All practice questions

Question 121Question →

In the $xy$ -plane, line $k$ passes through the points $(1, -2)$ and $(5, 6)$ . Line $m$ is the perpendicular bisector of the line segment connecting $(1, -2)$ and $(5, 6)$ . Which of the following statements must be true? Select all that apply.

Select all that apply

Line

m

passes through the point

(7, 0)

.

Line

m

has a

y

-intercept of

(0, 3.5)

.

The distance from the origin

(0, 0)

to the midpoint of the line segment is

\sqrt{13}

.

The slope of line

m

is

2

.

Line

m

is parallel to the line with equation

2x - y = 4

.

Show answer & explanation

Answer: Line $m$ passes through the point $(7, 0)$ .; Line $m$ has a $y$ -intercept of $(0, 3.5)$ .; The distance from the origin $(0, 0)$ to the midpoint of the line segment is $\sqrt{13}$ .

Answer

The statements asserting that line

m

passes through

(7, 0)

, that line

m

has a

y

-intercept of

(0, 3.5)

, and that the distance from the origin to the midpoint of the segment is

\sqrt{13}

are all true.

Line

k

has slope

\frac{6 - (-2)}{5 - 1} = 2

and segment midpoint

(3, 2)

. Line

m

, as the perpendicular bisector, has slope

-\frac{1}{2}

and equation

y = -\frac{1}{2}x + 3.5

. Substituting

x = 7

gives

y = 0

, so

(7, 0)

lies on line

m

. Substituting

x = 0

gives

y = 3.5

, confirming the

y

-intercept. Finally, the distance from

(0,0)

to the midpoint

(3,2)

is

\sqrt{3^2 + 2^2} = \sqrt{13}

. Thus, all three of these statements are correct.

Step-by-Step Solution

1

Find the midpoint and slope of the segment connecting

(1, -2)

and

(5, 6)

.

Midpoint

M = \left(\frac{1+5}{2}, \frac{-2+6}{2}\right) = (3, 2)

. Slope of segment

m_k = \frac{6 - (-2)}{5 - 1} = \frac{8}{4} = 2

.

Line

m

is the perpendicular bisector, so it passes through midpoint

M

and has a slope perpendicular to

m_k

.

2

Determine the slope and equation of line

m

.

Perpendicular slope

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

. Point-slope equation through

(3, 2)

:

y - 2 = -\frac{1}{2}(x - 3) \implies y = -\frac{1}{2}x + \frac{7}{2}

.

Perpendicular lines have negative reciprocal slopes.

3

Evaluate each given statement against the calculated values.

Line

m

contains

(7, 0)

because

0 = -\frac{1}{2}(7) + \frac{7}{2}

. The

y

-intercept is

(0, 3.5)

. Distance from

(0,0)

to

(3,2)

is

\sqrt{3^2 + 2^2} = \sqrt{13}

. Statements claiming slope is

2

or that line

m

is parallel to

2x - y = 4

are incorrect.

Determines which statements satisfy all mathematical conditions.

Key Concept

Perpendicular Bisectors, Slopes of Perpendicular Lines, and Distance Formula

Question 122Question →

The quadratic equation $2x^2 + px + q = 0$ , where $p$ and $q$ are constants, has roots $r$ and $s$ . The quadratic equation $x^2 + (p - 2)x + 24 = 0$ has roots $r + 2$ and $s + 2$ . What is the value of $q$ ?

Show answer & explanation

Answer: 32

Answer

The value of

q

is 32.

Applying Vieta's formulas to

2x^2 + px + q = 0

gives

r + s = -p/2

and

rs = q/2

. For the second equation

x^2 + (p-2)x + 24 = 0

, the sum of roots is

(r+2) + (s+2) = -(p-2)

, which simplifies to

(r+s) + 4 = 2 - p

. Substituting

r+s = -p/2

yields

-p/2 + 4 = 2 - p

, solving to

p = -4

and

r+s = 2

. The product of roots for the second equation is

(r+2)(s+2) = rs + 2(r+s) + 4 = 24

. Substituting

rs = q/2

and

r+s = 2

gives

q/2 + 4 + 4 = 24

, which simplifies to

q/2 = 16

and

q = 32

.

Step-by-Step Solution

1

Express the sum and product of roots

r

and

s

in terms of

p

and

q

using Vieta's formulas.

r + s = -\frac{p}{2}

and

rs = \frac{q}{2}

.

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

.

2

Relate the sum of the shifted roots

(r + 2)

and

(s + 2)

to the coefficients of the second quadratic equation.

(r + 2) + (s + 2) = -(p - 2) \implies (r + s) + 4 = 2 - p

.

The coefficient of

x

in

x^2 + (p - 2)x + 24 = 0

is

(p - 2)

, so the sum of its roots equals

-(p - 2)

.

3

Substitute

r + s = -\frac{p}{2}

into the sum relation to determine

p

.

-\frac{p}{2} + 4 = 2 - p \implies \frac{p}{2} = -2 \implies p = -4

.

Solving the linear equation for

p

yields

p = -4

, which means

r + s = 2

.

4

Expand the product of the shifted roots

(r + 2)(s + 2) = 24

and solve for

q

.

rs + 2(r + s) + 4 = 24 \implies \frac{q}{2} + 2(2) + 4 = 24 \implies \frac{q}{2} + 8 = 24 \implies q = 32

.

Substituting

rs = \frac{q}{2}

and

r + s = 2

isolates

q

, yielding

q = 32

.

Key Concept

Relating roots and coefficients of quadratic equations using Vieta's formulas and algebraic expansion.

Question 123Question →

A research study records five numerical observations: $12$ , $16$ , $20$ , $20$ , and $x$ , where $x$ is a positive integer. If the arithmetic mean of these five observations is equal to their median, which of the following values could be $x$ ? Select all such values.

Select all that apply

12

15

17

24

32

Show answer & explanation

Answer: $12$ ; $17$ ; $32$

Answer

The possible values of

x

are

12

,

17

, and

32

.

