Tüm alıştırma soruları

Soru 1981Soru →

A dataset consists of 7 integers: $5, 8, 12, 14, 16, 20,$ and $x$ . If the median of the dataset is strictly greater than the mean of the dataset, which of the following could be the value of $x$ ? Select all such values.

Geçerli olan tümünü seçin

4

10

13

18

25

Cevabı ve açıklamayı göster

Cevap: $4$ ; $13$ ; $18$

Cevap

The values of

x

that make the median strictly greater than the mean are

4

,

13

, and

18

.

The median of a 7-element dataset is its 4th element when arranged in ascending order. Testing the values shows that for

4

, the median (

12

) exceeds the mean (

11.29

); for

13

, the median (

13

) exceeds the mean (

12.57

); and for

18

, the median (

14

) exceeds the mean (

13.29

). All three satisfy the condition.

Adım Adım Çözüm

1

Express the mean of the dataset in terms of

x

.

The sum of the known 6 numbers is

5 + 8 + 12 + 14 + 16 + 20 = 75

. The total sum for 7 numbers is

75 + x

, making the mean

\text{Mean} = \frac{75 + x}{7}

.

The mean is calculated as the sum of all elements divided by the total count (7).

2

Analyze the median across different ranges of

x

.

For 7 numbers sorted in order, the median is the 4th number. If

x \le 12

, the sorted list starts with

x, 5, 8, 12

or similar, so the 4th element is

12

. If

12 < x < 14

, the 4th element is

x

. If

x \ge 14

, the 4th element is

14

.

The position of

x

relative to the known numbers determines which element falls into the middle (4th) spot.

3

Test the condition

\text{Median} > \text{Mean}

for each piecewise case.

Case 1 (

x \le 12

):

12 > \frac{75 + x}{7} \implies 84 > 75 + x \implies x < 9

. Thus,

x = 4

works, but

x = 10

does not.
Case 2 (

12 < x < 14

):

x > \frac{75 + x}{7} \implies 7x > 75 + x \implies 6x > 75 \implies x > 12.5

. Thus,

x = 13

works.
Case 3 (

x \ge 14

):

14 > \frac{75 + x}{7} \implies 98 > 75 + x \implies x < 23

. Thus,

x = 18

works, but

x = 25

does not.

Solving the inequality for each case yields all valid ranges for

x

:

x < 9

,

x = 13

, and

14 \le x < 23

.

Anahtar Kavram

Measures of Central Tendency (Mean vs. Median Analysis with Variables)

Soru 1982Soru →

Three decorative light signals flash at regular intervals of $15$ seconds, $20$ seconds, and $36$ seconds, respectively. If all three signals flash simultaneously at 12:00 PM, how many times will all three signals flash simultaneously between 12:01 PM and 1:00 PM, inclusive?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20

The three signals flash together every 180 seconds (3 minutes), which is the least common multiple of 15, 20, and 36. Within the 60-minute interval from 12:01 PM to 1:00 PM inclusive, simultaneous flashes occur at minute 3, 6, 9, ..., up to minute 60, resulting in exactly 20 simultaneous flashes.

Adım Adım Çözüm

1

Find the prime factorization of each interval in seconds

15 = 3 \cdot 5

,

20 = 2^2 \cdot 5

,

36 = 2^2 \cdot 3^2

Prime factorization allows systematic calculation of the least common multiple.

2

Calculate the LCM of 15, 20, and 36

\text{LCM} = 2^2 \cdot 3^2 \cdot 5 = 180

seconds

The LCM gives the minimum period of time after which all three signals flash together.

3

Convert the period into minutes and find the frequency in 60 minutes

180\text{ seconds} = 3\text{ minutes}

;

60 \div 3 = 20

flashes

The timeframe between 12:01 PM and 1:00 PM inclusive contains 60 minutes, yielding 20 multiples of 3 minutes.

Anahtar Kavram

Least Common Multiple (LCM) for periodic events

Soru 1983Soru →

In the $xy$ -plane, quadrilateral $ABCD$ has vertices $A(0, 0)$ , $B(6, 0)$ , $C(8, 4)$ , and $D(2, 4)$ . Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

Quadrilateral

ABCD

is a parallelogram.

The area of quadrilateral

ABCD

is

24

.

The diagonals

AC

and

BD

bisect each other at the point

(4, 2)

.

The length of diagonal

AC

is equal to the length of diagonal

BD

.

The diagonals

AC

and

BD

are perpendicular to each other.

Cevabı ve açıklamayı göster

Cevap: Quadrilateral $ABCD$ is a parallelogram.; The area of quadrilateral $ABCD$ is $24$ .; The diagonals $AC$ and $BD$ bisect each other at the point $(4, 2)$ .

Cevap

The correct statements are that quadrilateral

ABCD

is a parallelogram, its area is

24

, and its diagonals bisect each other at

(4, 2)

.

Quadrilateral

ABCD

is a parallelogram because both pairs of opposite sides are congruent and parallel (

AB = DC = 6

along the horizontal line, and

AD = BC = \sqrt{20}

). The area is base times height, which is

6 \times 4 = 24

. The diagonals bisect each other at their common midpoint

(4, 2)

.

Adım Adım Çözüm

1

Determine side lengths and slopes to classify the quadrilateral

Side

AB

is horizontal with length

6

; side

DC

is horizontal with length

6

. Side

AD

has components

(2,4)

and length

\sqrt{20}

; side

BC

has components

(2,4)

and length

\sqrt{20}

. Since opposite sides are parallel and congruent,

ABCD

is a parallelogram.

