Tüm alıştırma soruları

Soru 1581Soru →

For all real numbers $x \neq 0$ and $x \neq 1$ , the function $f$ is defined by $f(x) = \frac{x - 1}{x}$ . The composite function notation $f^n(x)$ is defined by $f^1(x) = f(x)$ and $f^n(x) = f\left(f^{n-1}(x)\right)$ for all integers $n \ge 2$ . What is the value of $f^{2026}(5)$ ?

-\frac{5}{4}

-\frac{1}{4}

\frac{4}{5}

\frac{5}{4}

5

Cevabı ve açıklamayı göster

Cevap: $\frac{4}{5}$

Cevap

\frac{4}{5}

Evaluating successive compositions of

f(x) = \frac{x-1}{x}

starting at

x = 5

yields

f^1(5) = \frac{4}{5}

,

f^2(5) = -\frac{1}{4}

, and

f^3(5) = 5

. This produces a repeating sequence of period 3. Since

2026 \equiv 1 \pmod 3

,

f^{2026}(5)

equals

f^1(5) = \frac{4}{5}

.

Adım Adım Çözüm

1

Evaluate the first iteration

f^1(5)

.

f^1(5) = f(5) = \frac{5 - 1}{5} = \frac{4}{5}

.

Substitute

x = 5

into the definition of

f(x)

.

2

Evaluate the second iteration

f^2(5) = f(f^1(5))

.

f^2(5) = f\left(\frac{4}{5}\right) = \frac{\frac{4}{5} - 1}{\frac{4}{5}} = \frac{-\frac{1}{5}}{\frac{4}{5}} = -\frac{1}{4}

.

Substitute

x = \frac{4}{5}

into the function definition.

3

Evaluate the third iteration

f^3(5) = f(f^2(5))

.

f^3(5) = f\left(-\frac{1}{4}\right) = \frac{-\frac{1}{4} - 1}{-\frac{1}{4}} = \frac{-\frac{5}{4}}{-\frac{1}{4}} = 5

.

Substitute

x = -\frac{1}{4}

into the function definition.

4

Identify the periodic pattern and evaluate

f^{2026}(5)

.

Since

f^3(5) = 5

, the function values repeat in a cycle of length 3:

(\frac{4}{5}, -\frac{1}{4}, 5)

. Dividing 2026 by 3 yields

2026 = 3 \times 675 + 1

, giving a remainder of 1. Therefore,

f^{2026}(5) = f^1(5) = \frac{4}{5}

.

The remainder determines the equivalent position in the 3-element repeating sequence.

Anahtar Kavram

Nested Function Composition and Periodicity

Soru 1582Soru →

If $x$ is a real number that satisfies $|3 - 2x| \le 9$ , what is the maximum possible value of $|x - 5|$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The maximum possible value of

|x - 5|

is 8.

Solving

|3 - 2x| \le 9

yields

-9 \le 3 - 2x \le 9

. Subtracting 3 gives

-12 \le -2x \le 6

. Dividing by

-2

and reversing inequality signs results in

-3 \le x \le 6

. To maximize

|x - 5|

, find the point in

[-3, 6]

that is farthest from 5. Evaluating at the endpoints gives

|-3 - 5| = 8

and

|6 - 5| = 1

. The maximum possible value is 8.

Adım Adım Çözüm

1

Solve the absolute value inequality

|3 - 2x| \le 9

.

-9 \le 3 - 2x \le 9

By definition of absolute value,

|u| \le k

means

-k \le u \le k

for

k \ge 0

.

2

Isolate

-2x

by subtracting 3 from each part.

-12 \le -2x \le 6

Subtracting a constant from all parts preserves the inequality direction.

3

Isolate

x

by dividing each part by

-2

.

-3 \le x \le 6

Dividing an inequality by a negative number flips the inequality signs.

4

Determine the maximum value of

|x - 5|

over the interval

[-3, 6]

.

Maximum occurs at

x = -3

, where

|-3 - 5| = 8

.

The expression

|x - 5|

represents the distance from

x

to 5 on the number line. The endpoint

x = -3

is furthest from 5.

Anahtar Kavram

Linear inequalities and absolute value as distance on a number line

Soru 1583Soru →

The table below shows the frequency distribution of daily rainfall, in millimeters, recorded across 50 weather monitoring stations during a storm event.

Daily Rainfall (mm)	Number of Stations
0 to under 10	10
10 to under 20	15
20 to under 30	17
30 to under 40	5
40 to under 50	3

What percentage of the stations that recorded at least 10 mm of daily rainfall recorded less than 30 mm of daily rainfall?

