All practice questions

Question 1961Question →

A renewable energy facility stores electricity in three separate battery modules: Module $X$ , Module $Y$ , and Module $Z$ . The ratio of the initial energy stored in Module $X$ to Module $Y$ is $2 : 3$ , and the ratio of the initial energy stored in Module $Y$ to Module $Z$ is $5 : 8$ . Each module discharges its stored energy at a constant individual rate. Operating alone, Module $Y$ can completely discharge its initial stored energy in $5$ hours. Module $X$ discharges energy at a rate $33\frac{1}{3}\%$ greater than Module $Y$ , and Module $Z$ discharges energy at a rate $25\%$ less than the combined discharge rate of Modules $X$ and $Y$ . If all three modules begin discharging simultaneously, how many hours will it take to completely discharge the total initial energy stored across all three modules?

3.5 hours

4 hours

5.25 hours

6 hours

7.5 hours

Show answer & explanation

Answer: 4 hours

Answer

4 hours

To find the total time required, first unify the two given ratios. Since

X : Y = 2 : 3

and

Y : Z = 5 : 8

, express both ratios with a common value for

Y

(

15

). This yields

X : Y : Z = 10 : 15 : 24

, so total energy is

10k + 15k + 24k = 49k

. Next, determine rates in terms of

k

: Module

Y

discharges

15k

in

5

hours, so its rate is

3k

units/hr. Module

X

discharges at a rate

33\frac{1}{3}\%

greater than Module

Y

, giving

3k \times \frac{4}{3} = 4k

units/hr. The combined rate of

X

and

Y

is

4k + 3k = 7k

units/hr. Module

Z

discharges at

25\%

less than this combined rate, giving

7k \times 0.75 = 5.25k

units/hr. The simultaneous discharge rate of all three modules is

4k + 3k + 5.25k = 12.25k

units/hr. Dividing total energy

49k

by

12.25k

gives exactly 4 hours.

Step-by-Step Solution

1

Determine the unified initial energy ratio among all three modules.

Energy ratio Module

X

: Module

Y

: Module

Z

=

10 : 15 : 24

, giving a total initial energy of

49k

units.

Module

X : Y = 2 : 3 = 10 : 15

and Module

Y : Z = 5 : 8 = 15 : 24

. Combining these gives

X : Y : Z = 10 : 15 : 24

for a common constant

k

.

2

Calculate the individual discharge rate of Module Y.

Discharge rate of Module

Y

=

3k

energy units per hour.

Module

Y

has

15k

energy units and discharges completely in

5

hours, so

\text{Rate}_Y = \frac{15k}{5} = 3k

.

3

Determine the discharge rates of Module X and Module Z.

Discharge rate of Module

X

=

4k

units per hour; discharge rate of Module

Z

=

5.25k

units per hour.

Module

X

rate is

33\frac{1}{3}\%

greater than Module

Y

:

\text{Rate}_X = 3k \times \left(1 + \frac{1}{3}\right) = 4k

. Combined rate of

X

and

Y

is

4k + 3k = 7k

. Module

Z

rate is

25\%

less than this combined rate:

\text{Rate}_Z = 7k \times 0.75 = 5.25k

.

4

Calculate total combined discharge rate and total time required.

Combined rate =

12.25k

units per hour; Total time =

4

hours.

Total rate

= 4k + 3k + 5.25k = 12.25k

units per hour. Total time

= \frac{\text{Total Energy}}{\text{Total Rate}} = \frac{49k}{12.25k} = 4

hours.

Key Concept

Combining multi-part ratios into a unified scale and calculating combined work rates with percentage adjustments

Question 1962Question →

Fill in the blanks in the passage below to maintain logical coherence and inter-blank structural dependency. What words best complete {{blank_1}} and {{blank_2}}?

Fill in the blanks below

Far from demonstrating that the kingdom's monetary reform was , recent numismatic analysis reveals that the currency debasement was in fact a strategy carefully calculated to stabilize trade during wartime.

Show answer & explanation

Answer

Blank 1 is completed by 'haphazard' (or synonyms such as 'arbitrary') and Blank 2 is completed by 'deliberate' (or synonyms such as 'calculated').

The sentence relies on inter-blank structural dependency mediated by the contrast signal 'Far from'. The trailing modifier 'carefully calculated to stabilize trade' dictates that Blank 2 must mean intentional or well-planned ('deliberate'). Because the opening phrase asserts that the monetary reform was 'far from' the quality described in Blank 1, Blank 1 must describe the opposite of a calculated strategy, namely being random or unplanned ('haphazard').

Step-by-Step Solution

1

Analyze the structural clues and contrast markers in the sentence.

The sentence begins with the contrast pivot 'Far from demonstrating', establishing an inverse relationship between the initial view in Blank 1 and the actual finding described in the second half of the sentence.

Tracking structural pivots is essential to determine whether the two clauses reinforce or contradict one another.

2

Solve Blank 2 using the contextual clue provided in the non-blank portion of the clause.

The phrase 'carefully calculated to stabilize trade' directly defines the nature of the strategy in Blank 2, indicating that it was intentional and planned ('deliberate').

Contextual clues downstream establish the semantic orientation for Blank 2 first.

3

Derive Blank 1 based on its dependency on Blank 2 and the opening pivot.

Since Blank 2 represents a planned strategy ('deliberate') and 'Far from' negates Blank 1 relative to Blank 2, Blank 1 must mean unplanned or random ('haphazard').

Inter-blank dependency requires that Blank 1 oppose the calculated nature confirmed in Blank 2.

Key Concept

Multi-Blank Dependency Tracking

Question 1963Question →

The greatest common divisor of a positive integer $n$ and $120$ is $24$ , and their least common multiple is $2520$ . How many positive divisors does $n$ have?