To verify if the quadrilateral is a parallelogram.

2

Calculate the area of the quadrilateral

Area =

\text{base} \times \text{height} = 6 \times 4 = 24

.

To evaluate the area statement.

3

Find the midpoints and lengths of diagonals

AC

and

BD

Midpoint of

AC = (4, 2)

and midpoint of

BD = (4, 2)

, so they bisect each other. Length

AC = \sqrt{8^2 + 4^2} = \sqrt{80} = 4\sqrt{5}

, and length

BD = \sqrt{(-4)^2 + 4^2} = \sqrt{32} = 4\sqrt{2}

.

To check diagonal bisection and length equality.

4

Determine the slopes of the diagonals to check for perpendicularity

Slope of

AC = \frac{4}{8} = \frac{1}{2}

; slope of

BD = \frac{4}{-4} = -1

. Product of slopes

= \frac{1}{2} \times (-1) = -\frac{1}{2} \neq -1

, so they are not perpendicular.

To verify whether the diagonals intersect at right angles.

Anahtar Kavram

Properties of quadrilaterals in the coordinate plane, including parallelogram identification, area calculation, midpoint theorem for diagonals, and perpendicular slope test.

Tahmini Süre:1m 30s

Soru 1984Soru →

During a quality assurance test of a dual-sensor monitoring device, Sensor $A$ operates independently of Sensor $B$ . The probability that Sensor $A$ detects a target signal during a test trial is $0.80$ , and the probability that Sensor $B$ detects the target signal during the same trial is $0.75$ . What is the probability that exactly one of the two sensors detects the target signal during a test trial?

Cevabı ve açıklamayı göster

Cevap: 0.35

Cevap

0.35

To find the probability that exactly one sensor detects the signal, we must evaluate two disjoint scenarios: Sensor

A

detects while Sensor

B

fails (

0.80 \times 0.25 = 0.20

), or Sensor

A

fails while Sensor

B

detects (

0.20 \times 0.75 = 0.15

). Summing these two probabilities gives

0.20 + 0.15 = 0.35

. Alternatively, subtracting the probability that both sensors detect (

0.80 \times 0.75 = 0.60

) from the probability that at least one detects (

1 - 0.20 \times 0.25 = 0.95

) yields

0.95 - 0.60 = 0.35

.

Adım Adım Çözüm

1

Determine the probabilities of individual event non-occurrences

P(A') = 0.20

and

P(B') = 0.25

The probability of an event not occurring is

1

minus the probability that it occurs.

2

Calculate joint probability for each mutually exclusive event outcome

P(A \text{ and } B') = 0.80 \times 0.25 = 0.20

and

P(A' \text{ and } B) = 0.20 \times 0.75 = 0.15

Since the sensors operate independently,

P(X \text{ and } Y) = P(X) \times P(Y)

.

3

Combine the independent outcomes that satisfy the condition

0.20 + 0.15 = 0.35

The scenarios (only

A

detects, or only

B

detects) are mutually exclusive, so their probabilities are added.

Anahtar Kavram

Probability of Independent and Mutually Exclusive Events

Soru 1985Soru →

Let $x$ and $y$ be integers such that $x < 0 < y$ . If $x$ and $y$ satisfy all of the following conditions:

1. $(-1)^{x y + x} = -1$
2. $(-1)^{x^2 y + y} = 1$
3. $y^2 - x^2 = 19$

What is the value of $x + y$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The value of

x + y

is 1.

Condition 1 dictates that

(-1)^{x y + x} = -1

, meaning

x y + x = x(y + 1)

is odd. For the product

x(y+1)

to be odd, both

x

and

y+1

must be odd, which means

x

is odd and

y

is even. Condition 3 factors as

(y - x)(y + x) = 19

. Given

x < 0 < y

, we know

y - x > y + x

. Since 19 is prime, its unique positive factor pair requires

y - x = 19

and

y + x = 1

. Solving this system yields

y = 10

and

x = -9

, which satisfies all sign and parity constraints. Thus,

x + y = -9 + 10 = 1

.

Adım Adım Çözüm

1

Analyze parity requirements from Condition 1

x

is odd and

y

is even

Since

(-1)^{x(y+1)} = -1

, the exponent

x(y+1)

must be odd, requiring both

x

and

y+1

to be odd.

2

Check consistency with Condition 2

Condition 2 is satisfied

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

is always even when

y

is even, making

(-1)^{x^2 y + y} = 1

true.

3

Factor difference of squares and set up system using sign rules

y - x = 19

and

y + x = 1

Since 19 is prime and

x < 0 < y

,

y - x > y + x > 0

, forcing the factor pair to be 19 and 1.

4

Solve for

x

and

y

and sum them

x = -9

,

y = 10

, giving

x + y = 1

Adding the system yields

2y = 20 \implies y = 10

, and substituting into

y + x = 1

yields

x = -9

.

Anahtar Kavram

Even-Odd Exponent Rules and Sign Properties of Integers

Soru 1986Soru →

If

x

is a real number such that

\sqrt{\frac{25^{x+1} - 25^x}{5^{x+2} + 5^{x+1}}} = 250

what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

x

is 7.