The mean of the dataset is given by

\frac{68+x}{5}

. Depending on where

x

falls relative to the sorted known values (

12, 16, 20, 20

), the median can be

16

(if

x \le 16

),

x

(if

16 \le x \le 20

), or

20

(if

x \ge 20

). Equating the mean to the median in each case yields three valid solutions:

x = 12

,

x = 17

, and

x = 32

.

Step-by-Step Solution

1

Express the arithmetic mean in terms of

x

.

The sum of the five numbers is

12 + 16 + 20 + 20 + x = 68 + x

. Therefore, the mean is

\frac{68 + x}{5}

.

The mean of

n

numbers is the total sum divided by

n

.

2

Analyze Case 1 where

x \le 16

.

The sorted dataset is

x, 12, 16, 20, 20

(or

12, x, 16, 20, 20

). The median (3rd value) is

16

. Setting mean = median gives

\frac{68 + x}{5} = 16 \implies 68 + x = 80 \implies x = 12

. Since

12 \le 16

,

x = 12

is valid.

The median of a 5-element set is the 3rd element when ordered.

3

Analyze Case 2 where

16 \le x \le 20

.

The sorted dataset is

12, 16, x, 20, 20

. The median is

x

. Setting mean = median gives

\frac{68 + x}{5} = x \implies 68 + x = 5x \implies 4x = 68 \implies x = 17

. Since

16 \le 17 \le 20

,

x = 17

is valid.

When

x

lies between 16 and 20,

x

itself becomes the 3rd element of the ordered set.

4

Analyze Case 3 where

x \ge 20

.

The sorted dataset is

12, 16, 20, 20, x

. The median is

20

. Setting mean = median gives

\frac{68 + x}{5} = 20 \implies 68 + x = 100 \implies x = 32

. Since

32 \ge 20

,

x = 32

is valid.

When

x \ge 20

, 20 is the 3rd element of the ordered set.

Key Concept

Solving for missing observations where mean equals median requires analyzing how the position of the variable affects the sorted order and the resulting median value.

Question 124Question →

For all real numbers $x$ and $y$ , the custom operation $\blacktriangle$ is defined by $x \blacktriangle y = 2x^2 - y^2 + xy$ . The function $g$ is defined for all real numbers $n$ by $g(n) = n \blacktriangle 3$ . If $k$ is a positive constant such that $g(k) = 0$ , what is the value of $k$ ?

\frac{2}{3}

1

\frac{3}{2}

3

9

Show answer & explanation

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

Answer

\frac{3}{2}

Evaluating

g(k) = k \blacktriangle 3

using the rule

x \blacktriangle y = 2x^2 - y^2 + xy

gives

2k^2 - 3^2 + 3k = 2k^2 + 3k - 9

. Setting this quadratic equal to zero gives

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

. The roots are

k = \frac{3}{2}

and

k = -3

. Because

k

is specified to be positive, the only valid solution is

\frac{3}{2}

.

Step-by-Step Solution

1

Apply the definition of the custom operation

\blacktriangle

to evaluate

g(k)

.

g(k) = k \blacktriangle 3 = 2(k)^2 - (3)^2 + k(3) = 2k^2 - 9 + 3k = 2k^2 + 3k - 9

Substitute

x = k

and

y = 3

into the given expression

x \blacktriangle y = 2x^2 - y^2 + xy

.

2

Set the expression for

g(k)

equal to

0

and factor the quadratic equation.

2k^2 + 3k - 9 = (2k - 3)(k + 3) = 0

We need to find values of

k

for which

g(k) = 0

.

3

Solve for

k

and apply the constraint that

k

is a positive constant.

2k - 3 = 0 \implies k = \frac{3}{2}

or

k + 3 = 0 \implies k = -3

. Since

k > 0

,

k = \frac{3}{2}

.

Disregard the negative root

k = -3

because the problem specifies that

k

must be positive.

Key Concept

Combining custom symbol operations with function notation to solve quadratic equations

Estimated Time:1m 30s

Question 125Question →

In regular octagon $ABCDEFGH$ with side length $2$ , diagonals $AD$ and $BE$ intersect at point $P$ . What is the area of quadrilateral $ABCP$ ?

2 + \sqrt{2}

2\sqrt{2}

4

4 + \sqrt{2}

1 + 2\sqrt{2}

Show answer & explanation

Answer: $2 + \sqrt{2}$

Answer

2 + \sqrt{2}

In regular octagon

ABCDEFGH

, each interior angle measures

135^\circ

. Segment

ABCD

forms an isosceles trapezoid where

AD

is parallel to

BC

. The perpendicular distance from

AD

to

BC

is

2 \sin(45^\circ) = \sqrt{2}

, and

AD = 2 + 2(2 \cos(45^\circ)) = 2 + 2\sqrt{2}

. Diagonal

BE

is parallel to side

CD

, so

\angle CBE = 45^\circ

, which implies

\angle ABE = 135^\circ - 45^\circ = 90^\circ

. Triangle

\triangle ABP

is therefore a right isosceles triangle with legs

AB = 2

and

BP = 2

, yielding hypotenuse

AP = 2\sqrt{2}

. Quadrilateral

ABCP

is a trapezoid with parallel sides

BC = 2

and

AP = 2\sqrt{2}

and height

h = \sqrt{2}

. Its area is

\frac{2 + 2\sqrt{2}}{2} \times \sqrt{2} = 2 + \sqrt{2}

.

Step-by-Step Solution

1

Calculate the interior angle of a regular octagon

Interior angle =

\frac{(8-2) \times 180^\circ}{8} = 135^\circ

Regular polygon interior angle formula.

2

Determine the orientation and length of diagonal

AD

AD \parallel BC

, height between them is

h = 2 \sin(45^\circ) = \sqrt{2}

, and base

AD = 2 + 2(2 \cos(45^\circ)) = 2 + 2\sqrt{2}

Quadrilateral

ABCD

forms an isosceles trapezoid with side length

2

and interior angles

135^\circ

.