6

9

12

24

48

Show answer & explanation

Answer: 24

Answer

24

First, write the prime factorizations of

120

, the

\text{gcd}

, and the

\text{lcm}

:
-

120 = 2^3 \cdot 3^1 \cdot 5^1

-

\text{gcd}(n, 120) = 24 = 2^3 \cdot 3^1

-

\text{lcm}(n, 120) = 2520 = 2^3 \cdot 3^2 \cdot 5^1 \cdot 7^1

For any prime

p

,

\min(v_p(n), v_p(120)) = v_p(\text{gcd})

and

\max(v_p(n), v_p(120)) = v_p(\text{lcm})

:
- For

p=2

:

\min(v_2(n), 3) = 3

and

\max(v_2(n), 3) = 3 \implies v_2(n) = 3

.
- For

p=3

:

\min(v_3(n), 1) = 1

and

\max(v_3(n), 1) = 2 \implies v_3(n) = 2

.
- For

p=5

:

\min(v_5(n), 1) = 0 \implies v_5(n) = 0

.
- For

p=7

:

\max(v_7(n), 0) = 1 \implies v_7(n) = 1

.

Thus,

n = 2^3 \cdot 3^2 \cdot 7^1

. The number of positive divisors of

n

is

(3+1)(2+1)(1+1) = 4 \cdot 3 \cdot 2 = 24

. Therefore, the option specifying 24 is correct.

Step-by-Step Solution

1

Find the prime factorizations of

120

,

\text{gcd}(n, 120)

, and

\text{lcm}(n, 120)

.

120 = 2^3 \cdot 3^1 \cdot 5^1

,

\text{gcd} = 24 = 2^3 \cdot 3^1

, and

\text{lcm} = 2520 = 2^3 \cdot 3^2 \cdot 5^1 \cdot 7^1

.

Expressing given quantities in terms of prime factors allows exact determination of the exponents in

n

.

2

Determine the prime factorization of

n = 2^a \cdot 3^b \cdot 5^c \cdot 7^d

.

a = 3

,

b = 2

,

c = 0

, and

d = 1

, giving

n = 2^3 \cdot 3^2 \cdot 7^1 = 504

.

Since

\min(a, 3) = 3

and

\max(a, 3) = 3

,

a=3

. Since

\min(b, 1) = 1

and

\max(b, 1) = 2

,

b=2

. Since

\min(c, 1) = 0

,

c=0

. Since

\max(d, 0) = 1

,

d=1

.

3

Calculate the number of positive divisors of

n

.

(3 + 1)(2 + 1)(1 + 1) = 4 \cdot 3 \cdot 2 = 24

.

The total number of positive divisors for an integer

p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}

is

(e_1 + 1)(e_2 + 1) \cdots (e_k + 1)

.

Key Concept

Prime Factorization, Greatest Common Divisor (GCD), Least Common Multiple (LCM), and Divisor Counting

Estimated Time:1m 30s

Question 1964Question →

Although the minimalist sculptor claimed her stark geometric installations were entirely devoid of narrative intent, recent archival discoveries reveal that her choices were surprisingly __________, deeply rooted in the political turmoil of her era. Which of the following options, when inserted into the blank, produce completed sentences that are equivalent in meaning? Select two choices.

Select all that apply

tendentious

austere

subversive

partisan

spartan

aesthetic

Show answer & explanation

Answer: tendentious; partisan

Answer

The correct options are 'tendentious' and 'partisan'.

The sentence uses the contrast signal 'Although' to set up an opposition between the sculptor's claim of having no narrative intent and the archival finding that her work was tied to political turmoil. The terms 'tendentious' (promoting a specific political cause or bias) and 'partisan' (supporting a political faction) both fill the blank appropriately and produce completed sentences with identical meanings.

Step-by-Step Solution

1

Analyze sentence syntax and structural signals

The opening concession clause ('Although...') indicates a contrast between the sculptor's public claim (that her work lacked narrative intent) and what archival discoveries actually revealed.

Identifying contrast signals establishes the required semantic polarity for the blank.

2

Determine the required contextual meaning for the blank

The clue 'deeply rooted in the political turmoil of her era' requires a word denoting political bias, ideological intent, or factional alignment.

The word filling the blank must oppose being 'devoid of narrative intent' and specifically describe a political orientation.

3

Evaluate candidate option pairs for synonymy and fit

'Tendentious' and 'partisan' both describe political bias or factional commitment, fitting the context and creating equivalent sentence meanings. 'Austere' and 'spartan' form a valid synonym pair but fail to fit the contrast required by the sentence logic.

Sentence equivalence requires selecting two choices that are both contextual synonyms and logically consistent with the sentence structure.

Key Concept

Eliminating false synonym traps and unpaired topical distractors in Sentence Equivalence

Question 1965Question →

A logistics company operates a fleet of $14$ vans consisting solely of Model $X$ and Model $Y$ vans.

- The ratio of the cargo volume capacity of one Model $X$ van to one Model $Y$ van is $3 : 5$ .
- The ratio of the energy consumption rate (in kWh per mile) of one Model $X$ van to one Model $Y$ van is $4 : 5$ .
- When fully loaded, a Model $X$ van carries $30\text{ m}^3$ of cargo and a Model $Y$ van carries $50\text{ m}^3$ of cargo.
- The entire fleet of $14$ vans, when fully loaded, transports a total of $620\text{ m}^3$ of cargo in one trip.
- A Model $X$ van consumes $2\text{ kWh}$ of energy per mile.

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

Select all that apply

There are exactly

4

Model

X

vans in the fleet.

Model

X

vans account for

\frac{3}{8}

of the total cargo capacity transported by the fleet.

The total energy consumed by the entire fleet per mile is

33\text{ kWh}

.

The average energy consumption rate per van in the fleet is

2.25\text{ kWh}

per mile.

If the cargo capacity of each Model

Y

van is increased by

20\%

, the total cargo capacity of the entire fleet would increase by

20\%

.

Show answer & explanation

Answer: There are exactly $4$ Model $X$ vans in the fleet.; The total energy consumed by the entire fleet per mile is $33\text{ kWh}$ .

Answer

The correct statements are that there are exactly 4 Model X vans in the fleet and that the total energy consumed by the entire fleet per mile is 33 kWh.