42.5\%

64\%

75\%

80\%

84\%

Cevabı ve açıklamayı göster

Cevap: $80\%$

Cevap

80\%

To find the desired percentage, first identify the total number of stations that recorded at least 10 mm of rainfall. This includes the intervals 10 to under 20, 20 to under 30, 30 to under 40, and 40 to under 50, giving

15 + 17 + 5 + 3 = 40

stations. Next, find how many of these 40 stations recorded less than 30 mm. These fall into the 10 to under 20 and 20 to under 30 intervals, which total

15 + 17 = 32

stations. Calculating the relative percentage gives

(32 / 40) \times 100\% = 80\%

.

Adım Adım Çözüm

1

Determine the conditional base (denominator)

Sum the frequencies for all intervals representing at least 10 mm:

15 + 17 + 5 + 3 = 40

stations.

The question specifies 'stations that recorded at least 10 mm of daily rainfall', restricting our population base to this subset.

2

Determine the target subgroup frequency (numerator)

Sum the frequencies for intervals within the base that are also less than 30 mm (the 10 to under 20 and 20 to under 30 intervals):

15 + 17 = 32

stations.

We need to count the stations that meet both conditions: at least 10 mm and less than 30 mm.

3

Calculate the percentage

\frac{32}{40} \times 100\% = 0.80 \times 100\% = 80\%

.

Divide the target subgroup count by the conditional base count and multiply by 100.

Anahtar Kavram

Grouped frequency table interpretation with conditional percentages

Tahmini Süre:1m 30s

Soru 1584Soru →

If $x$ is an integer that satisfies both $|2x + 1| \le 7$ and $\frac{5 - x}{-2} < -1$ , how many possible values of $x$ are there?

0

3

7

8

11

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

There are 7 possible integer values of

x

.

Solving

|2x + 1| \le 7

yields

-4 \le x \le 3

. Solving

\frac{5 - x}{-2} < -1

requires flipping the inequality symbol when multiplying by

-2

, resulting in

5 - x > 2

, which simplifies to

x < 3

. Taking the intersection of

-4 \le x \le 3

and

x < 3

gives the interval

-4 \le x < 3

. The integers in this range are

-4, -3, -2, -1, 0, 1, 2

, totaling 7 integers.

Adım Adım Çözüm

1

Solve the absolute value inequality

|2x + 1| \le 7

.

-7 \le 2x + 1 \le 7 \implies -8 \le 2x \le 6 \implies -4 \le x \le 3

.

An absolute value inequality of the form

|u| \le k

(for

k \ge 0

) expands to

-k \le u \le k

.

2

Solve the rational inequality

\frac{5 - x}{-2} < -1

.

5 - x > (-1)(-2) \implies 5 - x > 2 \implies -x > -3 \implies x < 3

.

Multiplying or dividing an inequality by a negative number requires reversing the direction of the inequality sign.

3

Find the intersection of the two solution sets and count integer solutions.

-4 \le x < 3

. The integer values satisfying this inequality are

-4, -3, -2, -1, 0, 1, 2

, which gives a total of 7 integer values.

The solution must satisfy both conditions simultaneously.

Anahtar Kavram

Solving combined linear and absolute value inequalities while correctly reversing inequality signs when multiplying or dividing by negative values.

Tahmini Süre:1m 30s

Soru 1585Soru →

Two water pumps, Pump A and Pump B, working simultaneously at their respective constant rates, can fill an empty storage tank in $4$ hours. If Pump A operates alone at its constant rate for $2$ hours and then Pump B operates alone at its constant rate for $7$ hours, the tank is also filled completely. How many hours would it take Pump A, working alone at its constant rate, to fill the entire storage tank?

\frac{10}{3}

hours

\frac{20}{3}

hours

10

hours

12

hours

15

hours

Cevabı ve açıklamayı göster

Cevap: $\frac{20}{3}$ hours

Cevap

\frac{20}{3}

hours (or

6\frac{2}{3}

hours)

The correct answer is derived by setting up two linear equations representing the total work accomplished:

4r_A + 4r_B = 1

and

2r_A + 7r_B = 1

. Solving this system gives Pump A's rate

r_A = \frac{3}{20}

tanks per hour. The time required for Pump A working alone is the reciprocal of its rate, which equals

\frac{20}{3}

hours.

Adım Adım Çözüm

1

Define variables and formulate the system of linear equations

Let

r_A

be the rate of Pump A (tanks/hour) and

r_B

be the rate of Pump B (tanks/hour).
Combined work equation:

4(r_A + r_B) = 1 \implies 4r_A + 4r_B = 1

Sequential work equation:

2r_A + 7r_B = 1

Work done equals rate multiplied by time, and completing one full tank corresponds to total work

= 1

.