3

Determine the direction of diagonal

BE

and find intersection point

P

\angle CBE = 180^\circ - 135^\circ = 45^\circ

, making

\angle ABE = 135^\circ - 45^\circ = 90^\circ

Quadrilateral

BCDE

is an isosceles trapezoid with

BE \parallel CD

.

4

Calculate the dimensions of right triangle

\triangle ABP

and segment

AP

\triangle ABP

is a right triangle at

B

with

\angle BAP = 45^\circ

,

AB = 2

,

BP = 2

, and hypotenuse

AP = 2\sqrt{2}

Since

AP \parallel BC

, alternate interior angle relationships yield

\angle BAP = 45^\circ

.

5

Calculate the area of quadrilateral

ABCP

\text{Area}(ABCP) = \frac{BC + AP}{2} \times h = \frac{2 + 2\sqrt{2}}{2} \times \sqrt{2} = (1 + \sqrt{2})\sqrt{2} = 2 + \sqrt{2}

Quadrilateral

ABCP

is a trapezoid with parallel bases

BC = 2

and

AP = 2\sqrt{2}

and height

\sqrt{2}

.

Key Concept

Properties of regular polygons, isosceles trapezoids, diagonal angles, and area decomposition.

Estimated Time:2m 30s

Question 126Question →

A quality control department recorded the processing time, in days, for a batch of requests. The table below displays the frequency distribution of the processing times:

Processing Time (Days)	Number of Requests
1	5
2	8
3	$k$
4	6
5	4

If $k$ is a positive integer and the median processing time for all requests in the batch is equal to 3 days, what is the minimum possible value of $k$ ?

2

3

4

5

6

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. There are

5 + 8 = 13

values less than

3

. For

3

to be the median of the dataset, the median position must be greater than

13

. With

k = 4

, the total number of items is

N = 23 + 4 = 27

. For an odd dataset of size

27

, the median is the

\frac{27 + 1}{2} = 14

th item. Since the first

13

items are less than

3

, the

14

th item is

3

, making

3

the median. Any value of

k \le 3

results in a median less than

3

.

Step-by-Step Solution

1

Calculate the cumulative frequency of values below 3 days.

Number of requests with processing times of 1 or 2 days is

5 + 8 = 13

.

To determine the median position, we first count how many data points lie strictly below 3 days.

2

Express the total number of requests

N

in terms of

k

.

N = 5 + 8 + k + 6 + 4 = 23 + k

.

The total number of requests determines whether

N

is odd or even and where the median position lies.

3

Determine the condition for 3 days to be the median.

For 3 days to be the median, the median position must be greater than 13 so that it falls into the category of 3 days.

Since 13 values are strictly less than 3, the median rank must be at least 14.

4

Test minimum integer values for

k

.

If

k = 3

,

N = 26

. The median is the average of the

13

th value (

2

) and the

14

th value (

3

), which is

2.5

. If

k = 4

,

N = 27

. The median is the

14

th value, which is

3

.

Testing

k = 4

yields

N = 27

where the

14

th value is

3

, satisfying the median requirement.

Key Concept

Median of a Frequency Distribution Table

Estimated Time:1m 30s

Question 127Question →

For all non-zero real numbers $x$ , the function $f$ satisfies the equation $f(x) + 2f\left(\frac{1}{x}\right) = 3x$ . What is the value of $f(2)$ ?

-1

0

1

2

3

Show answer & explanation

Answer: $-1$

Answer

-1

Evaluating the functional equation at

x = 2

gives

f(2) + 2f(1/2) = 6

. Evaluating the equation at

x = 1/2

gives

f(1/2) + 2f(2) = 3/2

. Substituting

f(1/2) = 3/2 - 2f(2)

into the first equation produces

-3f(2) + 3 = 6

, which simplifies to

f(2) = -1

.

Step-by-Step Solution

1

Substitute

x = 2

into the given functional equation

f(x) + 2f\left(\frac{1}{x}\right) = 3x

.

f(2) + 2f\left(\frac{1}{2}\right) = 6

This establishes a relation between the target value

f(2)

and the auxiliary value

f\left(\frac{1}{2}\right)

.

2

Substitute

x = \frac{1}{2}

into the original equation

f(x) + 2f\left(\frac{1}{x}\right) = 3x

.

f\left(\frac{1}{2}\right) + 2f(2) = 3\left(\frac{1}{2}\right) = \frac{3}{2}

This provides a second independent linear equation relating

f(2)

and

f\left(\frac{1}{2}\right)

.

3

Isolate

f\left(\frac{1}{2}\right)

in the second equation:

f\left(\frac{1}{2}\right) = \frac{3}{2} - 2f(2)

, and substitute this expression into the first equation.

f(2) + 2\left(\frac{3}{2} - 2f(2)\right) = 6 \implies f(2) + 3 - 4f(2) = 6 \implies -3f(2) + 3 = 6

Eliminating

f\left(\frac{1}{2}\right)

converts the system into a single linear equation in terms of

f(2)

.

4

Solve

-3f(2) + 3 = 6

for

f(2)

.

-3f(2) = 3 \implies f(2) = -1

Subtracting

3

from both sides gives

-3f(2) = 3

, and dividing by

-3

yields

f(2) = -1

.

Key Concept

Solving Functional Equations via Variable Substitution

Alternative Method

Generalize

f(x)

by setting

f(x) = Ax + \frac{B}{x}

. Substituting into

f(x) + 2f(1/x) = 3x

gives

(Ax + B/x) + 2(A/x + Bx) = (A + 2B)x + (2A + B)/x = 3x

. Equating coefficients:

A + 2B = 3

and

2A + B = 0 \implies B = -2A \implies A - 4A = 3 \implies A = -1, B = 2

. Thus

f(x) = -x + 2/x

, so

f(2) = -2 + 2/2 = -1

.

Estimated Time:1m 30s

Question 128Question →

A food processing plant packages grain using two automated lines, Line A and Line B. Line A operates at a constant rate of $40$ bags per minute, and Line B operates at a constant rate of $60$ bags per minute. Line A begins filling a batch of $4,200$ bags alone. After $15$ minutes, Line A's rate decreases by $25\%$ , and Line B is turned on to work alongside Line A until the batch is complete. How many total minutes does Line A operate to complete the batch?