The statement specifying that there are exactly 4 Model X vans is correct because solving the system

n_X + n_Y = 14

and

30 n_X + 50 n_Y = 620

gives

n_X = 4

. The statement asserting that total fleet energy consumption per mile is 33 kWh is also correct because

4(2\text{ kWh}) + 10(2.5\text{ kWh}) = 33\text{ kWh}

.

Step-by-Step Solution

1

Determine the number of Model X and Model Y vans using fleet totals.

Let

n_X

be the number of Model X vans and

n_Y

be the number of Model Y vans. Given

n_X + n_Y = 14

, substitute

n_Y = 14 - n_X

into the total cargo equation:

30 n_X + 50(14 - n_X) = 620 \implies 700 - 20 n_X = 620 \implies 20 n_X = 80 \implies n_X = 4

. Thus,

n_Y = 10

.

This establishes the exact composition of the fleet.

2

Evaluate the fraction of cargo carried by Model X vans.

Total cargo carried by Model X vans =

4 \times 30 = 120\text{ m}^3

. Fleet total cargo =

620\text{ m}^3

. Fraction =

\frac{120}{620} = \frac{6}{31}

.

Shows that the fleet cargo share depends on the van counts, not just the single-van ratio of 3 to 5.

3

Calculate individual consumption rates and total fleet consumption per mile.

The consumption rate ratio of Model X to Model Y is

4 : 5

. Given Model X rate is

2\text{ kWh/mile}

, Model Y rate is

2 \times \frac{5}{4} = 2.5\text{ kWh/mile}

. Total fleet consumption per mile =

4(2) + 10(2.5) = 8 + 25 = 33\text{ kWh}

.

Applies unit rates to the known number of each van model to find total rate.

4

Calculate the true weighted average consumption rate per van.

Average consumption rate per van =

\frac{\text{Total energy per mile}}{\text{Total vans}} = \frac{33\text{ kWh}}{14\text{ vans}} \approx 2.36\text{ kWh/mile}

.

Demonstrates why taking an unweighted mean

(2.0 + 2.5)/2 = 2.25

is incorrect when group sizes are unequal.

5

Analyze the impact of a 20% increase in Model Y capacity.

Initial Model Y total capacity =

10 \times 50 = 500\text{ m}^3

. A

20\%

increase adds

0.20 \times 500 = 100\text{ m}^3

. Percent increase in total fleet capacity =

\frac{100}{620} \approx 16.13\%

.

A percentage change in a part does not equal the percentage change in the whole.

Key Concept

Weighted averages and component vs. whole relationships in multi-part ratios and rates.

Question 1966Question →

In the $xy$ -plane, line $L$ passes through the origin $(0, 0)$ and the point $P(a, b)$ , where $a > 0$ and $b > 0$ . Line $M$ is perpendicular to line $L$ at point $P$ . If line $M$ has a $y$ -intercept at $(0, 10)$ and $b = 2$ , what is the value of $a$ ?

2\sqrt{2}

4

4\sqrt{2}

8

16

Show answer & explanation

Answer: $4$

Answer

The value of

a

is

4

.

Line

L

connects

(0,0)

to

(a,2)

, giving it a slope of

\frac{2}{a}

. Because line

M

is perpendicular to line

L

, its slope must be the negative reciprocal,

-\frac{a}{2}

. Line

M

also connects

(a,2)

to its

y

-intercept

(0,10)

, so its slope can independently be written as

\frac{10-2}{0-a} = -\frac{8}{a}

. Setting

-\frac{a}{2} = -\frac{8}{a}

yields

a^2 = 16

. Since

a > 0

,

a = 4

.

Step-by-Step Solution

1

Find the slope of line

L

Since line

L

passes through

(0, 0)

and

P(a, 2)

, its slope is

m_L = \frac{2 - 0}{a - 0} = \frac{2}{a}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Determine the slope of perpendicular line

M

The slope of line

M

is

m_M = -\frac{1}{m_L} = -\frac{a}{2}

.

Perpendicular lines have negative reciprocal slopes.

3

Calculate the slope of line

M

using given points

(a, 2)

and

(0, 10)

Using the slope formula:

m_M = \frac{10 - 2}{0 - a} = \frac{8}{-a} = -\frac{8}{a}

.

Line

M

passes through the point of intersection

P(a, 2)

and its

y

-intercept

(0, 10)

.

4

Equate the two slope expressions and solve for

a

-\frac{a}{2} = -\frac{8}{a} \implies a^2 = 16 \implies a = 4

(since

a > 0$).

Both expressions represent the slope of line

M

.

Key Concept

Perpendicular Slopes and Line Equations in Coordinate Geometry

Estimated Time:1m 30s

Question 1967Question →

Two events $E$ and $F$ in a sample space have probabilities $P(E) = 0.40$ and $P(F) = 0.50$ . Which of the following statements must be true? Select all such statements.

Select all that apply

If events

E

and

F

are mutually exclusive, then

P(E \text{ or } F) = 0.90

.

If events

E

and

F

are independent, then

P(E \text{ and } F) = 0.20

.

Events

E

and

F

cannot be both mutually exclusive and independent.

If events

E

and

F

are independent, then

P(E \text{ or } F) = 0.90

.

If events

E

and

F

are mutually exclusive, then

P(E \text{ and } F) = 0.90

.

Show answer & explanation

Answer: If events $E$ and $F$ are mutually exclusive, then $P(E \text{ or } F) = 0.90$ .; If events $E$ and $F$ are independent, then $P(E \text{ and } F) = 0.20$ .; Events $E$ and $F$ cannot be both mutually exclusive and independent.

Answer

The correct statements are: 'If events E and F are mutually exclusive, then P(E or F) = 0.90', 'If events E and F are independent, then P(E and F) = 0.20', and 'Events E and F cannot be both mutually exclusive and independent.'

The statement regarding mutually exclusive events correctly uses the addition rule P(E or F) = P(E) + P(F) = 0.90 because there is no overlap. The statement regarding independent events correctly applies the multiplication rule P(E and F) = P(E) * P(F) = 0.20. Finally, events with non-zero probabilities cannot be both mutually exclusive (requiring zero intersection) and independent (requiring positive intersection), making the impossibility statement correct.