2

Express

r_B

in terms of

r_A

using the first equation

r_A + r_B = \frac{1}{4} \implies r_B = \frac{1}{4} - r_A

Simplifying the combined rate equation allows substitution into the second linear equation.

3

Substitute

r_B

into the second equation and solve for

r_A

2r_A + 7\left(\frac{1}{4} - r_A\right) = 1 \implies 2r_A + \frac{7}{4} - 7r_A = 1 \implies -5r_A = 1 - \frac{7}{4} = -\frac{3}{4} \implies r_A = \frac{3}{20}

Eliminating

r_B

isolates

r_A

as a single-variable linear equation.

4

Calculate the time required for Pump A to fill the tank alone

\text{Time}_A = \frac{1}{r_A} = \frac{1}{\frac{3}{20}} = \frac{20}{3}\text{ hours}

The total time to complete 1 unit of work is the reciprocal of the unit work rate.

Anahtar Kavram

Formulating and solving systems of two linear equations in two variables derived from work-rate relationships.

Soru 1586Soru →

In the $xy$ -plane, line $\ell_1$ is defined by the equation $3x - 4y = 12$ . Line $\ell_2$ is perpendicular to line $\ell_1$ and intersects the positive $y$ -axis at the point $(0, d)$ . If the area of the triangular region bounded by line $\ell_1$ , line $\ell_2$ , and the $y$ -axis is $24$ , what is the value of $d$ ?

5

6

7

9

13

Cevabı ve açıklamayı göster

Cevap: $7$

Cevap

The value of

d

is

7

.

The line

\ell_1

has equation

y = \frac{3}{4}x - 3

, placing its

y

-intercept at

(0, -3)

. Line

\ell_2

is perpendicular, so its slope is

-\frac{4}{3}

, giving the equation

y = -\frac{4}{3}x + d

. The vertical base of the triangle along the

y

-axis spans from

(0, -3)

to

(0, d)

, with a length of

d + 3

. The intersection of the two lines occurs at an

x

-coordinate of

\frac{12(d+3)}{25}

, which serves as the height of the triangle. Setting the area

\frac{1}{2} \times (d+3) \times \frac{12(d+3)}{25} = 24

simplifies to

(d+3)^2 = 100

. Because

d > 0

,

d + 3 = 10

, giving

d = 7

.

Adım Adım Çözüm

1

Find the slope and

y

-intercept of line

\ell_1

.

Converting

3x - 4y = 12

into slope-intercept form yields

y = \frac{3}{4}x - 3

. The slope of

\ell_1

is

m_1 = \frac{3}{4}

and its

y

-intercept is

(0, -3)

.

Knowing the slope and

y

-intercept of

\ell_1

is essential to determine the equation of line

\ell_2

and the vertices of the triangular region along the

y

-axis.

2

Determine the equation of line

\ell_2

.

Since

\ell_2

is perpendicular to

\ell_1

, its slope is the negative reciprocal of

\frac{3}{4}

, which is

m_2 = -\frac{4}{3}

. Given that

\ell_2

intersects the

y

-axis at

(0, d)

, its equation is

y = -\frac{4}{3}x + d

.

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Find the

x

-coordinate of the intersection point of

\ell_1

and

\ell_2

.

Set the two equations equal:

\frac{3}{4}x - 3 = -\frac{4}{3}x + d \implies \left(\frac{3}{4} + \frac{4}{3}\right)x = d + 3 \implies \frac{25}{12}x = d + 3 \implies x = \frac{12(d+3)}{25}

.

The

x

-coordinate of the intersection point represents the horizontal altitude (height) of the triangle with respect to the vertical base along the

y

-axis.

4

Express the area of the triangular region in terms of

d

and solve for

d

.

The vertical base along the

y

-axis stretches from

(0, -3)

to

(0, d)

, having length

d - (-3) = d + 3

. The height is

h = \frac{12(d+3)}{25}

. Using the area formula

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

, we set

\frac{1}{2}(d+3)\left(\frac{12(d+3)}{25}\right) = 24 \implies \frac{6(d+3)^2}{25} = 24 \implies (d+3)^2 = 100

. Since

d > 0

,

d + 3 = 10

, so

d = 7

.

The problem states that the area of the triangle is

24

.

Anahtar Kavram

Perpendicular line slopes and geometric area calculations using coordinate geometry

Tahmini Süre:2m 30s

Soru 1587Soru →

A coffee roaster creates a custom blend by combining Grade X beans costing $\$p$ per pound with Grade Y beans costing $\$q$ per pound, where $0 < p < q$ . The batch contains a total of $M$ pounds, consisting of $x$ pounds of Grade X and $y$ pounds of Grade Y. The total cost of the batch is $C$ dollars, and the average cost per pound of the blend is $A$ dollars. Which of the following algebraic relationships MUST be true? Indicate all such statements.