Factoring out common powers in the numerator and denominator yields

25^x(25-1) = 24 \cdot 5^{2x}

and

5^{x+1}(5+1) = 6 \cdot 5^{x+1}

. Simplifying their ratio inside the square root gives

\frac{24 \cdot 5^{2x}}{6 \cdot 5^{x+1}} = 4 \cdot 5^{x-1}

. Taking the square root gives

2 \cdot 5^{(x-1)/2}

. Setting this equal to 250 yields

5^{(x-1)/2} = 125 = 5^3

. Equating the exponents gives

(x-1)/2 = 3

, which solves to

x = 7

.

Adım Adım Çözüm

1

Factor the numerator and express terms with a common base of 5

25^{x+1} - 25^x = 25^x(25 - 1) = 24 \cdot (5^2)^x = 24 \cdot 5^{2x}

Factoring out

25^x

simplifies the difference into a single term with base 5.

2

Factor the denominator using base 5

5^{x+2} + 5^{x+1} = 5^{x+1}(5 + 1) = 6 \cdot 5^{x+1}

Factoring out the common power

5^{x+1}

simplifies the sum into a single term.

3

Simplify the fraction inside the square root

\frac{24 \cdot 5^{2x}}{6 \cdot 5^{x+1}} = 4 \cdot 5^{2x - (x+1)} = 4 \cdot 5^{x-1}

Dividing coefficients (24/6 = 4) and applying exponent rules for division (

a^m / a^n = a^{m-n}

).

4

Take the square root of the simplified expression

\sqrt{4 \cdot 5^{x-1}} = \sqrt{4} \cdot \sqrt{5^{x-1}} = 2 \cdot 5^{\frac{x-1}{2}}

Using radical rules

\sqrt{ab} = \sqrt{a}\sqrt{b}

and

\sqrt{a^k} = a^{k/2}

.

5

Set the simplified radical expression equal to 250 and solve for x

2 \cdot 5^{\frac{x-1}{2}} = 250 \implies 5^{\frac{x-1}{2}} = 125 \implies 5^{\frac{x-1}{2}} = 5^3 \implies \frac{x-1}{2} = 3 \implies x = 7

Dividing both sides by 2 gives

5^{(x-1)/2} = 125 = 5^3

. Equating exponents yields

(x-1)/2 = 3

, so

x = 7

.

Anahtar Kavram

Exponent rules, base conversion, factoring exponential terms, and radical simplification

Tahmini Süre:2m 0s

Soru 1987Soru →

A cyclist completes a journey consisting of three distinct segments: an uphill segment, a flat segment, and a downhill segment. The ratio of the distances of the uphill, flat, and downhill segments is $2 : 3 : 5$ , respectively. The cyclist's average speed on the flat segment is twice her average speed on the uphill segment, and her average speed on the downhill segment is three times her average speed on the uphill segment. If the cyclist's overall average speed for the entire journey is $30$ miles per hour, what is her average speed, in miles per hour, on the flat segment?

Cevabı ve açıklamayı göster

Cevap: 31

Cevap

31

The correct average speed on the flat segment is 31 miles per hour. Setting up segment distances as

2x

,

3x

, and

5x

(total distance

10x

) and segment speeds as

v

,

2v

, and

3v

, the segment times are

t_1 = \frac{2x}{v}

,

t_2 = \frac{3x}{2v}

, and

t_3 = \frac{5x}{3v}

. The total travel time is

T = \frac{31x}{6v}

. Dividing total distance

10x

by total time

T

yields an overall average speed of

\frac{60v}{31} = 30

. Solving for

v

gives

v = 15.5

miles per hour. Thus, the average speed on the flat segment is

2v = 31

miles per hour.

Adım Adım Çözüm

1

Define segment distances using ratio multipliers.

Distances are

d_1 = 2x

,

d_2 = 3x

, and

d_3 = 5x

, giving total distance

D = 10x

.

The distances of the three segments are in the ratio

2 : 3 : 5

.

2

Express segment speeds relative to the uphill speed

v

.

Uphill speed is

v

, flat speed is

2v

, and downhill speed is

3v

.

The problem states flat speed is twice uphill speed, and downhill speed is three times uphill speed.

3

Calculate the time spent on each segment.

t_1 = \frac{2x}{v}

,

t_2 = \frac{3x}{2v}

, and

t_3 = \frac{5x}{3v}

.

Time equals distance divided by speed (

t = \frac{d}{v}

).

4

Calculate total travel time by summing individual segment times.

T = \frac{x}{v} \left(2 + \frac{3}{2} + \frac{5}{3}\right) = \frac{31x}{6v}

.

Combining fractions with a common denominator of 6 gives

\frac{12 + 9 + 10}{6} = \frac{31}{6}

.

5

Relate total distance and total time to the overall average speed.

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{10x}{\frac{31x}{6v}} = \frac{60v}{31} = 30

.

Overall average speed is defined as total distance divided by total time.

6

Solve for

v

and calculate the flat segment speed

2v

.

v = 15.5

mph, so flat segment speed

= 2(15.5) = 31

mph.

Solving

\frac{60v}{31} = 30

yields

v = 15.5

, making

2v = 31

.

Anahtar Kavram

Weighted Average Speed and Multi-Segment Distance-Rate-Time Ratios

Soru 1988Soru →

Which of the following values are solutions to the equation $(x - 3)^2 = 16$ ? Select all that apply.

Geçerli olan tümünü seçin

-7

-1

5

7

19

Cevabı ve açıklamayı göster

Cevap: $-1$ ; $7$

Cevap

The solutions to the equation are

-1

and

7

.

Taking the square root of both sides of

(x - 3)^2 = 16

gives

x - 3 = 4

or

x - 3 = -4

. Solving these equations yields

x = 7

and

x = -1

.

