

Aşağıdaki boşlukları doldurun

While early economic historians asserted that the severe seventeenth-century agrarian downturn was an inevitable outcome of climatic cooling, recent demographic data reveal that regional grain shortages were frequently by dynamic market distribution, demonstrating that adaptive social infrastructure could the adverse effects of ecological volatility.

Cevabı ve açıklamayı göster

Cevap

Blank 1 requires a word meaning lessened or buffered (e.g., 'mitigated'), and Blank 2 requires a verb meaning weaken or reduce (e.g., 'attenuate').

The introductory contrast signal 'While...' signals a reversal of the traditional view that climate alone dictated agrarian disaster. Consequently, the demographic evidence must show that trade networks softened or reduced local shortages (e.g., 'mitigated'), which directly leads to the conclusion that adaptive social institutions could weaken or counter (e.g., 'attenuate') ecological shocks.

Adım Adım Çözüm

1

Analyze the structural pivot at the start of the sentence ('While early economic historians asserted...').

Identify that the main clause must contrast with the deterministic view that climate cooling inevitably caused agrarian ruin.

The introductory concession 'While...' establishes a contrast between environmental determinism and the mitigating role of human infrastructure.

2

Track inter-blank dependency between Blank 1 and Blank 2.

Recognize that if dynamic market distribution reduced the severity of grain shortages (Blank 1: mitigated/ameliorated), it demonstrates that social infrastructure served to lessen environmental impacts (Blank 2: attenuate/offset).

Both blanks must align in polarity to support the idea that social systems counteracted climate stress rather than amplifying it.

Anahtar Kavram

Multi-Blank Dependency Tracking

Soru 2015Soru →

If $x > 0$ and $x^2 - 6x - 16 = 0$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The correct value of

x

is 8.

Factoring

x^2 - 6x - 16 = 0

gives

(x - 8)(x + 2) = 0

. Setting each factor to zero yields

x = 8

and

x = -2

. Given the constraint

x > 0

, the only valid answer is 8.

Adım Adım Çözüm

1

Factor the quadratic expression

(x - 8)(x + 2) = 0

Find two numbers that multiply to

-16

and add to

-6

, which are

-8

and

2

.

2

Solve for possible values of

x

x = 8

or

x = -2

By the zero-product property, if

(x - 8)(x + 2) = 0

, then either

x - 8 = 0

or

x + 2 = 0

.

3

Apply the given constraint

x > 0

x = 8

The solution

x = -2

is rejected because

x

must be positive.

Anahtar Kavram

Factoring Quadratic Equations

Tahmini Süre:45s

Soru 2016Soru →

The daily water consumption per household in a municipal district is normally distributed with a mean of $320$ liters and a standard deviation of $40$ liters. A local utility company classifies households into three consumption categories:

- Efficient: Daily consumption below $240$ liters
- Moderate: Daily consumption between $240$ liters and $400$ liters
- High: Daily consumption above $400$ liters

Based on the $68\text{--}95\text{--}99.7$ empirical rule for normal distributions, in a random sample of $10,000$ households, how many more households are classified as Moderate than are classified as Efficient?

5,200

7,900

9,000

9,250

9,500

Cevabı ve açıklamayı göster

Cevap: $9,250$

Cevap

9,250

households

The boundary values

240

liters and

400

liters represent

z = -2

and

z = +2

, respectively. According to the empirical rule,

95\%

of the population falls between these limits, yielding

9,500

Moderate households. The remaining

5\%

is split equally into the upper and lower tails (

2.5\%

each), meaning

2.5\%

(

250

households) are Efficient. Subtracting

250

from

9,500

yields

9,250

.

Adım Adım Çözüm

1

Calculate the z-scores for the boundaries of the Moderate and Efficient categories.

The boundary

240

liters corresponds to

z = \frac{240 - 320}{40} = -2.0

. The boundary

400

liters corresponds to

z = \frac{400 - 320}{40} = +2.0

.

Standardizing values into z-scores allows the direct application of the empirical rule.

2

Determine the percentage and count of households in the Moderate category.

By the

68\text{--}95\text{--}99.7

rule,

95\%

of the data lies between

z = -2.0

and

z = +2.0

. Count =

0.95 \times 10,000 = 9,500

households.

The Moderate category spans from

\mu - 2\sigma

to

\mu + 2\sigma

.

3

Determine the percentage and count of households in the Efficient category.

The area below

z = -2.0

is

\frac{100\% - 95\%}{2} = 2.5\%

. Count =

0.025 \times 10,000 = 250

households.

The normal distribution is symmetric, so the remaining

5\%

outer area is evenly split into two tails of

2.5\%

each.

4

Calculate the difference between Moderate and Efficient household counts.

9,500 - 250 = 9,250

households.

Subtracting the count of Efficient households from Moderate households satisfies the core question prompt.

Anahtar Kavram

Normal distribution z-score standardization and asymmetric region calculations using the 68-95-99.7 empirical rule.

Tahmini Süre:2m 30s

Soru 2017Soru →

In a regular polygon, the ratio of the measure of an interior angle to the measure of an exterior angle is $7:2$ . What is the total number of diagonals of this polygon?