Geçerli olan tümünü seçin

The weight of Grade X beans in pounds is given by

x = \frac{qM - C}{q - p}

.

The weight of Grade X beans in pounds is given by

x = \frac{C - pM}{q - p}

.

The average cost per pound of the blend satisfies

A = q - \frac{x(q - p)}{M}

.

If the weight of Grade X in the blend is increased by

20\%

while keeping the weight of Grade Y constant, the total cost of the batch increases by

0.20C

dollars.

The ratio of the weight of Grade X to the weight of Grade Y in the blend is equal to

\frac{q - A}{A - p}

.

Cevabı ve açıklamayı göster

Cevap: The weight of Grade X beans in pounds is given by $x = \frac{qM - C}{q - p}$ .; The average cost per pound of the blend satisfies $A = q - \frac{x(q - p)}{M}$ .; The ratio of the weight of Grade X to the weight of Grade Y in the blend is equal to $\frac{q - A}{A - p}$ .

Cevap

The true statements are that the weight of Grade X beans is

x = \frac{qM - C}{q - p}

, the average cost per pound satisfies

A = q - \frac{x(q - p)}{M}

, and the ratio of Grade X weight to Grade Y weight is

\frac{q - A}{A - p}

.

The correct statements correctly model the system of equations

x + y = M

and

C = px + qy

. Solving for

x

in terms of total cost gives

x = \frac{qM - C}{q - p}

. Dividing the expression for total cost by total mass

M

yields

A = q - \frac{x(q - p)}{M}

. Finally, setting total cost

C = A(x+y) = px + qy

and rearranging gives the ratio

\frac{x}{y} = \frac{q - A}{A - p}

.

Adım Adım Çözüm

1

Set up equations for total mass

M

and total cost

C

using individual weights

x

and

y

.

x + y = M \implies y = M - x

and

C = px + qy = px + q(M - x)

.

Relating the two variables through total weight eliminates

y

to express cost purely in terms of

x

.

2

Solve the total cost equation for

x

.

C = qM - (q - p)x \implies (q - p)x = qM - C \implies x = \frac{qM - C}{q - p}

.

Isolating

x

confirms the valid algebraic formula for the weight of Grade X.

3

Calculate average cost

A = \frac{C}{M}

by substituting the simplified expression for

C

.

A = \frac{qM - (q - p)x}{M} = q - \frac{x(q - p)}{M}

.

Dividing total cost by total weight

M

yields the weighted average cost per pound.

4

Determine the ratio of component weights

\frac{x}{y}

in terms of unit costs

p, q

and average cost

A

.

A(x + y) = px + qy \implies Ax + Ay = px + qy \implies y(q - A) = x(A - p) \implies \frac{x}{y} = \frac{q - A}{A - p}

.

Rearranging the weighted average equation isolates the ratio of the quantities of the two components.

Anahtar Kavram

Linear weighted averages and multi-variable system modeling in mixture word problems.

Tahmini Süre:2m 30s

Soru 1588Soru →

A municipal utility company charges residential customers according to two seasonal rate structures. During the peak summer season, electricity costs $\$0.24$ per kilowatt-hour ( $\text{kWh}$ ). During the off-peak season, electricity costs $\$0.15$ per $\text{kWh}$ . A homeowner consumed a total of $14,000\text{ kWh}$ of electricity over the entire year, resulting in a total annual bill of $\$2,640$ . How many kilowatt-hours of electricity did the homeowner consume during the peak summer season?

6,000

8,000

2,250

11,000

7,000

Cevabı ve açıklamayı göster

Cevap: $6,000$

Cevap

6,000\text{ kWh}

The correct answer of

6,000\text{ kWh}

is found by setting up a linear model for total energy consumption (

x + y = 14,000

) and total cost (

0.24x + 0.15y = 2,640

). Substituting

y = 14,000 - x

gives

0.09x = 540

, which solves to

x = 6,000\text{ kWh}

.

Adım Adım Çözüm

1

Define variables for the two seasons.

Let

x

be the number of kilowatt-hours consumed during the peak summer season, and let

y

be the number of kilowatt-hours consumed during the off-peak season.

Establishing clear variables allows modeling the problem as a system of linear equations.

2

Set up the total consumption equation and express

y

in terms of

x

.

x + y = 14,000 \implies y = 14,000 - x

The total consumption across both seasons equals

14,000\text{ kWh}

.