Adım Adım Çözüm

1

Take the square root of both sides of the quadratic equation

(x - 3)^2 = 16

.

x - 3 = \pm \sqrt{16} = \pm 4

Applying the square root property yields both a positive and a negative root.

2

Solve the equation corresponding to the positive root:

x - 3 = 4

.

x = 4 + 3 = 7

Adding

3

to both sides isolates

x

.

3

Solve the equation corresponding to the negative root:

x - 3 = -4

.

x = -4 + 3 = -1

Adding

3

to both sides isolates

x

for the negative root.

Anahtar Kavram

Solving quadratic equations of the form

(x - a)^2 = k

using the square root property.

Tahmini Süre:1m 0s

Soru 1989Soru →

A non-profit organization hosted two fundraising events, Event A and Event B. Event A charged a ticket price of $\$30$ per person and collected an additional fixed donation of $\$250$ from a local sponsor. Event B charged a ticket price of $\$45$ per person and collected a fixed donation of $\$400$ from a corporate sponsor. The number of attendees at Event B was $10$ fewer than twice the number of attendees at Event A. If the total revenue raised from both events combined was $\$8,600$ , how many people attended Event B?

60

70

130

140

150

Cevabı ve açıklamayı göster

Cevap: 130

Cevap

130 people attended Event B.

Setting up the linear equation for total revenue gives

(30x + 250) + (45(2x - 10) + 400) = 8,600

, where

x

is the attendance at Event A. Expanding and combining like terms yields

120x + 200 = 8,600

, which solves to

x = 70

. Substituting

x = 70

into the expression for Event B attendance (

2x - 10

) gives

2(70) - 10 = 130

.

Adım Adım Çözüm

1

Define variables for the unknown quantities.

Let

x

be the number of attendees at Event A. The number of attendees at Event B is

2x - 10

.

Event B has 10 fewer attendees than twice Event A.

2

Express the revenue generated by each event in terms of

x

.

Event A revenue:

30x + 250

; Event B revenue:

45(2x - 10) + 400 = 90x - 450 + 400 = 90x - 50

.

Revenue equals ticket price times attendees plus fixed sponsor donations.

3

Set up and simplify the linear equation for total combined revenue.

(30x + 250) + (90x - 50) = 8,600 \implies 120x + 200 = 8,600

.

The sum of revenues from both events equals the total revenue of $8,600.

4

Solve the linear equation for

x

.

120x = 8,400 \implies x = 70

.

Subtract 200 from both sides and divide by 120 to isolate

x

.

5

Calculate the number of attendees at Event B.

Event B attendees

= 2(70) - 10 = 140 - 10 = 130

.

Substitute

x = 70

into the expression

2x - 10

.

Anahtar Kavram

Linear Equations in One Variable

Tahmini Süre:1m 30s

Soru 1990Soru →

A sequence of positive real numbers $a_1, a_2, a_3, \dots$ is defined by $a_1 = 3$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all integers $n \ge 1$ . Which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

The sequence

a_1, a_2, a_3, \dots

is strictly increasing.

The term

a_{50}

is strictly greater than

14

.

When

a_{50}

is rounded to the nearest integer, the result is

15

.

The sequence of consecutive term differences

d_n = a_{n+1} - a_n

is strictly decreasing for

n \ge 1

.

The term

a_{100}

is strictly greater than

25

.

Cevabı ve açıklamayı göster

Cevap: The sequence $a_1, a_2, a_3, \dots$ is strictly increasing.; The term $a_{50}$ is strictly greater than $14$ .; The sequence of consecutive term differences $d_n = a_{n+1} - a_n$ is strictly decreasing for $n \ge 1$ .

Cevap

The correct statements are that the sequence is strictly increasing, the term

a_{50}

is strictly greater than 14, and the sequence of consecutive term differences

d_n = a_{n+1} - a_n

is strictly decreasing.

The statement asserting strict monotonicity is correct because

a_{n+1} - a_n = \frac{2}{a_n} > 0

for positive terms. The statement regarding

a_{50} > 14

is correct because

a_{50}^2 > 3^2 + 4(49) = 205 > 196

. The statement concerning consecutive differences is correct because

\frac{2}{a_n}

strictly decreases as

a_n

increases.

Adım Adım Çözüm

1

Analyze monotonicity of the sequence

a_{n+1} - a_n = \frac{2}{a_n} > 0

for all

n \ge 1

Since

a_1 = 3 > 0

, all terms remain positive, making each term strictly larger than the previous.

2

Analyze the sequence of term differences

d_n = \frac{2}{a_n}

decreases as

a_n

increases

Since

a_n

grows strictly monotonically, its reciprocal strictly decreases, so

d_{n+1} < d_n

.

3

Derive a lower bound for

a_{50}

using quadratic expansion

a_{50}^2 > 205 \implies a_{50} > 14

Expanding

a_{k+1}^2 = a_k^2 + 4 + \frac{4}{a_k^2} > a_k^2 + 4

and summing from

k=1

to

49

gives

a_{50}^2 > 3^2 + 4(49) = 205 > 196 = 14^2

.

4

Derive an upper bound for

a_{50}

to test the rounding statement

a_{50} < 14.45

, so rounding to the nearest integer yields 14

Using

\frac{4}{a_k^2} < \frac{4}{4k+5}

, the sum of error terms is bounded above by

\int_0^{49} \frac{4}{4x+5} dx = \ln(40.2) \approx 3.7

. Thus

a_{50}^2 < 208.7 < 14.5^2

, so

a_{50}

rounds to 14, not 15.