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

The total number of diagonals of the regular polygon is

27

.

The correct answer is

27

. An interior angle and an exterior angle of a polygon are supplementary (

180^\circ

). Given the ratio

7:2

, the exterior angle is

\frac{2}{9} \times 180^\circ = 40^\circ

. Since the sum of exterior angles of any convex polygon is

360^\circ

, the number of sides is

n = \frac{360^\circ}{40^\circ} = 9

. Using the formula for the number of diagonals,

\frac{n(n-3)}{2}

, we obtain

\frac{9(9-3)}{2} = 27

.

Adım Adım Çözüm

1

Determine the measure of the exterior angle using the given interior-to-exterior ratio.

Exterior angle measure =

40^\circ

At any vertex of a polygon, the interior angle and exterior angle sum to

180^\circ

. With a ratio of

7:2

, the exterior angle represents

\frac{2}{7+2} = \frac{2}{9}

of the total

180^\circ

.

2

Calculate the number of sides (

n

) of the regular polygon.

n = 9

The sum of the exterior angles of any convex polygon is

360^\circ

. Since all exterior angles in a regular polygon are equal,

n = \frac{360^\circ}{40^\circ} = 9

.

3

Calculate the total number of diagonals using the formula

\frac{n(n-3)}{2}

.

Number of diagonals =

27

Substituting

n = 9

into

\frac{n(n-3)}{2}

yields

\frac{9 \times 6}{2} = 27

.

Anahtar Kavram

Interior and exterior angle properties of regular polygons, and the diagonal counting formula for convex polygons.

Soru 2018Soru →

In his analysis of late nineteenth-century industrial monopolies, historian Julian Vance argues that traditional regulatory measures did not merely fail to curb corporate expansion; rather, they inadvertently provided these cartels with an unexpected "shield against the elements." By establishing standardized compliance codes that only capital-rich conglomerates could afford to maintain, state legislatures effectively insulated incumbent firms from the chaotic, unpredictable pressures of emergent competitors. Far from exposing monopolists to rigorous oversight, the new legal framework acted as a protective barrier, allowing established enterprises to weather economic recessions while smaller, more agile ventures withered under the administrative burden. Consequently, what proponents celebrated as public accountability functioned instead as an artificial market sanctuary, neutralizing the very competitive forces intended to keep corporate power in check.

In the passage, the author uses the phrase "shield against the elements" in order to make which of the following points?

describe physical barriers constructed by industrial firms to protect their manufacturing facilities from severe weather conditions

assert that regulatory compliance codes completely eliminated financial volatility across every sector of the national economy

convey how compliance regulations insulated established corporate cartels from the unpredictable pressures of market competition

accuse state legislators of engaging in an explicit corrupt conspiracy to intentionally destroy small businesses

highlight a defensive risk-management strategy that smaller, agile ventures successfully deployed during economic downturns

Cevabı ve açıklamayı göster

Cevap: convey how compliance regulations insulated established corporate cartels from the unpredictable pressures of market competition

Cevap

The author uses the phrase to convey how compliance regulations insulated established corporate cartels from the unpredictable pressures of market competition.

The author uses the metaphorical phrase 'shield against the elements' to illustrate how onerous regulatory standards inadvertently protected large incumbent conglomerates from competitive market forces ('the elements'), serving as a sanctuary rather than a restrictive oversight mechanism.

Adım Adım Çözüm

1

Analyze the target phrase in its immediate context within the passage.

The phrase 'shield against the elements' refers to regulatory measures providing corporate cartels with protection.

Understanding surrounding sentence clues is necessary to decode figurative language.

2

Identify the metaphorical meaning of 'the elements' and 'shield' based on surrounding elaboration.

'The elements' refers to 'the chaotic, unpredictable pressures of emergent competitors' and economic recessions, while the 'shield' refers to high compliance costs that only large conglomerates could afford.

Figurative expressions rely on context clues to define what the metaphor stands for.

3

Match the contextual metaphorical meaning to the option that captures this specific relationship without overgeneralizing or taking the phrase literally.

The option stating that compliance regulations insulated established cartels from competitive market pressures perfectly matches the passage intent.

Direct contextual alignment confirms the correct choice.

Anahtar Kavram

Interpreting Idiomatic Usage and Figurative Language in Context

Soru 2019Soru →

Three positive integers $x$ , $y$ , and $z$ satisfy $\gcd(x, y) = 12$ , $\gcd(y, z) = 18$ , and $\gcd(x, z) = 30$ , where $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b$ . What is the minimum possible value of the least common multiple $\text{lcm}(x, y, z)$ ?

90

180

360

540

1080

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

180

The correct answer is 180. Expressing the pairwise GCDs in prime factorized form shows that

\gcd(x, y) = 2^2 \cdot 3^1

,

\gcd(y, z) = 2^1 \cdot 3^2

, and

\gcd(x, z) = 2^1 \cdot 3^1 \cdot 5^1

. To satisfy these simultaneously while minimizing the overall LCM, the maximum power of 2 across the numbers must be

2^2

, the maximum power of 3 must be

3^2

, and the maximum power of 5 must be

5^1

. Thus,

\text{lcm}(x, y, z) = 2^2 \cdot 3^2 \cdot 5^1 = 180

.