3

Set up the total cost equation using the respective seasonal rates.

0.24x + 0.15y = 2,640

Peak electricity costs

\$0.24/\text{kWh}

and off-peak costs

\$0.15/\text{kWh}

, summing to the total annual cost of

\$2,640

.

4

Substitute

y = 14,000 - x

into the cost equation and solve for

x

.

0.24x + 0.15(14,000 - x) = 2,640 \implies 0.24x + 2,100 - 0.15x = 2,640 \implies 0.09x = 540 \implies x = 6,000

Simplifying the single-variable linear equation isolates the peak season consumption.

Anahtar Kavram

Linear Modeling and Systems of Linear Equations

Tahmini Süre:1m 30s

Soru 1589Soru →

In the $xy$ -plane, line $k$ passes through the origin $(0,0)$ and the point $(4, 3)$ . Line $m$ is perpendicular to line $k$ and intersects line $k$ at point $(4, 3)$ . If line $m$ intersects the $x$ -axis at point $Q$ and the $y$ -axis at point $R$ , what is the distance between point $Q$ and point $R$ ?

\frac{35}{12}

\frac{25}{12}

10

\frac{125}{12}

\frac{175}{12}

Cevabı ve açıklamayı göster

Cevap: $\frac{125}{12}$

Cevap

\frac{125}{12}

The correct answer is derived by finding the slope of line

k

as

\frac{3}{4}

, taking its negative reciprocal

-\frac{4}{3}

as the slope of perpendicular line

m

, using the point

(4,3)

to form line

m

's equation

y = -\frac{4}{3}x + \frac{25}{3}

, identifying intercepts

Q\left(\frac{25}{4}, 0\right)

and

R\left(0, \frac{25}{3}\right)

, and applying the distance formula to obtain

\sqrt{\left(\frac{25}{4}\right)^2 + \left(\frac{25}{3}\right)^2} = \frac{125}{12}

.

Adım Adım Çözüm

1

Determine the slope of line

k

The slope of line

k

is

m_k = \frac{3 - 0}{4 - 0} = \frac{3}{4}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Find the equation of line

m

Line

m

has slope

m_m = -\frac{4}{3}

and equation

y - 3 = -\frac{4}{3}(x - 4) \implies y = -\frac{4}{3}x + \frac{25}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other (

m_m = -\frac{1}{m_k}

). Point-slope form is used with point

(4, 3)

.

3

Calculate the intercepts

Q

and

R

x

-intercept

Q = \left(\frac{25}{4}, 0\right)

and

y

-intercept

R = \left(0, \frac{25}{3}\right)

.

Setting

y = 0

in

y = -\frac{4}{3}x + \frac{25}{3}

gives

x = \frac{25}{4}

. Setting

x = 0

gives

y = \frac{25}{3}

.

4

Compute the distance between

Q

and

R

Distance

d = \sqrt{\left(\frac{25}{4}\right)^2 + \left(\frac{25}{3}\right)^2} = 25 \sqrt{\frac{1}{16} + \frac{1}{9}} = 25 \sqrt{\frac{25}{144}} = 25 \times \frac{5}{12} = \frac{125}{12}

.

Applying the distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

for points

Q

and

R

.

Anahtar Kavram

Perpendicular slopes and distance formula in coordinate geometry

Tahmini Süre:2m 0s

Soru 1590Soru →

Dataset $W$ consists of $101$ distinct real numbers arranged in strictly increasing order: $x_1 < x_2 < \dots < x_{101}$ . The interquartile range of $W$ is $Q$ , its range is $R$ , and its standard deviation is $\sigma$ . A new dataset $W'$ is constructed by adding a positive constant $k$ to each of the highest $25$ observations (that is, $x_{77}, x_{78}, \dots, x_{101}$ become $x_{77}+k, x_{78}+k, \dots, x_{101}+k$ ), while the first $76$ observations remain unchanged. Which of the following statements MUST be true regarding the statistics of $W'$ compared to dataset $W$ ?

The interquartile range of

W'

is equal to

Q + \frac{k}{2}

.

The interquartile range of

W'

is equal to

Q + k

.

The interquartile range of

W'

is equal to

Q

.

The standard deviation of

W'

is equal to

\sigma

.

The median of

W'

is equal to the median of

W

plus

\frac{k}{2}

.

Cevabı ve açıklamayı göster

Cevap: The interquartile range of $W'$ is equal to $Q + \frac{k}{2}$ .

Cevap

The statement specifying that the interquartile range of

W'

is equal to

Q + \frac{k}{2}

must be true.