5

Estimate

a_{100}

to test the magnitude bound

a_{100} < \sqrt{409.4} \approx 20.23 < 25

Summing the squared recurrence up to 100 terms gives

a_{100}^2 < 405 + \ln(80.2) \approx 409.4

, showing

a_{100}

cannot exceed 25.

Anahtar Kavram

Estimation of non-linear recursive sequences using squared bounds and integral comparison.

Tahmini Süre:3m 0s

Soru 1991Soru →

In avian biomechanics, the structural design of bird wings frequently demonstrates evolutionary trade-offs rather than pure optimization for a single flight mode. Pelagic seabirds such as albatrosses possess exceptionally high-aspect-ratio wings that maximize aerodynamic efficiency during long-distance soaring. However, this specialized geometry severely limits their maneuverability in turbulent coastal environments and complicates takeoff from flat ground. When forced to navigate dense cliffside wind gradients or initiate sudden evasive maneuvers, the rigid, elongated wing structure can compromise the bird's survival capacity by reducing its agility and increasing structural strain under unexpected turbulence. Researchers evaluating these morphological constraints emphasize that functional adaptations are rarely without penalty; evolutionary pressures do not yield flawless mechanical designs, but rather viable configurations where extreme specialization in one domain predictably impairs performance in another.

In the context of the passage, the word 'compromise' most nearly means:

jeopardize

settle by mutual concession

harmonize

permanently forfeit

arbitrate

Cevabı ve açıklamayı göster

Cevap: jeopardize

Cevap

In the context of the passage, 'compromise' means to imperil or jeopardize, as the wing structure weakens or impairs the bird's survival capacity in turbulent conditions.

The correct answer is the option meaning 'jeopardize'. In the passage, the author discusses how specialized wing morphology impairs maneuverability in turbulence, thereby endangering or undermining the bird's survival capacity under those specific conditions.

Adım Adım Çözüm

1

Locate the target word in the passage and analyze the surrounding clause.

The target word 'compromise' modifies 'the bird's survival capacity by reducing its agility and increasing structural strain under unexpected turbulence.'

Contextual clues in the immediate sentence describe negative physical consequences (reduced agility, increased strain).

2

Examine surrounding thematic signals in the passage.

The preceding sentence notes that specialized geometry 'severely limits maneuverability,' and the subsequent sentence notes that 'extreme specialization in one domain predictably impairs performance in another.'

The author establishes a clear negative polarity showing that structural trade-offs harm or imperil performance.

3

Evaluate the answer choices against the contextual meaning of impairment/endangerment.

The word 'jeopardize' precisely matches the sense of imperiling or exposing to risk.

It captures the secondary definition of 'compromise' (to expose to danger or weaken) while rejecting the primary conversational meaning (to negotiate terms).

Anahtar Kavram

Reading Comprehension: Passage-Based Word in Context Analysis

Tahmini Süre:1m 0s

Soru 1992Soru →

Two deep-space radio signals have power measurements of $S_1 = 4.5 \times 10^{-7}$ watts and $S_2 = 3.5 \times 10^{-6}$ watts. A combined signal power is defined as $S_3 = S_1 + S_2$ . When $S_3$ is written in scientific notation as $a \times 10^n$ , where $1 \leq a < 10$ and $n$ is an integer, what is the value of $a + n$ ?

-5.00

-2.05

2.00

9.95

32.50

Cevabı ve açıklamayı göster

Cevap: $-2.05$

Cevap

The value of

a + n

is

-2.05

.

To add numbers in scientific notation, first rewrite them with identical powers of 10. Expressing

4.5 \times 10^{-7}

as

0.45 \times 10^{-6}

allows direct addition with

3.5 \times 10^{-6}

, resulting in

3.95 \times 10^{-6}

. Here

a = 3.95

(which satisfies

1 \leq a < 10

) and

n = -6

. Adding

a + n

yields

3.95 + (-6) = -2.05

.

Adım Adım Çözüm

1

Express both quantities with a common power of 10 to allow addition.

S_1 = 4.5 \times 10^{-7} = 0.45 \times 10^{-6}

watts.

Before adding numbers in scientific notation, their powers of 10 must match.

2

Add the coefficients while maintaining the common power of 10.

S_3 = (0.45 + 3.5) \times 10^{-6} = 3.95 \times 10^{-6}

watts.

Distributive property allows adding coefficients once exponents match.

3

Verify proper scientific notation form

a \times 10^n

where

1 \leq a < 10

.

a = 3.95

and

n = -6

.

The coefficient

3.95

satisfies

1 \leq 3.95 < 10

, so no further decimal shift is needed.

4

Calculate the requested sum

a + n

.

3.95 + (-6) = -2.05

.

Adding the coefficient

3.95

to the negative exponent

-6

gives

-2.05

.

Anahtar Kavram

Scientific Notation Addition and Place Value Alignment

Soru 1993Soru →

If $x$ and $y$ are positive integers such that $3^{x+2} \cdot 4^y - 3^x \cdot 4^{y+1} = 11,520$ , what is the value of $x + y$ ?

5

6

7

8

10

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

Factoring

3^x \cdot 4^y

from the expression

3^{x+2} \cdot 4^y - 3^x \cdot 4^{y+1}

yields

3^x \cdot 4^y (3^2 - 4^1) = 5 \cdot 3^x \cdot 4^y

. Setting this equal to 11,520 and dividing by 5 gives

3^x \cdot 4^y = 2,304

. Prime factorization of 2,304 gives

3^2 \cdot 4^4

, so

x = 2

and

y = 4

. The sum

x + y

is equal to 6.