Step-by-Step Solution

1

Evaluate the union probability for mutually exclusive events.

P(E or F) = P(E) + P(F) = 0.40 + 0.50 = 0.90.

By definition of mutually exclusive events, P(E and F) = 0.

2

Evaluate the joint probability for independent events.

P(E and F) = P(E) * P(F) = 0.40 * 0.50 = 0.20.

The multiplication rule applies directly to independent events.

3

Evaluate whether mutually exclusive events with non-zero probabilities can be independent.

They cannot be both mutually exclusive and independent.

Mutual exclusivity requires P(E and F) = 0, whereas independence requires P(E and F) = 0.20, which are contradictory.

4

Evaluate the union probability for independent events.

P(E or F) = 0.40 + 0.50 - 0.20 = 0.70.

The general addition rule P(E or F) = P(E) + P(F) - P(E and F) must account for the overlap.

Key Concept

Distinction between independent events and mutually exclusive events in probability

Question 1968Question →

A sequence $a_1, a_2, a_3, \dots$ is defined by $a_1 = 3$ and $a_{n+1} = \sqrt{a_n + 12}$ for all integers $n \geq 1$ . If $S_{100} = \sum_{k=1}^{100} a_k$ , what is the value of $S_{100}$ rounded to the nearest integer?

396

398

399

400

401

Show answer & explanation

Answer: 399

Answer

399

The sequence terms approach the fixed limit

L = 4

defined by

L = \sqrt{L + 12}

. Each term

a_k

is strictly less than 4, with difference

\epsilon_k = 4 - a_k

. Evaluating the first few differences yields

\epsilon_1 = 1

,

\epsilon_2 \approx 0.127

,

\epsilon_3 \approx 0.016

,

\epsilon_4 \approx 0.002

, and subsequent terms are negligible. The total sum

S_{100} = 100 \times 4 - \sum_{k=1}^{100} \epsilon_k \approx 400 - 1.145 = 398.855

. Rounding

398.855

to the nearest integer gives 399.

Step-by-Step Solution

1

Determine the limit/fixed point of the recursive sequence.

Solving

L = \sqrt{L + 12}

yields

L^2 - L - 12 = 0 \implies (L - 4)(L + 3) = 0

. Since terms are positive,

L = 4

.

Understanding the upper limit

L = 4

helps set up a upper bound for all terms in the sequence.

2

Calculate the first few terms and their deficits

\epsilon_k = 4 - a_k

from the limit 4.

\epsilon_1 = 4 - 3 = 1

;

\epsilon_2 = 4 - \sqrt{15} \approx 0.12702

;

\epsilon_3 = 4 - \sqrt{15.87298} \approx 0.01591

;

\epsilon_4 \approx 0.00199

;

\epsilon_5 \approx 0.00025

.

Because convergence is extremely fast (each error term shrinks by a factor greater than 8), only the first few deficit terms contribute meaningfully to the sum.

3

Sum the total deficit

\sum_{k=1}^{100} \epsilon_k

.

Total deficit

\approx 1 + 0.12702 + 0.01591 + 0.00199 + 0.00025 + \dots \approx 1.145

.

Adding all error terms gives the precise deviation of

S_{100}

from

100 \times 4 = 400

.

4

Compute

S_{100}

and round to the nearest integer.

S_{100} = 400 - 1.145 = 398.855

. Rounded to the nearest integer,

398.855

rounds to

399

.

Since

398.855 \geq 398.5

, rounding to the nearest integer yields 399.

Key Concept

Fixed-point sequence convergence, summation bounding, and rounding

Estimated Time:2m 30s

Question 1969Question →

In a convex polygon, the sum of the measures of all interior angles except one is $1,200^\circ$ . If the measure of the remaining interior angle is an integer degree, what is the number of sides of the polygon?

7

8

9

10

11

Show answer & explanation

Answer: 9

Answer

The number of sides of the polygon is 9.

The interior angle sum of an

n

-sided polygon is

(n - 2) \times 180^\circ

. Because the polygon is convex, the missing angle

x

must be between

0^\circ

and

180^\circ

. Adding

x

to

1,200^\circ

gives a total sum between

1,200^\circ

and

1,380^\circ

. The only multiple of

180^\circ

in this range is

1,260^\circ

, which corresponds to

(n - 2) = 7

, or

n = 9

sides.

Step-by-Step Solution

1

Set up the inequality for the interior angle sum of a convex polygon

The sum of the interior angles of an

n

-sided convex polygon is

S = (n - 2) \times 180^\circ

. Let

x

be the measure of the remaining interior angle, where

0^\circ < x < 180^\circ

.

Every interior angle of a convex polygon must be strictly greater than

0^\circ

and strictly less than

180^\circ

.

2

Formulate the bounds for

(n - 2) \times 180^\circ

Since

S = 1,200^\circ + x

, we have

1,200^\circ < (n - 2) \times 180^\circ < 1,200^\circ + 180^\circ = 1,380^\circ

.

Adding the bounds of

x

(

0^\circ

to

180^\circ

) to the given sum of

1,200^\circ

establishes the valid range for the total angle sum.

3

Solve for the integer value of

n

Dividing the inequality by

180^\circ

gives

6.67 < n - 2 < 7.67

. Since

n - 2

must be an integer,

n - 2 = 7

, which means

n = 9

.

The only integer in the range

(6.67, 7.67)

is 7, corresponding to a 9-sided polygon with a remaining angle of

x = 1,260^\circ - 1,200^\circ = 60^\circ

.

Key Concept

Polygon Interior Angle Sum Theorem

Alternative Method

Divide

1,200^\circ

by

180^\circ

to get

6.67

. Since

(n - 2)

must be an integer and the remaining angle is positive and less than

180^\circ

, round

6.67

up to the next integer, 7. Thus,

n - 2 = 7

, so

n = 9

.

Estimated Time:1m 15s

Question 1970Question →

Read the passage below and determine the contextual meaning of the highlighted word.