For an ordered dataset of 101 elements, the first quartile

Q_1

is the median of the lower 50 values (

\frac{x_{25} + x_{26}}{2}

) and the third quartile

Q_3

is the median of the upper 50 values (

\frac{x_{76} + x_{77}}{2}

). Since elements

x_1

through

x_{76}

are unchanged,

Q_1

remains identical. However,

x_{77}

increases by

k

, causing

Q_3

to increase by

\frac{k}{2}

. Consequently, the new interquartile range is

Q' = Q_3' - Q_1' = Q + \frac{k}{2}

.

Adım Adım Çözüm

1

Determine the position of quartiles for a 101-element dataset.

For

101

sorted values, the median is

x_{51}

. The lower half consists of

50

values (

x_1

through

x_{50}

), whose median

Q_1

is

\frac{x_{25} + x_{26}}{2}

. The upper half consists of

50

values (

x_{52}

through

x_{101}

), whose median

Q_3

is

\frac{x_{76} + x_{77}}{2}

.

Establishing the exact index formulas for

Q_1

and

Q_3

is required to evaluate changes in IQR.

2

Analyze how

Q_1

changes in dataset

W'

.

x_{25}

and

x_{26}

are among the first

76

observations, so they are unchanged. Thus,

Q_1' = Q_1

.

Since the modified values start at index 77, the first quartile remains completely unaffected.

3

Analyze how

Q_3

changes in dataset

W'

.

In

W'

,

x_{76}' = x_{76}

and

x_{77}' = x_{77} + k

. Therefore,

Q_3' = \frac{x_{76} + (x_{77} + k)}{2} = \frac{x_{76} + x_{77}}{2} + \frac{k}{2} = Q_3 + \frac{k}{2}

.

One of the two values used to calculate

Q_3

is shifted by

k

, so their average increases by

\frac{k}{2}

.

4

Compute the new interquartile range

Q'

.

Q' = Q_3' - Q_1' = \left(Q_3 + \frac{k}{2}\right) - Q_1 = (Q_3 - Q_1) + \frac{k}{2} = Q + \frac{k}{2}

.

Subtracting

Q_1'

from

Q_3'

yields the new interquartile range directly.

Anahtar Kavram

Quartile Interpolation and Selective Data Transformations

Soru 1591Soru →

The quadratic equation $x^2 - px + q = 0$ , where $p$ and $q$ are real constants with $p > 0$ , has two real roots $r_1$ and $r_2$ . If $r_1^2 + r_2^2 = 29$ and $r_1^3 + r_2^3 = 117$ , what is the value of $q$ ?

Cevabı ve açıklamayı göster

Cevap: -10

Cevap

The value of

q

is

-10

.

By Vieta's formulas,

r_1 + r_2 = p

and

r_1 r_2 = q

. Using the identity

r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2 = 29

, we get

q = \frac{p^2 - 29}{2}

. Using the sum of cubes identity

r_1^3 + r_2^3 = (r_1 + r_2)(r_1^2 - r_1 r_2 + r_2^2) = 117

, we obtain

p(29 - q) = 117

. Substituting

q

into this equation yields

p^3 - 87p + 234 = 0

. Factoring this cubic equation yields

p = 3

as a valid root, leading to

q = -10

. Evaluating the discriminant

p^2 - 4q = 9 - 4(-10) = 49 > 0

confirms that real roots exist. The second positive root for

p

gives a negative discriminant, making

-10

the unique correct value of

q

.

Adım Adım Çözüm

1

Express symmetric sums of roots in terms of equation coefficients

p

and

q

using Vieta's formulas.

r_1 + r_2 = p

and

r_1 r_2 = q

. Therefore,

r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2 = p^2 - 2q = 29

, which gives

q = \frac{p^2 - 29}{2}

.

Relating root sums and products directly to coefficients simplifies the algebraic system.

2

Express the sum of cubes in terms of

p

and

q

and substitute

q

.

r_1^3 + r_2^3 = (r_1 + r_2)(r_1^2 - r_1 r_2 + r_2^2) = p(29 - q) = 117

. Substituting

q = \frac{p^2 - 29}{2}

yields

p\left(29 - \frac{p^2 - 29}{2}\right) = 117 \implies p(87 - p^2) = 234 \implies p^3 - 87p + 234 = 0

.

Combining symmetric expressions forms a single polynomial equation in variable

p

.

3

Factor the cubic equation

p^3 - 87p + 234 = 0

to find candidate values for

p > 0

.

Testing factor

p = 3

gives

3^3 - 87(3) + 234 = 27 - 261 + 234 = 0

. Factoring out

(p - 3)

gives

(p - 3)(p^2 + 3p - 78) = 0

.