Adım Adım Çözüm

1

Factor out the greatest common exponential factor

3^x \cdot 4^y

from the left side of the equation.

3^x \cdot 4^y (3^2 - 4^1) = 11,520

By exponent rules,

3^{x+2} = 3^x \cdot 3^2

and

4^{y+1} = 4^y \cdot 4^1

.

2

Evaluate the constant factor inside the parentheses.

3^2 - 4^1 = 9 - 4 = 5

, so

5 \cdot 3^x \cdot 4^y = 11,520

Simplifying numerical exponents.

3

Divide both sides of the equation by 5.

3^x \cdot 4^y = 2,304

Isolating the variable exponential terms.

4

Determine the prime factorization of 2,304 into powers of 3 and 4.

2,304 = 9 \cdot 256 = 3^2 \cdot 4^4

, which implies

x = 2

and

y = 4

Unique factorization for integer bases.

5

Calculate the sum

x + y

.

x + y = 2 + 4 = 6

Answering the explicit prompt.

Anahtar Kavram

Factoring Exponential Expressions and Unique Factorization

Soru 1994Soru →

Although the early paleontologist presented her fossil evidence with apparent hesitation, historical analyses reveal that her caution stemmed not from ideological doubts, but rather from a deep-seated __________ to engage in public controversies. Which two of the following options, when inserted into the blank, produce completed sentences that are logically equivalent?

Geçerli olan tümünü seçin

disinclination

trepidation

aversion

polemic

apprehension

skepticism

Cevabı ve açıklamayı göster

Cevap: disinclination; aversion

Cevap

The correct options are 'disinclination' and 'aversion', as both words denote reluctance or unwillingness, appropriately completing the sentence's contrast structure.

The sentence structure creates a contrast: the scientist's behavior was NOT caused by fear or ideological doubts, BUT RATHER by a simple reluctance to engage in public debate. The words 'disinclination' and 'aversion' both convey a strong unwillingness or preference against doing something, fitting the logic of the sentence while producing equivalent meanings.

Adım Adım Çözüm

1

Analyze structural transition and contextual clues in the sentence stem.

The phrase 'stemmed not from ideological doubts, but rather from...' signals a contrast between fear/doubt and a different underlying motive for her behavior.

Identifying the reversal clue establishes that the blank requires a word meaning reluctance or preference against public disputes rather than fear.

2

Evaluate candidate choices for semantic synonym pairs.

Two distinct synonym pairs emerge: 'disinclination' / 'aversion' (reluctance/unwillingness) and 'trepidation' / 'apprehension' (fear/anxiety). 'Polemic' and 'skepticism' are unpaired topical distractors.

GRE Sentence Equivalence requires both contextual fit and semantic pair equivalence.

3

Eliminate false synonym pairs and topical traps to select the contextually fitting pair.

'Trepidation' and 'apprehension' mean fear, which contradicts 'not from ideological doubts'. Thus, 'disinclination' and 'aversion' are selected.

Only 'disinclination' and 'aversion' fulfill both the structural contrast constraint and semantic equivalence requirements.

Anahtar Kavram

Eliminating False Synonyms and Topically Related Distractors

Soru 1995Soru →

For two positive integers $m$ and $n$ , the greatest common divisor is $\gcd(m, n) = 15$ and the least common multiple is $\text{lcm}(m, n) = 900$ . If $m = 75$ , what is the total number of positive divisors of $n$ ?

18

9

5

12

3

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

The correct answer is 18. First, use the relation

m \cdot n = \gcd(m, n) \cdot \text{lcm}(m, n)

to find

n = \frac{15 \cdot 900}{75} = 180

. Next, write 180 in prime factorized form:

180 = 2^2 \cdot 3^2 \cdot 5^1

. The total number of positive divisors is found by adding 1 to each exponent and multiplying the results:

(2 + 1)(2 + 1)(1 + 1) = 3 \cdot 3 \cdot 2 = 18

.

Adım Adım Çözüm

1

Calculate the value of

n

using the fundamental product identity for GCD and LCM.

n = \frac{\gcd(m, n) \cdot \text{lcm}(m, n)}{m} = \frac{15 \cdot 900}{75} = 180

.

For any two positive integers, the product of the integers equals the product of their GCD and LCM.

2

Find the prime factorization of

180

.

180 = 2^2 \cdot 3^2 \cdot 5^1

.

Breaking down 180 into prime powers allows determination of the total count of positive divisors.

3

Apply the divisor counting formula by adding 1 to each prime exponent and multiplying.

(2+1)(2+1)(1+1) = 3 \cdot 3 \cdot 2 = 18

.

If an integer has prime factorization

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

, the number of positive divisors is

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

.

Anahtar Kavram

GCD and LCM fundamental identity (

a \cdot b = \gcd(a,b) \cdot \text{lcm}(a,b)

) combined with the prime factorization divisor counting formula.

Tahmini Süre:1m 30s

Soru 1996Soru →

In Year 1, a software company had a total of 400 subscribers divided between a Basic plan and a Premium plan, with $70\%$ of the subscribers enrolled in the Basic plan. In Year 2, the number of Basic plan subscribers decreased by $20\%$ , while the total number of subscribers across both plans increased by $10\%$ . What was the percentage increase in the number of Premium plan subscribers from Year 1 to Year 2?