40 minutes

51 minutes

55 minutes

63 minutes

70 minutes

Show answer & explanation

Answer: 55 minutes

Answer

55 minutes

In the first 15 minutes, Line A fills

40 \times 15 = 600

bags, leaving

4,200 - 600 = 3,600

bags remaining. Line A's rate then drops to

40 \times 0.75 = 30

bags per minute. Working together with Line B (

60

bags per minute), their combined rate is

30 + 60 = 90

bags per minute. Filling the remaining

3,600

bags takes

\frac{3,600}{90} = 40

minutes. Thus, Line A operates for a total of

15 + 40 = 55

minutes.

Step-by-Step Solution

1

Calculate the number of bags filled by Line A during the first 15 minutes.

Line A filled

40 \text{ bags/min} \times 15 \text{ min} = 600 \text{ bags}

.

Line A worked alone for 15 minutes at its initial constant rate.

2

Determine the remaining number of bags to be filled.

Remaining bags =

4,200 - 600 = 3,600 \text{ bags}

.

Subtract the bags already completed from the total batch size.

3

Calculate Line A's reduced rate and the new combined rate of both lines.

Line A's new rate =

40 \times (1 - 0.25) = 30 \text{ bags/min}

. Combined rate =

30 + 60 = 90 \text{ bags/min}

.

Line A's filling rate drops by 25%, and Line B contributes 60 bags per minute simultaneously.

4

Calculate the time needed for both lines to finish the remaining bags.

Time for second phase =

\frac{3,600 \text{ bags}}{90 \text{ bags/min}} = 40 \text{ minutes}

.

Divide remaining work by the combined filling rate.

5

Calculate the total time Line A was operating.

Total operating time =

15 \text{ min} + 40 \text{ min} = 55 \text{ minutes}

.

Line A operated during both the initial 15-minute period and the subsequent 40-minute period.

Key Concept

Linear Modeling and Piecewise Work-Rate Scenarios

Estimated Time:2m 0s

Question 129Question →

For all real numbers $a$ and $b$ such that $b \neq 0$ and $a \neq -b$ , which of the following expressions is equivalent to $\frac{a^3b - ab^3}{a^2b + 2ab^2 + b^3}$ ?

\frac{a(a - b)}{a + b}

a - b

\frac{a(a + b)}{a - b}

\frac{a^2 + b^2}{a + b}

\frac{a - b}{a + b}

Show answer & explanation

Answer: $\frac{a(a - b)}{a + b}$

Answer

\frac{a(a - b)}{a + b}

Factoring the numerator yields

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

, and factoring the denominator yields

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

. Dividing both the numerator and the denominator by the common factor

b(a + b)

produces

\frac{a(a - b)}{a + b}

.

Step-by-Step Solution

1

Factor out the greatest common monomial factor from the numerator and denominator.

The numerator

a^3b - ab^3

becomes

ab(a^2 - b^2)

, and the denominator

a^2b + 2ab^2 + b^3

becomes

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

.

Extracting common terms simplifies polynomials prior to applying polynomial identities.

2

Apply polynomial algebraic identities to factor the remaining terms.

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

and

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

. Thus, the expression becomes

\frac{ab(a - b)(a + b)}{b(a + b)^2}

.

The difference of squares and perfect square trinomial identities allow complete factorization.

3

Cancel non-zero common factors from the numerator and denominator.

Canceling

b

and one factor of

(a + b)

leaves

\frac{a(a - b)}{a + b}

.

Dividing both numerator and denominator by

b(a + b)

yields the simplified expression.

Key Concept

Simplifying rational expressions using common monomial factoring, difference of squares, and perfect square trinomial identities.

Estimated Time:1m 30s

Question 130Question →

If $x$ is a real number that satisfies the equation $(x - 3)^2 = 16$ , which of the following could be the value of $x^2 - 4x$ ? Select all such values.

Select all that apply

-5

5

21

45

77

Show answer & explanation

Answer: $5$ ; $21$

Answer

The possible values of

x^2 - 4x

are

5

and

21

.

Taking the square root of both sides of

(x - 3)^2 = 16

yields

x - 3 = \pm 4

. This gives two solutions for

x

:

x = 7

and

x = -1

. Substituting

x = 7

into

x^2 - 4x

gives

49 - 28 = 21

. Substituting

x = -1

into

x^2 - 4x

gives

(-1)^2 - 4(-1) = 1 + 4 = 5

. Thus, both

5

and

21

are valid values.

Step-by-Step Solution

1

Solve the quadratic equation for

x

x - 3 = 4 \implies x = 7

or

x - 3 = -4 \implies x = -1

Taking the square root of both sides of

(x - 3)^2 = 16

yields both positive and negative roots.

2

Substitute the first root into the expression

x^2 - 4x

7^2 - 4(7) = 49 - 28 = 21

Evaluating the target expression for

x = 7

.

3

Substitute the second root into the expression

x^2 - 4x

(-1)^2 - 4(-1) = 1 + 4 = 5

Evaluating the target expression for

x = -1

.

Key Concept

Quadratic Equations and Factoring

Question 131Question →

If $x$ is a real number such that $(x^2 - 4)^2 - 3(x^2 - 4) - 10 = 0$ , which of the following could be the value of $x$ ? Select all such values.

Select all that apply

-3

-\sqrt{2}

2

3

\sqrt{5}

Show answer & explanation

Answer: $-3$ ; $-\sqrt{2}$ ; $3$

Answer

The values that satisfy the equation are

-3

,

-\sqrt{2}

, and

3

.