Finding the roots of the polynomial identifies candidate values for coefficient

p

.

4

Verify the real root constraint

p^2 - 4q > 0

for candidates of

p

.

For

p = 3

,

q = \frac{3^2 - 29}{2} = -10

, giving discriminant

p^2 - 4q = 9 - 4(-10) = 49 > 0

(real roots

5

and

-2

). For the positive root

p = \frac{\sqrt{321}-3}{2} \approx 7.458

of

p^2 + 3p - 78 = 0

, the discriminant is

p^2 - 4q = 3p - 29 \approx -6.63 < 0

, which yields non-real complex roots.

The question specifies real roots, so candidate solutions resulting in negative discriminants must be discarded.

Anahtar Kavram

Quadratic Equations and Factoring (Vieta's Formulas, Symmetric Polynomials, and Real Root Conditions)

Soru 1592Soru →

An altitude of an acute triangle divides its base into two adjacent segments of lengths $5$ and $9$ . If the area of the triangle is $84$ square units, what is the perimeter of the triangle?

28

36

42

48

52

Cevabı ve açıklamayı göster

Cevap: $42$

Cevap

The perimeter of the triangle is

42

.

The base of the triangle is the sum of the two adjacent segments,

5 + 9 = 14

. Setting up the area formula gives

84 = \frac{1}{2} \times 14 \times h

, which yields an altitude height of

h = 12

. The altitude creates two right triangles: one with legs

5

and

12

(hypotenuse

\sqrt{5^2 + 12^2} = 13

) and another with legs

9

and

12

(hypotenuse

\sqrt{9^2 + 12^2} = 15

). Adding all three side lengths (

14 + 13 + 15

) yields

42

.

Adım Adım Çözüm

1

Find the total base length and calculate the height (altitude) of the triangle.

Base

= 5 + 9 = 14

. Height

h = \frac{2 \times \text{Area}}{\text{base}} = \frac{2 \times 84}{14} = 12

.

The area of a triangle is given by

A = \frac{1}{2} b h

.

2

Calculate the lengths of the two non-base sides using the Pythagorean theorem on the two right triangles formed by the altitude.

Left side

= \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = 13

. Right side

= \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = 15

.

An altitude drawn to the base creates two right triangles sharing the altitude as a common leg.

3

Sum all three boundary sides to find the perimeter.

Perimeter

= 14 + 13 + 15 = 42

.

Perimeter is the total length around the outside of the triangle.

Anahtar Kavram

Area and perimeter of triangles split by an altitude using the Pythagorean theorem.

Tahmini Süre:1m 30s

Soru 1593Soru →

The running times, in minutes, for a fleet of delivery vehicles have a 25th percentile of $134$ minutes and a 75th percentile of $242$ minutes. To analyze the data in hours, a manager converts each vehicle's running time to hours by dividing by $60$ and then adds a fixed delay of $1.25$ hours to every converted value. What is the interquartile range, in hours, of the adjusted running times?

Cevabı ve açıklamayı göster

Cevap: 1.8

Cevap

The interquartile range of the adjusted running times is 1.8 hours.

The interquartile range (IQR) measures the spread of the middle 50% of a dataset (

Q_3 - Q_1

). For the original dataset in minutes,

IQR = 242 - 134 = 108

minutes. When data is transformed linearly via

Y = aX + b

, scaling all values by a positive constant

a

scales the IQR by

a

, whereas adding a constant

b

shifts all values equally and does not change the distance between percentiles. Here

a = \frac{1}{60}

and

b = 1.25

. Therefore, the new IQR in hours is

108 \times \frac{1}{60} = 1.8

hours.

Adım Adım Çözüm

1

Find the original interquartile range (IQR) in minutes

IQR = 242 - 134 = 108 minutes

The interquartile range is defined as the difference between the 75th percentile (Q3) and the 25th percentile (Q1).

2

Apply the scaling factor for unit conversion

108 / 60 = 1.8 hours

Dividing every value by 60 scales the data by a factor of 1/60, which scales measures of dispersion like IQR by |1/60|.

3

Evaluate the effect of adding the constant shift

Adjusted IQR remains 1.8 hours

Adding a constant shift to every data point shifts the entire distribution without changing the spread or distance between percentiles.

Anahtar Kavram

Linear Transformations on Measures of Dispersion

Soru 1594Soru →

Three adjacent faces of a right rectangular prism have surface areas of $24$ , $36$ , and $54$ square centimeters. What is the volume of the prism, in cubic centimeters?

Cevabı ve açıklamayı göster

Cevap: 216

Cevap

The volume of the rectangular prism is

216

cubic centimeters.