Adım Adım Çözüm

1

Express the given GCD values in their prime factorized forms.

\gcd(x, y) = 12 = 2^2 \cdot 3^1 \cdot 5^0

,

\gcd(y, z) = 18 = 2^1 \cdot 3^2 \cdot 5^0

, and

\gcd(x, z) = 30 = 2^1 \cdot 3^1 \cdot 5^1

.

Analyzing prime factor exponents determines the minimum required power of each prime in

x

,

y

, and

z

.

2

Determine the minimum necessary exponents for prime 2 in

x

,

y

, and

z

.

Since

\gcd(x, y)

has

2^2

, both

x

and

y

must contain at least

2^2

. Thus, the maximum exponent of 2 across the integers is at least 2.

The least common multiple takes the maximum exponent for each prime factor among all three numbers.

3

Determine the minimum necessary exponents for prime 3 in

x

,

y

, and

z

.

Since

\gcd(y, z)

has

3^2

, both

y

and

z

must contain at least

3^2

. Thus, the maximum exponent of 3 across the integers is at least 2.

To satisfy

\gcd(y, z) = 18

, the prime factor 3 must appear with an exponent of at least 2 in both

y

and

z

.

4

Determine the minimum necessary exponents for prime 5 in

x

,

y

, and

z

.

Since

\gcd(x, z)

has

5^1

, both

x

and

z

must contain at least

5^1

. Thus, the maximum exponent of 5 across the integers is at least 1.

To satisfy

\gcd(x, z) = 30

, the prime factor 5 must appear with an exponent of at least 1 in both

x

and

z

.

5

Calculate the minimum least common multiple

\text{lcm}(x, y, z)

.

\text{lcm}(x, y, z) = 2^2 \cdot 3^2 \cdot 5^1 = 4 \cdot 9 \cdot 5 = 180

. (A valid assignment is

x = 60

,

y = 36

,

z = 90

).

Multiplying the minimum maximum prime factor powers yields the smallest possible value for the LCM.

Anahtar Kavram

Prime Factor Exponents in Pairwise GCD and LCM Relations

Tahmini Süre:2m 0s

Soru 2020Soru →

In January, a coffee roasting facility produced two specialty blends: Roast A and Roast B. Roast A sold for $\$10$ per pound and Roast B sold for $\$30$ per pound. A distributor purchased a total of $100$ pounds of coffee consisting of these two blends at an average price of $\$18$ per pound. In February, the price of Roast A increased by $20\%$ , while the price of Roast B decreased by $4\%$ . If the distributor purchased the exact same quantities of Roast A and Roast B in February as in January, by what percent did the total cost of the order increase?

3.85\%

4\%

8\%

10.4\%

16\%

Cevabı ve açıklamayı göster

Cevap: $4\%$

Cevap

The total cost of the order increased by

4\%

.

The choice stating

4\%

is correct because the total cost in January is

\$1,800

(

60

lbs of Roast A at

\$10

and

40

lbs of Roast B at

\$30

). In February, the price of Roast A rises to

\$12

and Roast B drops to

\$28.80

, bringing the total cost to

\$1,872

. The change of

\$72

represents a

4\%

increase relative to the baseline January cost of

\$1,800

.

Adım Adım Çözüm

1

Determine the quantities of Roast A and Roast B purchased in January.

Roast A:

60

lbs, Roast B:

40

lbs.

Let

a

be the pounds of Roast A and

b

be the pounds of Roast B. We are given

a + b = 100

and

10a + 30b = 18 \times 100 = 1800

. Substituting

b = 100 - a

gives

10a + 30(100 - a) = 1800 \implies -20a = -1200 \implies a = 60

and

b = 40

.

2

Calculate the total cost of the order in January.

Total January Cost =

\$1,800

.

Total cost =

(60 \text{ lbs} \times \$10/\text{lb}) + (40 \text{ lbs} \times \$30/\text{lb}) = \$600 + \$1,200 = \$1,800

.

3

Calculate the new prices and total cost in February.

Total February Cost =

\$1,872

.

New price of Roast A =

\$10 \times (1 + 0.20) = \$12/\text{lb}

. New price of Roast B =

\$30 \times (1 - 0.04) = \$28.80/\text{lb}

. Total February Cost =

(60 \times \$12) + (40 \times \$28.80) = \$720 + \$1,152 = \$1,872

.

4

Calculate the percentage increase in total cost from January to February.

Percent Increase =

4\%

.

Percent Change =

\frac{\text{February Cost} - \text{January Cost}}{\text{January Cost}} \times 100\% = \frac{\$1,872 - \$1,800}{\$1,800} \times 100\% = \frac{\$72}{\$1,800} \times 100\% = 4\%

.

Anahtar Kavram

Weighted Percent Change and Systems of Equations

Tahmini Süre:1m 30s

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