24\%

30\%

50\%

80\%

180\%

Cevabı ve açıklamayı göster

Cevap: $80\%$

Cevap

The percentage increase in Premium plan subscribers from Year 1 to Year 2 was

80\%

.

In Year 1,

70\%

of the 400 subscribers were on the Basic plan (

0.70 \times 400 = 280

), which leaves 120 subscribers on the Premium plan. In Year 2, the total subscriber count increased by

10\%

to 440 (

400 \times 1.10 = 440

), while Basic subscribers decreased by

20\%

to 224 (

280 \times 0.80 = 224

). Subtracting 224 Basic subscribers from the total 440 gives 216 Premium subscribers in Year 2. The increase in Premium subscribers is

216 - 120 = 96

. To find the percentage increase, divide the increase of 96 by the original Premium subscriber count of 120, yielding

\frac{96}{120} = 0.80

, or

80\%

.

Adım Adım Çözüm

1

Calculate the number of Basic and Premium subscribers in Year 1

Basic subscribers =

0.70 \times 400 = 280

; Premium subscribers =

400 - 280 = 120

Determining the starting baseline values for each category is necessary to evaluate subsequent changes.

2

Calculate the total subscribers and Basic subscribers in Year 2

Total subscribers in Year 2 =

400 \times (1 + 0.10) = 440

; Basic subscribers in Year 2 =

280 \times (1 - 0.20) = 224

Apply the given percentage changes to find the updated group totals in Year 2.

3

Find the number of Premium subscribers in Year 2 and the absolute increase

Premium subscribers in Year 2 =

440 - 224 = 216

; Absolute increase =

216 - 120 = 96

Subtracting the Basic subscriber count from the total count yields the Premium subscriber count for Year 2.

4

Compute the percentage increase for Premium subscribers

Percentage increase =

\frac{96}{120} \times 100\% = 80\%

Divide the absolute increase by the initial Year 1 Premium subscriber count (the correct base value) and convert to a percentage.

Anahtar Kavram

Successive percentage changes and base shifting in multi-part totals

Tahmini Süre:1m 30s

Soru 1997Soru →

If $x$ and $y$ are real numbers such that $|x - 3| \le 2$ and $|y + 1| \le 4$ , which of the following could be the value of $|x - y|$ ? Select all such values.

Geçerli olan tümünü seçin

-2

0

5

10

12

Cevabı ve açıklamayı göster

Cevap: $0$ ; $5$ ; $10$

Cevap

The possible values of

|x - y|

are 0, 5, and 10.

Solving the inequalities yields

1 \le x \le 5

and

-5 \le y \le 3

. The expression

|x - y|

represents the distance between

x

and

y

on the number line. Because the intervals overlap between 1 and 3,

x

and

y

can be equal, making the minimum distance 0. The maximum distance occurs at the extreme points

x = 5

and

y = -5

, giving a distance of

|5 - (-5)| = 10

. Therefore, any value from 0 to 10 inclusive is possible, making 0, 5, and 10 valid values.

Adım Adım Çözüm

1

Solve the absolute value inequality for

x

.

-2 \le x - 3 \le 2 \implies 1 \le x \le 5

Unwrapping

|x - 3| \le 2

gives the bounds for

x

on the real number line.

2

Solve the absolute value inequality for

y

.

-4 \le y + 1 \le 4 \implies -5 \le y \le 3

Unwrapping

|y + 1| \le 4

gives the bounds for

y

on the real number line.

3

Determine the minimum and maximum possible values for

|x - y|

.

Minimum value is 0 (since the intervals

[1, 5]

and

[-5, 3]

overlap at

[1, 3]

). Maximum value is

|5 - (-5)| = 10

. Thus,

0 \le |x - y| \le 10

.

The absolute value

|x - y|

represents the distance between

x

and

y

on the number line, which can take any real value from 0 to 10.

4

Evaluate the given choices against the range

[0, 10]

.

The values 0, 5, and 10 fall within

[0, 10]

, whereas

-2

is impossible for absolute values and

12

exceeds the maximum bound.

Determines which specific options are valid outcomes for

|x - y|

.

Anahtar Kavram

Absolute value as distance on the real number line and range of differences between bounded real variables

Tahmini Süre:1m 30s

Soru 1998Soru →

For all real numbers

x

such that

x \neq 3

and

x \neq -3

, the algebraic expression

\frac{\frac{x^4 - 81}{x^2 - 9} \cdot \frac{x^3 - 27}{x^2 + 3x + 9} - x(x - 3)^2}{x^2 - 9}

simplifies to a constant value. What is the value of this constant?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The simplified expression evaluates to the constant value

3

.

Factoring the numerator components using difference of squares and difference of cubes simplifies the product term to

x^3 - 3x^2 + 9x - 27

. Subtracting

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

simplifies the entire numerator to

3x^2 - 27 = 3(x^2 - 9)

. Dividing by the denominator

(x^2 - 9)

cancels out the variable terms entirely, yielding the constant value 3.

Adım Adım Çözüm

1

Simplify the first rational component using the difference of squares identity

\frac{x^4 - 81}{x^2 - 9} = \frac{(x^2 - 9)(x^2 + 9)}{x^2 - 9} = x^2 + 9

Since

x \neq \pm 3

,

x^2 - 9 \neq 0

, allowing direct cancellation of

(x^2 - 9)

.

2

Simplify the second rational component using the difference of cubes identity

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

Applying

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

where

a = x

and

b = 3

allows cancellation of the quadratic factor.