Fill in the blanks below

In his landmark 1974 treatise on architectural phenomenology, Gaston Bachelard argued that spatial configurations do not merely contain human experience; rather, they actively inform the psychological orientation of their inhabitants. Where mid-century functionalists viewed domestic spaces as passive containers designed solely to satisfy pragmatic needs, Bachelard asserted that subtle structural choices��such as the verticality of a staircase or the enclosure of a cellar—impart a specific imaginative coherence to daily life. Consequently, to suggest that an edifice merely holds its occupants is to misunderstand how architectural geometry continuously shapes the interior landscape of the mind.

As used in the passage, the word inform most nearly means .

Show answer & explanation

Answer

The word 'inform' in this context means 'shape' or 'give form to'.

In the passage, the author uses 'inform' to describe how spatial configurations actively give structure, form, or character to an inhabitant's psychological orientation. This is supported by parallel expressions in the passage, such as structural choices that 'impart... imaginative coherence' and architectural geometry that 'shapes the interior landscape of the mind'.

Step-by-Step Solution

1

Analyze the sentence structure and surrounding context.

The text contrasts 'merely contain human experience' with 'actively inform the psychological orientation'. Later, the passage notes that structural choices 'impart a specific imaginative coherence' and 'continuously shapes the interior landscape'.

Contextual clues and parallel phrasing ('impart', 'shapes') establish that the word refers to molding or giving structure to psychological orientation, rather than transmitting factual information.

2

Differentiate between the primary definition and the secondary contextual meaning.

While 'inform' commonly means to communicate facts or notify, here it operates in its secondary sense of giving intrinsic form, shape, or character to something.

The passage focuses on how physical architecture actively forms mental states, making 'shape' or 'give form to' the precise contextual meaning.

Key Concept

Reading Comprehension: Passage-Based Word in Context Analysis

Question 1971Question →

A manufacturing facility produces precision metal components. Each component has a mass of $8.4 \times 10^{-4}$ kilograms. If a shipment container holds a batch of these components with a total mass of $1.05$ kilograms, how many components are in the container?

Show answer & explanation

Answer: 1250

Answer

The container holds 1,250 components.

To find the number of components, divide the total batch mass by the mass per component:

\frac{1.05 \text{ kg}}{8.4 \times 10^{-4} \text{ kg}}

. Expressing

\frac{1}{10^{-4}}

as

10^4

transforms the expression into

\frac{1.05}{8.4} \times 10^4

. Dividing

1.05

by

8.4

yields

0.125

. Finally,

0.125 \times 10^4 = 1250

.

Step-by-Step Solution

1

Set up the ratio of total mass to single component mass

Number of components =

\frac{1.05}{8.4 \times 10^{-4}}

Dividing total mass by individual component mass gives the total count.

2

Apply exponent rules to move the power of 10 to the numerator

\frac{1.05}{8.4} \times 10^4

Since

\frac{1}{10^{-4}} = 10^4

, shifting the negative exponent to the numerator changes its sign.

3

Divide the decimal coefficients

\frac{1.05}{8.4} = 0.125

Simplifying

\frac{105}{840}

reduces to

\frac{1}{8} = 0.125

.

4

Evaluate the product with the place value shift

0.125 \times 10,000 = 1250

Multiplying by

10^4

shifts the decimal point 4 places to the right.

Key Concept

Division with scientific notation and decimal place value adjustment

Estimated Time:1m 30s

Question 1972Question →

In GRE Sentence Equivalence items, test-takers must evaluate structural contrast clues to eliminate false synonym traps and topically related distractors. Consider the following sentence context:

"Far from presenting an entirely __________ ecosystem, the hydrothermal vent community exhibited an extraordinary degree of biodiversity that defied early marine biological models."

Match each candidate option pair below with its correct functional distractor classification.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings match "barren" and "desolate" to the Correct Contextual Pair; "teeming" and "abundant" to the False Synonym Trap; "abyssal" and "benthic" to Topically Related Distractors; and "sparse" and "prolific" to the Antonymic Pair.

Matching "barren" and "desolate" to the Correct Contextual Pair is correct because both words mean devoid of life, satisfying the structural contrast initiated by "Far from" and completing the sentence with equivalent meanings. "Teeming" and "abundant" form a False Synonym Trap as they reverse sentence logic. "Abyssal" and "benthic" represent Topically Related Distractors since they pertain to oceanography but lack pair equivalence. "Sparse" and "prolific" form an Antonymic Pair because their meanings directly oppose one another.

Step-by-Step Solution

1

Analyze the sentence context and structural transition signals.

The phrase "Far from presenting an entirely __________ ecosystem" establishes a contrast with "exhibited an extraordinary degree of biodiversity." The blank requires words meaning empty or lifeless.

Identifying sentence valence and structural clues determines the target meaning required for the blank.

2

Evaluate the pair "barren" and "desolate".

Both words mean lacking vegetation or life. When plugged into the sentence, both assert that the ecosystem was far from empty, matching the high biodiversity context.

This establishes the pair as the Correct Contextual Pair.

3

Evaluate the pair "teeming" and "abundant".

These words are true synonyms meaning plentiful, but using them creates a logical contradiction due to the contrast marker "Far from".

Synonyms that fit the subject matter but violate sentence logic constitute a False Synonym Trap.

4

Evaluate "abyssal" and "benthic" alongside "sparse" and "prolific".

"Abyssal" and "benthic" share a marine theme but are not synonymous in this context. "Sparse" and "prolific" are antonyms.

"Abyssal" and "benthic" are Topically Related Distractors, whereas "sparse" and "prolific" form an Antonymic Pair.

Key Concept

Eliminating False Synonyms and Topically Related Distractors

Question 1973Question →

A commercial property was purchased at an initial value of $V$ dollars. In its first year, the value of the property increased by $25\%$ . In its second year, the property's value decreased by $20\%$ relative to its value at the end of the first year. In its third year, the value of the property increased by $15\%$ relative to its value at the end of the second year. If the value of the property at the end of the third year was $\$276,000$ , what was the initial purchase value $V$ , in dollars, of the property?