The given equation is in quadratic form with respect to

(x^2 - 4)

. Letting

u = x^2 - 4

, we factor the equation as

(u - 5)(u + 2) = 0

, giving

u = 5

or

u = -2

. Setting

x^2 - 4 = 5

gives

x^2 = 9

, which yields

x = 3

or

x = -3

. Setting

x^2 - 4 = -2

gives

x^2 = 2

, which yields

x = \sqrt{2}

or

x = -\sqrt{2}

. Thus,

-3

,

-\sqrt{2}

, and

3

are all valid solutions.

Step-by-Step Solution

1

Use algebraic substitution to simplify the quadratic form.

Let

u = x^2 - 4

. The equation becomes

u^2 - 3u - 10 = 0

.

Recognizing the structure as a quadratic equation in terms of

(x^2 - 4)

simplifies factoring.

2

Factor the quadratic equation in terms of

u

.

(u - 5)(u + 2) = 0

, giving solutions

u = 5

or

u = -2

.

Finding the roots of the substituted variable

u

leads to the possible values for

x^2 - 4

.

3

Substitute back

x^2 - 4

for

u

and solve for

x

in each case.

For

u = 5

:

x^2 - 4 = 5 \implies x^2 = 9 \implies x = 3

or

x = -3

.
For

u = -2

:

x^2 - 4 = -2 \implies x^2 = 2 \implies x = \sqrt{2}

or

x = -\sqrt{2}

.

Taking both positive and negative square roots provides all valid real solutions.

Key Concept

Quadratic Form Substitution and Factoring

Estimated Time:1m 40s

Question 132Question →

If $x$ is a negative real number such that $(x + 5)^2 = 36$ , what is the value of $x^2 - 2x$ ?

-1

3

-99

143

99

Show answer & explanation

Answer: $143$

Answer

The value of

x^2 - 2x

is

143

.

Taking the square root of both sides of

(x + 5)^2 = 36

gives

x + 5 = 6

or

x + 5 = -6

. Solving for

x

yields

x = 1

or

x = -11

. Because

x

is given as a negative real number,

x = -11

. Evaluating the target expression at

x = -11

yields

(-11)^2 - 2(-11) = 121 + 22 = 143

.

Step-by-Step Solution

1

Solve the quadratic equation for

x

by applying the square root property.

x + 5 = \pm \sqrt{36} = \pm 6

Taking the square root of both sides yields both positive and negative root options.

2

Determine the two potential solutions for

x

.

x = 6 - 5 = 1

or

x = -6 - 5 = -11

Isolating

x

gives two real solutions.

3

Select the valid solution based on the problem constraints.

x = -11

The stem specifies that

x

must be a negative real number, so

x = 1

is rejected.

4

Substitute

x = -11

into the expression

x^2 - 2x

.

(-11)^2 - 2(-11) = 121 + 22 = 143

Squaring a negative number yields a positive result, and subtracting a negative quantity adds to the total.

Key Concept

Solving quadratic equations using square roots and evaluating algebraic expressions

Question 133Question →

Consider the following GRE Issue prompt claim:

"To foster true innovation in scientific research, government funding bodies should allocate financial resources exclusively to unconstrained, curiosity-driven inquiry rather than targeted, application-oriented projects."

An essay writer aims to deconstruct this prompt by identifying its primary unstated assumption to form a critical foundation for their thesis. Which of the following best identifies the essential unstated assumption that underlies the prompt's central recommendation?

Targeted, application-oriented projects rely entirely on existing scientific paradigms and are inherently incapable of yielding foundational breakthroughs that redefine a field.

Government funding bodies possess the objective institutional competency required to accurately distinguish pure curiosity-driven proposals from application-oriented proposals.

Private industry investment will automatically expand to fully cover all necessary funding for targeted, application-oriented research if public funding is withdrawn.

Curiosity-driven research always produces measurable technological advancements more rapidly than research constrained by specific commercial or practical goals.

Public policy decisions regarding scientific grants are consistently distorted by political interests whenever research goals are tied to specific applied outcomes.

Show answer & explanation

Answer: Targeted, application-oriented projects rely entirely on existing scientific paradigms and are inherently incapable of yielding foundational breakthroughs that redefine a field.

Answer

The essential unstated assumption is that targeted, application-oriented projects rely entirely on existing paradigms and are inherently incapable of yielding foundational breakthroughs that redefine a field.

The prompt advocates an absolute policy rule: funding must go exclusively to curiosity-driven research in order to foster true innovation. For this recommendation to hold logically, one must assume that application-oriented research is fundamentally incapable of producing true innovation. If applied research could lead to fundamental breakthroughs, the prompt's recommendation for total exclusivity would crumble. Therefore, identifying that targeted research is assumed to be incapable of foundational breakthroughs directly uncovers the prompt's central unstated premise.

Step-by-Step Solution

1

Analyze the prompt's core claim and recommendation structure.

The prompt claims that fostering 'true innovation' requires exclusive funding for 'curiosity-driven inquiry' over 'application-oriented projects.'

Deconstructing an Issue prompt requires isolating the prescriptive assertion and its scope.

2

Identify the logical bridge required to justify the absolute exclusion of applied research.

To justify excluding application-oriented research completely in pursuit of true innovation, the arguer must assume that application-oriented research contributes zero or negligible true innovation.

If applied research could also produce true innovation, exclusive funding of curiosity-driven research would not be logically necessary.

3

Evaluate the option choices to select the statement representing this fundamental presupposition.

The statement asserting that application-oriented projects are inherently incapable of yielding foundational breakthroughs directly exposes this underlying premise.

Exposing this hidden assumption provides the essay writer with the strongest leverage to construct a nuanced counterargument.

Key Concept

Deconstructing Issue Prompts via Underlying Assumptions

Question 134Question →

Paleoclimatologists analyzing speleothem oxygen isotope ratios from subterranean caves in the Amazon basin have posited that a prolonged megadrought precipitated the collapse of pre-Columbian urban settlements around 1350 CE. Proponents of this climate-deterministic model point to a sharp, synchronous elevation in $\delta^{18}\text{O}$ signatures across multiple cave sites as unequivocal evidence of widespread precipitation failure. However, recent high-resolution palynological evidence from adjacent lacustrine sediment cores reveals that agricultural cultivation and forest clearing continued unabated throughout the purported drought interval. This persistence of intensive land use strongly suggests that indigenous socio-ecological systems possessed sufficient hydrologic resilience to buffer against regional rainfall deficits. Consequently, attributing the abandonment of these complex settlements primarily to exogenous climatic forcing oversimplifies the intricate interplay between anthropogenic environmental modification and institutional governance.