3

Multiply the simplified expressions

(x^2 + 9)(x - 3) = x^3 - 3x^2 + 9x - 27

Distribute each term of the binomials to get the expanded polynomial.

4

Expand the subtracted term in the numerator

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

Expand

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

and distribute

x

.

5

Subtract the two expanded expressions to simplify the entire numerator

(x^3 - 3x^2 + 9x - 27) - (x^3 - 6x^2 + 9x) = 3x^2 - 27 = 3(x^2 - 9)

Combine like terms;

x^3

and

9x

terms cancel out, leaving

3x^2 - 27

.

6

Divide the simplified numerator by the main denominator

\frac{3(x^2 - 9)}{x^2 - 9} = 3

Cancel the common factor

(x^2 - 9)

from numerator and denominator.

Anahtar Kavram

Simplifying complex algebraic expressions via polynomial factoring (difference of squares and difference of cubes) and combining like terms.

Soru 1999Soru →

In a sample space of a random experiment, $A$ and $B$ are independent events such that $P(A) = 0.35$ and $P(A \cup B) = 0.74$ . Event $C$ is mutually exclusive with Event $A$ . If the conditional probability $P(C \mid B) = 0.20$ , what is the probability that Event $B$ occurs, but neither Event $A$ nor Event $C$ occurs?

0.19

0.27

0.39

0.48

0.51

Cevabı ve açıklamayı göster

Cevap: 0.27

Cevap

0.27

To find the probability that Event

B

occurs without

A

or

C

, we must isolate the region of

B

that does not overlap with

A

or

C

. Since

A

and

B

are independent,

P(A \cup B) = P(A) + P(B) - P(A)P(B)

, which allows us to solve for

P(B) = 0.60

and

P(A \cap B) = 0.21

. Next, using the conditional probability

P(C \mid B) = 0.20

, we find

P(B \cap C) = 0.20 \times 0.60 = 0.12

. Because

A

and

C

are mutually exclusive, the intersections

A \cap B

and

B \cap C

do not overlap. Subtracting both intersection probabilities from

P(B)

yields

0.60 - 0.21 - 0.12 = 0.27

.

Adım Adım Çözüm

1

Calculate the probability of Event

B

,

P(B)

, using the independence of

A

and

B

.

P(B) = 0.60

Since

A

and

B

are independent,

P(A \cap B) = P(A)P(B)

. Substituting into the union formula

P(A \cup B) = P(A) + P(B) - P(A)P(B)

gives

0.74 = 0.35 + P(B)(1 - 0.35)

, so

0.39 = 0.65 P(B)

, yielding

P(B) = 0.60

.

2

Find the joint probability

P(A \cap B)

.

P(A \cap B) = 0.21

By independence,

P(A \cap B) = P(A) \times P(B) = 0.35 \times 0.60 = 0.21

.

3

Calculate the joint probability

P(B \cap C)

using the conditional probability formula.

P(B \cap C) = 0.12

From the definition of conditional probability,

P(C \mid B) = \frac{P(B \cap C)}{P(B)}

, so

P(B \cap C) = P(C \mid B) \cdot P(B) = 0.20 \times 0.60 = 0.12

.

4

Determine the probability that

B

occurs but neither

A

nor

C

occurs.

P(B \cap A^c \cap C^c) = 0.27

Since

A

and

C

are mutually exclusive, events

(A \cap B)

and

(B \cap C)

are disjoint subsets of

B

. Therefore,

P(B \cap A^c \cap C^c) = P(B) - P(A \cap B) - P(B \cap C) = 0.60 - 0.21 - 0.12 = 0.27

.

Anahtar Kavram

Probability Rules for Independent, Dependent, and Mutually Exclusive Events

Soru 2000Soru →

In the $xy$ -plane, line $k$ passes through the point $(3, 5)$ and has a slope of $-3$ . Line $m$ is perpendicular to line $k$ and also passes through the point $(3, 5)$ . What is the $x$ -intercept of line $m$ ?

-12

4

18

\frac{14}{3}

\frac{4}{3}

Cevabı ve açıklamayı göster

Cevap: $-12$

Cevap

-12

The slope of line

k

is

-3

, so the slope of perpendicular line

m

is its negative reciprocal,

\frac{1}{3}

. Substituting slope

\frac{1}{3}

and point

(3, 5)

into point-slope form gives

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

, which simplifies to

y = \frac{1}{3}x + 4

. Setting

y = 0

to find the

x

-intercept gives

0 = \frac{1}{3}x + 4

, yielding

x = -12

.

Adım Adım Çözüm

1

Determine the slope of line

m

Since line

m

is perpendicular to line

k

, its slope is the negative reciprocal of

-3

, which is

\frac{1}{3}

.

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

2

Find the equation of line

m

Using point-slope form with point

(3, 5)

and slope

\frac{1}{3}

:

y - 5 = \frac{1}{3}(x - 3) \implies y = \frac{1}{3}x + 4

.

A line with slope

m

passing through

(x_1, y_1)

follows

y - y_1 = m(x - x_1)

.

3

Calculate the

x

-intercept of line

m

Set

y = 0

:

0 = \frac{1}{3}x + 4 \implies \frac{1}{3}x = -4 \implies x = -12

.

The

x

-intercept is the

x

-coordinate where the line intersects the

x

-axis (

y = 0

).

Anahtar Kavram

Perpendicular Slopes and Line Intercepts

Tahmini Süre:1m 30s

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