Show answer & explanation

Answer: 240000

Answer

240000

Each percent change applies to the value at the end of the previous period. A

25\%

increase scales the value by

1.25

, a

20\%

decrease scales it by

0.80

, and a

15\%

increase scales it by

1.15

. The net multiplier is

1.25 \times 0.80 \times 1.15 = 1.15

. Solving

1.15V = 276,000

gives

V = 240,000

.

Step-by-Step Solution

1

Convert percent changes to growth and decay multiplier factors

Year 1 factor = 1.25, Year 2 factor = 0.80, Year 3 factor = 1.15

Percent changes are compounded by multiplying successive growth/decay factors.

2

Multiply the factors sequentially to express the final value in terms of initial value V

Final Value = V × 1.25 × 0.80 × 1.15 = 1.15V

Combining successive multipliers simplifies the relationship between original and final amounts.

3

Set the algebraic expression equal to $276,000 and solve for V

V = 276,000 / 1.15 = 240,000

Dividing the final value by the overall multiplier determines the original base value.

Key Concept

Successive Percent Changes and Base Shift

Estimated Time:1m 30s

Question 1974Question →

At an automated agricultural sorting facility, harvested apples are inspected for two independent quality conditions: surface bruising and internal core rot. The probability that a randomly chosen apple has surface bruising is $\frac{1}{6}$ , and the probability that it has internal core rot is $\frac{1}{5}$ . What is the probability that a randomly chosen apple has at least one of these two quality conditions?

\frac{1}{30}

\frac{2}{11}

\frac{1}{3}

\frac{11}{30}

\frac{2}{5}

Show answer & explanation

Answer: $\frac{1}{3}$

Answer

\frac{1}{3}

The probability of at least one of two independent events occurring is given by

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

. Since the events are independent,

P(B \cap R) = P(B) \times P(R) = \frac{1}{6} \times \frac{1}{5} = \frac{1}{30}

. Substituting these values gives

\frac{1}{6} + \frac{1}{5} - \frac{1}{30} = \frac{5}{30} + \frac{6}{30} - \frac{1}{30} = \frac{10}{30} = \frac{1}{3}

. Alternatively, using the complement rule,

P(\text{at least one}) = 1 - P(\text{neither}) = 1 - \left(1 - \frac{1}{6}\right)\left(1 - \frac{1}{5}\right) = 1 - \left(\frac{5}{6} \times \frac{4}{5}\right) = 1 - \frac{4}{6} = \frac{1}{3}

.

Step-by-Step Solution

1

Identify the given probabilities and event relationship.

Let

B

be the event that an apple has surface bruising, so

P(B) = \frac{1}{6}

. Let

R

be the event that an apple has internal core rot, so

P(R) = \frac{1}{5}

. The events are given as independent.

Establishing the mathematical definitions and given conditions is necessary before applying probability rules.

2

Calculate the joint probability of both events occurring,

P(B \cap R)

.

P(B \cap R) = P(B) \times P(R) = \frac{1}{6} \times \frac{1}{5} = \frac{1}{30}

.

For independent events, the probability that both occur simultaneously is the product of their individual probabilities.

3

Apply the addition rule of probability (inclusion-exclusion principle) to find

P(B \cup R)

.

P(B \cup R) = P(B) + P(R) - P(B \cap R) = \frac{1}{6} + \frac{1}{5} - \frac{1}{30} = \frac{5}{30} + \frac{6}{30} - \frac{1}{30} = \frac{10}{30} = \frac{1}{3}

.

The probability of at least one event occurring requires subtracting the intersection so that the overlapping outcome is not counted twice.

Key Concept

Probability of the Union of Independent Events

Estimated Time:1m 30s

Question 1975Question →

A chemical synthesis plant uses two reaction vessels, Vessel 1 and Vessel 2, to produce a liquid solution composed entirely of Compound X, Compound Y, and water.

- Vessel 1 produces the solution at a constant rate of $60$ liters per hour, with Compound X, Compound Y, and water in the volume ratio $2 : 1 : 3$ , respectively.
- Vessel 2 produces the solution at a constant rate of $90$ liters per hour, with Compound X, Compound Y, and water in the volume ratio $1 : 3 : 2$ , respectively.

If both vessels operate simultaneously to fill an initially empty storage tank, which of the following statements must be true regarding the mixture in the storage tank after operating for any duration $H$ hours ( $H > 0$ )? Select all such statements.

Select all that apply

Compound Y accounts for exactly

\frac{11}{30}

of the total volume of solution in the storage tank.

The ratio of the volume of Compound X to the volume of water in the storage tank is

7 : 12

.

The volume of Compound X in the storage tank is greater than the volume of Compound Y in the storage tank.

Water constitutes strictly more than

40\%

of the total volume of solution in the storage tank.

Compound X represents exactly

\frac{7}{18}

of the total volume of solution in the storage tank.

Show answer & explanation

Answer: Compound Y accounts for exactly $\frac{11}{30}$ of the total volume of solution in the storage tank.; The ratio of the volume of Compound X to the volume of water in the storage tank is $7 : 12$ .

Answer

The statements confirming that Compound Y accounts for exactly 11/30 of the total solution volume and that the ratio of Compound X to water is 7:12 are both correct.

The total volume rate of solution produced by both vessels combined is 150 liters per hour. Within this total, Compound X is produced at 35 liters per hour, Compound Y at 55 liters per hour, and water at 60 liters per hour. Consequently, Compound Y accounts for 55 / 150 = 11/30 of the total volume, and the ratio of Compound X to water is 35 : 60 = 7 : 12. Both of these statements must be true.

Step-by-Step Solution

1

Calculate the hourly output rates for each component from Vessel 1.

Total rate = 60 L/hr. Ratio X:Y:Water = 2:1:3 (sum of parts = 6). Compound X = 60 * (2/6) = 20 L/hr; Compound Y = 60 * (1/6) = 10 L/hr; Water = 60 * (3/6) = 30 L/hr.

Decompose the total production rate into component rates using part-to-whole fractions.