Which sentence in the passage presents empirical evidence that directly counters the climate-deterministic explanation for the settlement collapse?

Show answer & explanation

Answer: Sentence 3; 3; the third sentence; However, recent high-resolution palynological evidence from adjacent lacustrine sediment cores reveals that agricultural cultivation and forest clearing continued unabated throughout the purported drought interval.

Answer

The third sentence ('However, recent high-resolution palynological evidence from adjacent lacustrine sediment cores reveals that agricultural cultivation and forest clearing continued unabated throughout the purported drought interval.') provides the empirical counterevidence.

The sentence stating that high-resolution palynological evidence reveals uninterrupted agricultural cultivation during the purported drought provides concrete physical data that directly challenges the premise that climate deterioration forced the abandonment of the settlements.

Step-by-Step Solution

1

Identify the central claim of the climate-deterministic model.

The main claim (introduced in Sentence 1 and supported in Sentence 2) is that a severe megadrought caused the collapse of pre-Columbian urban settlements.

Understanding the target hypothesis is essential before searching for empirical evidence that weakens or refutes it.

2

Scan the passage for empirical data that contradicts this claim.

Sentence 3 introduces fossil pollen (palynological) data showing uninterrupted agricultural activity and land modification during the exact period of the alleged drought.

Continued farming directly challenges the assertion that environmental conditions collapsed societal infrastructure.

3

Distinguish empirical findings from subsequent interpretations or conclusions.

Sentence 3 contains the actual physical data (palynological core evidence), whereas Sentence 4 and Sentence 5 present logical inferences and overall conclusions drawn from that evidence.

The prompt specifically asks for the sentence presenting empirical evidence rather than the author's synthesis or conclusion.

Key Concept

Identifying empirical counterevidence within Reading Comprehension argument structures

Question 135Question →

A municipal power authority introduced automated smart meters across residential districts to detect voltage anomalies and isolate localized power outages automatically. During the six months following installation, the total duration of power outages experienced by residents in these districts decreased by 35 percent compared to the same period in previous years. Municipal officials concluded that the automatic anomaly detection of the smart meters was directly responsible for reducing outage durations.

Which of the following, if true, provides the strongest support for the municipal officials' conclusion?

Simultaneously with the smart meter rollout, the power authority completed major structural upgrades to main transmission cables serving the same districts.

The average monthly energy cost per household in districts with smart meters did not decrease after the devices were installed.

During the six-month period following installation, the region experienced weather patterns and storm frequencies typical of previous years.

Commercial districts in adjacent municipalities that did not install smart meters experienced a substantial increase in customer service inquiries regarding billing during the same period.

Most residents in the affected districts were aware that the new meters were designed to report power fluctuations directly to repair technicians.

Show answer & explanation

Answer: During the six-month period following installation, the region experienced weather patterns and storm frequencies typical of previous years.

Answer

The argument is most strengthened by the statement showing that weather patterns and storm frequencies during the post-installation period were typical of previous years.

The conclusion attributes the 35 percent drop in outage durations directly to the smart meters' anomaly detection capability. The statement confirming that storm frequencies and weather patterns were typical of prior years eliminates an alternative explanation—namely, that a period of unusually mild weather reduced grid damage and caused the drop in outage times. By ruling out this key alternative cause, the statement significantly strengthens the causal connection proposed by municipal officials.

Step-by-Step Solution

1

Identify the premise and conclusion of the argument

Premise: Outage durations dropped 35 percent after installing automated smart meters. Conclusion: Smart meter anomaly detection directly caused the reduction.

Understanding the precise causal claim is essential for evaluating statements that strengthen or weaken it.

2

Identify potential logical vulnerabilities in the causal claim

The main vulnerability is that an external factor (such as unusually mild weather or grid infrastructure repairs) might be the true cause of the reduced outage durations.

A standard way to strengthen a causal conclusion is to eliminate competing alternative causes.

3

Evaluate the options for a statement that addresses these vulnerabilities

The option noting that weather conditions were typical of previous years confirms that mild weather was not responsible for the decrease, thereby supporting the smart meters as the true cause.

Ruling out a plausible confounding variable increases the probability that the author's stated cause is correct.

Key Concept

Strengthening Causal Claims by Ruling Out Alternative Causes

Question 136Question →

Read the passage below:

Conventional economic models long assumed that consumer decisions are driven entirely by rational utility maximization. However, recent empirical work in behavioral economics demonstrates that cognitive biases systematically alter financial choices. For instance, a 2018 study observed that individuals consistently value immediate rewards over significantly larger delayed gains, a phenomenon known as hyperbolic discounting. Critics contend that such laboratory findings fail to account for real-world market mechanisms that attenuate irrational behavior. Nonetheless, these findings have prompted policymakers to redesign default savings plans to nudge individuals toward optimal long-term outcomes.

Which sentence in the passage provides specific empirical evidence supporting the claim that cognitive biases influence financial decision-making?

Conventional economic models long assumed that consumer decisions are driven entirely by rational utility maximization.

However, recent empirical work in behavioral economics demonstrates that cognitive biases systematically alter financial choices.

For instance, a 2018 study observed that individuals consistently value immediate rewards over significantly larger delayed gains, a phenomenon known as hyperbolic discounting.

Critics contend that such laboratory findings fail to account for real-world market mechanisms that attenuate irrational behavior.

Nonetheless, these findings have prompted policymakers to redesign default savings plans to nudge individuals toward optimal long-term outcomes.

Show answer & explanation

Answer: For instance, a 2018 study observed that individuals consistently value immediate rewards over significantly larger delayed gains, a phenomenon known as hyperbolic discounting.

Answer

The sentence stating, 'For instance, a 2018 study observed that individuals consistently value immediate rewards over significantly larger delayed gains, a phenomenon known as hyperbolic discounting.'

The sentence referencing the 2018 study directly provides specific empirical evidence by citing observational data on hyperbolic discounting to demonstrate how cognitive biases affect financial choices.

Step-by-Step Solution

1

Analyze the prompt requirement

The prompt requires selecting a sentence that supplies specific empirical evidence (data, study, or observed phenomenon) rather than a theoretical assertion or policy outcome.

Select-in-passage questions require identifying the exact sentence fulfilling a specific rhetorical function.

2

Evaluate the rhetorical role of each sentence in the passage

The sentence referencing the 2018 study introduces specific experimental observation ('hyperbolic discounting') supporting the broader claim.

Concrete study details serve as empirical evidence for general theoretical claims.

Key Concept

Select-in-Passage Question Analysis: Identifying Supporting Empirical Evidence

Question 137Question →

Based on the passage below, what structural role does the detail regarding third-party debt assignments play in Vance's overall argument? Complete the following statement with an appropriate functional verb:

The reference to legally enforceable third-party debt assignments serves to {{blank_1}} the objection that Flemish manorial ledgers were merely informal administrative record-keeping systems rather than genuine financial instruments.

Fill in the blanks below

For decades, economic historians attributing the rise of early modern European banking institutions primarily to Mediterranean commercial networks treated Northern European agrarian arrangements as largely static background conditions. However, recent reassessments of medieval manor accounting led by Dr. Evelyn Vance suggest that institutional credit instruments were independently synthesized in lowland agrarian estates long before Mediterranean techniques proliferated northward. Vance points out that non-monetized credit ledgers in fourteenth-century Flemish manors performed functionally equivalent clearinghouse operations, settling seasonal debt obligations without specie transfers. Critics of Vance's thesis contend that these estate ledgers were merely informal administrative record-keeping systems lacking negotiable instruments, thus failing to constitute true financial intermediation. To substantiate her position against these detractors, Vance emphasizes that the Flemish ledgers permitted third-party debt assignments that were legally enforceable in local manorial courts. Ultimately, while Vance does not argue that rural manors possessed fully developed capital markets, her evidence demonstrates that key functional components of modern banking emerged autonomously in agrarian contexts.

Sentence to complete:
The reference to legally enforceable third-party debt assignments serves to the objection that Flemish manorial ledgers were merely informal administrative record-keeping systems rather than genuine financial instruments.

Show answer & explanation

Answer

rebut (or counter / refute / undermine / challenge)

The sentence citing legally enforceable third-party debt assignments directly follows the critics' argument that Flemish ledgers lacked negotiability. By providing empirical evidence of negotiability and legal enforcement, the detail serves rhetorically to rebut, counter, or refute the critics' objection, defending Vance's central position.

Step-by-Step Solution

1

Analyze the structural flow and argumentative trajectory of the passage.

The text moves from a traditional scholarly view to Vance's revisionist thesis, followed by an objection from critics, Vance's counter-evidence, and finally a qualified broader conclusion.

Understanding the contextual placement of each claim helps determine how individual sentences function relative to surrounding counterarguments.

2

Examine the specific relationship between the critics' objection and the highlighted detail.

The critics claim the ledgers lacked negotiable instruments and were thus not true financial intermediation. Vance responds by introducing third-party debt assignments, showing legal negotiability existed.

Offering empirical facts that directly contradict a criticism functions rhetorically as a rebuttal or refutation of that objection.

3

Select a precise functional verb that captures this rhetorical role.

Verbs such as 'rebut', 'refute', 'counter', or 'undermine' precisely describe introducing evidence to dismantle a opposing counterargument.

These verbs denote active refutation within academic discourse.

Key Concept

Rhetorical Role of Evidence in Rebutting Counterarguments

Estimated Time:2m 30s

Question 138Question →

Recent investigations into urban microclimates have challenged the long-standing assumption that city green spaces act uniformly as cooling zones. While expansive municipal parks undeniably reduce ambient temperatures through evapotranspiration, smaller pockets of vegetation—such as roadside bioswales and residential gardens—exhibit far more variable thermal effects. Microclimate researchers demonstrate that the cooling efficacy of small-scale urban greenery is contingent upon canopy density, soil moisture retention, and local wind corridors. In densely built environments with high-rise structures, sparse vegetation can occasionally trap heat by reducing surface air movement without offering sufficient shade. Consequently, urban planners are urged to move beyond aggregate metric goals for total green coverage and instead adopt site-specific spatial strategies to optimize thermal relief.

Based on the passage, which of the following statements correctly capture a main idea or primary purpose of the author? Select all options that apply.

Select all that apply

Arguing that the thermal mitigation benefits of urban greenery depend on specific contextual factors rather than uniform presence alone.

Urging urban planners to transition from broad coverage targets to tailored, site-specific vegetation strategies.

Demonstrating that small residential gardens invariably raise local temperatures in metropolitan areas.

Explaining the exact biophysical process by which evapotranspiration functions in large municipal parks.

Recommending an immediate moratorium on high-rise building construction until thermal modeling improves.

Show answer & explanation

Answer: Arguing that the thermal mitigation benefits of urban greenery depend on specific contextual factors rather than uniform presence alone.; Urging urban planners to transition from broad coverage targets to tailored, site-specific vegetation strategies.

Answer

The passage primarily aims to demonstrate that urban greenery's cooling effects depend on local environmental conditions and to recommend site-specific planning over aggregate coverage goals.

The passage centers on showing that cooling benefits from small-scale urban greenery depend heavily on structural and environmental context (such as canopy density and wind corridors) rather than just presence, and concludes by urging urban planners to adopt site-specific strategies rather than relying on aggregate green space targets. Thus, both statements reflecting these two points accurately capture the main idea and primary purpose.

Step-by-Step Solution

1

Identify the primary claim and scope of the passage.

The author contrasts the uniform cooling assumption of large parks with the variable effectiveness of small vegetation pockets, establishing that efficacy depends on local context.