2

Calculate the hourly output rates for each component from Vessel 2.

Total rate = 90 L/hr. Ratio X:Y:Water = 1:3:2 (sum of parts = 6). Compound X = 90 * (1/6) = 15 L/hr; Compound Y = 90 * (3/6) = 45 L/hr; Water = 90 * (2/6) = 30 L/hr.

Decompose the second vessel's production rate into component rates.

3

Combine the component rates from both vessels.

Total solution rate = 60 + 90 = 150 L/hr. Compound X = 20 + 15 = 35 L/hr; Compound Y = 10 + 45 = 55 L/hr; Water = 30 + 30 = 60 L/hr.

Find the overall composition delivered to the storage tank per hour.

4

Evaluate each candidate statement against the combined component rates.

Fraction of Y = 55 / 150 = 11/30 (True). Ratio X to Water = 35 : 60 = 7 : 12 (True). X (35 L/hr) > Y (55 L/hr) (False). Water percentage = 60 / 150 = 40% (Not strictly greater than 40%, so False). Fraction of X = 35 / 150 = 7/30 != 7/18 (False).

Determine which statements are mathematically true for any operating duration H > 0.

Key Concept

Weighted rate aggregation and multi-component ratio analysis

Question 1976Question →

Read the passage below:

Although early art conservators viewed the anomalous chemical compounds detected in the Renaissance painting as unequivocal evidence of modern (i)________, recent non-destructive spectroscopic analyses demonstrated that these compounds resulted from gradual ambient degradation over centuries, thereby (ii)________ long-standing allegations of deliberate forgery.

Which of the following word pairs best completes the blanks in the passage so that the text maintains logical coherence and proper inter-blank structural dependency?

(i) chicanery ... (ii) debunking

(i) chicanery ... (ii) substantiating

(i) authenticity ... (ii) debunking

(i) provenance ... (ii) exacerbating

(i) erudition ... (ii) reinforcing

Show answer & explanation

Answer: (i) chicanery ... (ii) debunking

Answer

The pair '(i) chicanery ... (ii) debunking' correctly completes the passage by establishing inter-blank logical dependency.

The correct answer pair '(i) chicanery ... (ii) debunking' creates complete semantic coherence across both clauses. In Blank (i), 'chicanery' (deception or trickery) reflects the initial belief that anomalous pigments signified modern forgery. In Blank (ii), 'debunking' (disproving) logically follows the discovery that the chemical compounds were actually produced by natural degradation, thereby disproving the allegations of forgery.

Step-by-Step Solution

1

Analyze structural transition signals and contextual clues.

The concession signal 'Although' establishes a contrast between initial suspicions and recent scientific findings.

The initial perception must contrast with the revelation that the compounds formed naturally over centuries.

2

Determine the inter-blank dependency between Blank (i) and Blank (ii).

If Blank (i) represents suspected deception (e.g., forgery/chicanery), then proving the compounds were natural means Blank (ii) must mean disproving or exposing those allegations as false.

The discovery of natural ambient degradation refutes the hypothesis of modern tampering.

3

Evaluate option combinations against inter-blank polarity.

Only '(i) chicanery ... (ii) debunking' maintains aligned negative initial suspicion paired with positive disproof of forgery allegations.

'Chicanery' accurately denotes deception/forgery, and 'debunking' accurately captures the invalidation of the allegations.

Key Concept

Multi-Blank Dependency Tracking

Question 1977Question →

In the $xy$ -plane, line $k$ passes through the points $(-3, 5)$ and $(3, 1)$ . Line $m$ is perpendicular to line $k$ and passes through the point $(1, -1)$ . Which of the following statements must be true? Indicate all such statements.

Select all that apply

The slope of line

m

is

\frac{3}{2}

.

Line

m

passes through Quadrants I, III, and IV.

The

x

-intercept of line

k

is

\left(\frac{9}{2}, 0\right)

.

The slope of line

k

is

-\frac{3}{2}

.

Line

k

and line

m

intersect at the point

(3, 1)

.

Show answer & explanation

Answer: The slope of line $m$ is $\frac{3}{2}$ .; Line $m$ passes through Quadrants I, III, and IV.; The $x$ -intercept of line $k$ is $\left(\frac{9}{2}, 0\right)$ .

Answer

The statements confirming that the slope of line

m

is

\frac{3}{2}

, line

m

passes through Quadrants I, III, and IV, and the

x

-intercept of line

k

is

\left(\frac{9}{2}, 0\right)

are all correct.

The slope of line

k

is

-\frac{2}{3}

, making line

m

's perpendicular slope

\frac{3}{2}

. The equation for line

m

is

y = \frac{3}{2}x - \frac{5}{2}

, which crosses the

y

-axis at

\left(0, -\frac{5}{2}\right)

and the

x

-axis at

\left(\frac{5}{3}, 0\right)

, thus entering Quadrants I, III, and IV. The

x

-intercept of line

k

is found by setting

y = 0

in

y = -\frac{2}{3}x + 3

, yielding

x = \frac{9}{2}

.

Step-by-Step Solution

1

Calculate the slope of line

k

using the slope formula

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

.

Slope of line

k

is

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

.

Determining the slope of line

k

is required to find perpendicular slopes and line equations.

2

Determine the slope and equation of line

m

.

Perpendicular slope is

m_m = -\frac{1}{m_k} = \frac{3}{2}

. Using point-slope form with point

(1, -1)

,

y - (-1) = \frac{3}{2}(x - 1) \Rightarrow y = \frac{3}{2}x - \frac{5}{2}

.

Perpendicular lines have negative reciprocal slopes.

3

Find the

x

-intercept of line

k

and analyze the quadrant passage of line

m

.

Line

k

equation is

y = -\frac{2}{3}x + 3

; setting

y = 0

yields

x = \frac{9}{2}

. For line

m

, positive slope and negative

y

-intercept

\left(0, -\frac{5}{2}\right)

mean it traverses Quadrants I, III, and IV.

Setting

y=0

gives the

x

-intercept, and evaluating the line equation across negative, zero, and positive

x

-values identifies quadrant coverage.