Establishing the core argument requires synthesizing the main topic across the passage.

2

Identify the author's primary recommendation.

The final sentence explicitly advises urban planners to shift from aggregate metrics to site-specific strategies.

The conclusion explicitly states the practical purpose toward which the rest of the text builds.

3

Evaluate the choices against the identified main idea and primary purpose.

The statements highlighting contextual dependence and the recommendation for tailored urban planning accurately express the main ideas of the text.

Correct options must align with the passage's overall argument without overstating claims or focusing on peripheral details.

Key Concept

Main Idea and Primary Purpose

Question 139Question →

Consider the following draft body paragraph from an essay responding to a GRE Issue prompt on municipal governance:

"Municipalities should prioritize reserving vacant downtown lots for public parks rather than selling them for commercial development. Opponents might argue that commercial developments generate immediate property tax revenues necessary to fund municipal services. However, this perspective overlooks the fact that public parks increase neighboring property values over time, thereby indirectly expanding the tax base while simultaneously improving community health."

Which of the following statements accurately describe the structural integration of the counterargument and rebuttal in this excerpt? (Select all that apply.)

Select all that apply

The paragraph introduces an opposing perspective using a clear transition to distinguish the counterargument from the author's primary position.

The rebuttal directly responds to the central premise of the counterargument by offering an alternative economic reasoning.

The paragraph places the counterargument in complete isolation from the main line of reasoning, creating a structural disconnect.

The author dismisses the opposing viewpoint superficially without offering substantive counter-reasoning or evidence.

The excerpt exhibits a one-sided line of reasoning that fails to acknowledge potential objections to the author's thesis.

Show answer & explanation

Answer: The paragraph introduces an opposing perspective using a clear transition to distinguish the counterargument from the author's primary position.; The rebuttal directly responds to the central premise of the counterargument by offering an alternative economic reasoning.

Answer

The statements recognizing the clear transitional introduction of the opposing perspective and the direct economic engagement of the rebuttal are both correct.

The paragraph successfully integrates the counterargument by first signaling it with clear transitional phrasing ('Opponents might argue that...') and then directly countering its economic premise regarding tax revenues with a logical counter-explanation ('public parks increase neighboring property values over time, thereby indirectly expanding the tax base').

Step-by-Step Solution

1

Analyze how the counterargument is introduced within the paragraph structure.

The author uses 'Opponents might argue that...' to explicitly signal a shift from the main claim to an opposing viewpoint.

Clear transitional framing ensures the reader distinguishes between the author's core thesis and alternative perspectives.

2

Evaluate the structural and logical connection between the counterargument and the rebuttal.

The author transitions with 'However' and refutes the tax revenue concern by demonstrating an alternative, long-term economic mechanism (property value appreciation).

An effective rebuttal must engage directly with the premise of the counterargument rather than shifting to an unrelated topic.

Key Concept

Effective structural integration requires clear transitional signaling of opposing claims followed by a direct, logically aligned rebuttal.

Question 140Question →

Consider the following GRE Issue prompt and analytical task:

Prompt: "As artificial intelligence and automated systems become increasingly integrated into public governance, society must prioritize efficiency over human oversight in municipal decision-making."

Task: Write a response in which you discuss the extent to which you agree or disagree with the claim. In developing and supporting your position, address the specific scope and operational definitions of key concepts central to the claim.

Which of the following thesis statements best operationalizes key terms and establishes an effective scope for an essay responding to this prompt?

While automated systems streamline routine administrative procedures such as transit scheduling, human oversight remains indispensable when algorithms impact civil liberties or welfare distribution, requiring 'efficiency' to be defined not merely as processing speed, but as equitable service delivery within specific governance domains.

Efficiency must be the absolute priority in all municipal decision-making because human oversight invariably introduces bureaucratic delay, bias, and administrative incompetence across every level of public governance.

Artificial intelligence and automated systems are changing modern society rapidly, so municipal governments must adapt their decision-making processes to keep pace with technological progress.

There are compelling reasons both to prioritize efficiency and to preserve human oversight, so municipal leaders should carefully weigh the advantages and disadvantages of each approach depending on circumstances.

Human oversight should never be compromised for efficiency under any circumstances, as automated decision-making algorithms lack moral accountability and empathy.

Show answer & explanation

Answer: While automated systems streamline routine administrative procedures such as transit scheduling, human oversight remains indispensable when algorithms impact civil liberties or welfare distribution, requiring 'efficiency' to be defined not merely as processing speed, but as equitable service delivery within specific governance domains.

Answer

The statement that qualifies the role of automation by distinguishing routine administration from high-stakes civil rights decisions and redefines 'efficiency' as equitable service delivery.

The correct response effectively defines the key term 'efficiency' as equitable service delivery rather than mere speed, and establishes clear scope boundaries by contrasting routine municipal administration (where automation excels) with sensitive areas like civil liberties and welfare allocation (where human oversight is crucial). This directly satisfies the GRE AWA requirement for defining scope and operationalizing terms.

Step-by-Step Solution

1

Analyze the prompt requirements for key term definition and scope establishment.

Identified the central concepts requiring definition ('efficiency', 'human oversight', 'municipal decision-making') and the need for qualifying scope boundaries.

Effective GRE Issue response framing requires defining ambiguous terms and delineating where the claim holds true versus where it breaks down.

2

Evaluate each candidate thesis against the prompt task instructions.

Distinguished between vague, absolutist, or one-sided statements and the statement that offers operational definitions and contextual boundaries.

A high-scoring GRE Issue thesis must avoid rigid dogmatism and vague summaries while establishing explicit criteria for evaluation.

3

Confirm the option that best fulfills both key term operationalization and scope setting.

Selected the thesis that defines 'efficiency' beyond speed and restricts automation to routine administrative contexts while reserving human oversight for high-stakes decisions.

This framing directly answers the task prompt by establishing clear domain limits and explicit operational definitions.

Key Concept

Defining Key Terms & Establishing Essay Scope

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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