Key Concept

Coordinate Geometry: Line equations, slopes of perpendicular lines, intercepts, and quadrant passage.

Question 1978Question →

A technical consultant charges a one-time setup fee of $\$150$ plus a standard rate of $\$45$ per hour for the first $20$ hours of work on a project. For any hours worked beyond $20$ hours, the consultant charges an increased hourly rate that is $20\%$ higher than the standard rate. If the total bill for a project was $\$1,320$ , how many total hours did the consultant work on the project?

Show answer & explanation

Answer: 25

Answer

The consultant worked a total of 25 hours on the project.

To find the total hours worked, first subtract the setup fee (

150) and the cost of the first 20 hours (20 *

45 =

900) from the total bill of

1,320. This leaves

1,320 -

1,050 =

270. The hourly rate after 20 hours increases by 20% to

45 * 1.20 =

54 per hour. Dividing the remaining

270 by $54 gives 5 additional hours. Adding these 5 hours to the initial 20 hours gives a total of 25 hours.

Step-by-Step Solution

1

Determine the base cost for the setup fee and the initial 20 hours.

Base cost =

150 + (20 *

45) = $1,050.

The initial cost tier applies up to 20 hours of work.

2

Calculate the higher hourly rate applied to additional hours worked beyond 20.

Increased rate =

45 * 1.20 =

54 per hour.

The rate increases by 20% over the standard rate of $45 per hour.

3

Formulate and solve a linear equation for total hours

h

.

1050 + 54(h - 20) = 1320 \implies 54(h - 20) = 270 \implies h - 20 = 5 \implies h = 25

.

Subtracting the base cost leaves

270 for overtime hours, which divides by

54 per hour to yield 5 additional hours, for 25 hours total.

Key Concept

Linear Equations in One Variable

Question 1979Question →

In contemporary cognitive neuroscience, early models of lexical retrieval posited a strictly modular architecture wherein visual word recognition operated independently of sentence-level context. Recent neuroimaging evidence, however, has undermined this rigid framework; far from functioning as an autonomous processor, the visual word form area exhibits rapid sensitivity to top-down contextual expectations. Indeed, far from remaining static or fixed, the neural representations governing early orthographic processing are remarkably plastic, dynamically altering their selectivity to align with surrounding semantic signals. Consequently, researchers now view early visual processing as an integrative rather than encapsulated stage of reading.

In the passage above, which of the following best captures the meaning of the word 'plastic' as used by the author?

synthetic

adaptable

impressionable

immutable

superficial

Show answer & explanation

Answer: adaptable

Answer

adaptable

The term 'adaptable' correctly captures the meaning of 'plastic' in this context. The passage employs the contrast signal 'far from remaining static or fixed' paired with the continuation clue 'dynamically altering their selectivity'. These structural markers indicate that the neural representations are capable of change and responsiveness to context, making 'adaptable' the exact semantic fit.

Step-by-Step Solution

1

Locate the target word and identify its immediate sentence environment.

The word 'plastic' is used to describe 'neural representations governing early orthographic processing' in the sentence beginning with 'Indeed...'

Contextual meaning must be derived from surrounding syntactical structures rather than prior memory of the dictionary definition.

2

Analyze structural transition signals and contrast clues in the sentence.

The phrase 'far from remaining static or fixed' creates a direct contrast with 'plastic', while 'dynamically altering their selectivity' provides an elaborative continuation clue.

Structural signals establish that 'plastic' signifies the opposite of 'static or fixed' and is synonymous with dynamic alteration.

3

Evaluate the options against the identified structural meaning.

The term 'adaptable' directly matches the concept of dynamically altering selectivity and contrasting with fixed states.

Choosing 'adaptable' preserves the semantic coherence of both the sentence and the author's broader argument regarding integrative neural processing.

Key Concept

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Question 1980Question →

If $x$ is a real number such that $0 < x < 1$ , which of the following expressions is equivalent to $\sqrt{\frac{x^{-2} - 1}{x^{-2} + 2 + x^2}}$ ?

\frac{\sqrt{1-x^2}}{1+x^2}

\frac{1-x}{1+x^2}

\frac{1-x^2}{1+x^2}

\frac{\sqrt{1-x^2}}{(1+x)^2}

\frac{1-x}{1+x}

Show answer & explanation

Answer: $\frac{\sqrt{1-x^2}}{1+x^2}$

Answer

The expression

\frac{\sqrt{1-x^2}}{1+x^2}

is equivalent to the given radical expression.

Expressing

x^{-2}

as

\frac{1}{x^2}

allows the numerator to be rewritten as

\frac{1-x^2}{x^2}

and the denominator as

\frac{(1+x^2)^2}{x^2}

. Dividing these fractions cancels out

x^2

, leaving

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

under the radical. Taking the square root of the numerator and denominator separately gives

\frac{\sqrt{1-x^2}}{1+x^2}

.

Step-by-Step Solution

1

Rewrite negative exponents as fractions in both the numerator and denominator.

Numerator:

x^{-2} - 1 = \frac{1}{x^2} - 1 = \frac{1-x^2}{x^2}

. Denominator:

x^{-2} + 2 + x^2 = \frac{1}{x^2} + 2 + x^2 = \frac{1 + 2x^2 + x^4}{x^2}

.

Converting negative exponents into positive fractional exponents allows common denominators to be established.

2

Factor the perfect square trinomial in the denominator.

The numerator of the denominator expression is

1 + 2x^2 + x^4 = (1 + x^2)^2

. Thus, the entire denominator is

\frac{(1+x^2)^2}{x^2}

.

Recognizing

1 + 2x^2 + x^4

as

(1+x^2)^2

simplifies taking the square root.

3

Simplify the quotient inside the radical.

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

Canceling the common factor of

x^2

in the denominators reduces the nested fraction.

4

Apply the square root rule

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

.

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

Since

1+x^2 > 0

for all real

x

,

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

.

Key Concept

Simplifying radical expressions containing negative powers and algebraic fractions

Estimated Time:2m 0s

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