Let the edge lengths of the right rectangular prism be

l

,

w

, and

h

. The areas of three adjacent faces meeting at a common vertex are given by

lw = 24

,

wh = 36

, and

lh = 54

. Multiplying these three equations together yields

(lw)(wh)(lh) = (lwh)^2 = 24 \times 36 \times 54 = 46,656

. Since the volume

V

of a right rectangular prism is equal to

lwh

, taking the positive square root gives

V = \sqrt{46,656} = 216

cubic centimeters.

Adım Adım Çözüm

1

Define variables for the three dimensions of the rectangular prism.

Let

l

,

w

, and

h

represent the length, width, and height of the prism.

The area of each rectangular face is the product of two adjacent dimensions.

2

Express the given face areas in terms of the dimensions.

lw = 24

,

wh = 36

, and

lh = 54

.

Adjacent faces share pairwise dimensions.

3

Multiply all three face area equations together.

(lw)(wh)(lh) = l^2 w^2 h^2 = (lwh)^2 = 24 \times 36 \times 54 = 46,656

.

Multiplying the three face areas yields the square of the prism's volume (

V = lwh

).

4

Calculate the square root of the product to determine the volume.

V = lwh = \sqrt{46,656} = 216

.

Taking the square root gives the volume directly without needing to solve for individual edge lengths.

Anahtar Kavram

For a right rectangular prism with adjacent face areas

A_1

,

A_2

, and

A_3

, the volume is given by

V = \sqrt{A_1 \times A_2 \times A_3}

.

Tahmini Süre:1m 15s

Soru 1595Soru →

Group A consists of $15$ test scores with an arithmetic mean of $80$ and a median of $82$ . Group B consists of $25$ test scores with an arithmetic mean of $90$ and a median of $86$ . The two groups are combined into a single dataset of $40$ test scores. Which of the following statements MUST be true about the combined dataset? Indicate all such statements.

Geçerli olan tümünü seçin

The mean of the combined dataset is equal to

86.25

.

The median of the combined dataset is at least

82

.

The median of the combined dataset is at most

86

.

The median of the combined dataset is equal to

84

.

The mode of the combined dataset must lie between

80

and

90

.

Cevabı ve açıklamayı göster

Cevap: The mean of the combined dataset is equal to $86.25$ .; The median of the combined dataset is at least $82$ .; The median of the combined dataset is at most $86$ .

Cevap

The statements asserting that the combined mean is

86.25

, that the combined median is at least

82

, and that the combined median is at most

86

must all be true.

The combined mean is calculated directly from the total sum divided by total elements, giving exactly

86.25

. For the median of

40

combined values (the average of the

20\text{th}

and

21\text{st}

values in sorted order), counting how many values are guaranteed to be above or below specific thresholds shows that at least

21

values are

\ge 82

and at least

21

values are

\le 86

. Therefore, the median must lie within

[82, 86]

.

Adım Adım Çözüm

1

Calculate the weighted arithmetic mean for the combined dataset.

Combined Mean =

\frac{15 \times 80 + 25 \times 90}{15 + 25} = \frac{1200 + 2250}{40} = \frac{3450}{40} = 86.25

.

The mean of combined sets is the total sum of all values divided by the total number of items.

2

Analyze the lower bound for the median of the combined set.

At least

21

values in the combined set of

40

items are greater than or equal to

82

.

Since the median of Group A (

15

items) is

82

, at least

8

items are

\ge 82

. Since the median of Group B (

25

items) is

86

, at least

13

items are

\ge 86 \ge 82

. In total,

8 + 13 = 21

items are

\ge 82

. In a sorted list of

40

items, having at least

21

items

\ge 82

implies that the

20\text{th}

and

21\text{st}

items must be

\ge 82

, so the median

\ge 82

.

3

Analyze the upper bound for the median of the combined set.

At least

21

values in the combined set of

40

items are less than or equal to

86

.

Group A has at least

8

items

\le 82 \le 86

. Group B has at least

13

items

\le 86

. Thus, at least

8 + 13 = 21

items are

\le 86

. In a sorted list of

40

items, having at least

21

items

\le 86

ensures the

20\text{th}

and

21\text{st}

items are

\le 86

, so the median

\le 86

.

4

Evaluate the remaining statements regarding the exact median and mode.

Neither the exact median value of

84

nor the existence/location of a mode is guaranteed.

Medians do not combine linearly like means, and mode depends entirely on specific frequency counts which are not given.

Anahtar Kavram

Weighted averages and order statistic bounds for combined datasets

Soru 1596Soru →

For all real numbers $x$ , the function $f$ is defined by $f(x) = x^2 - 4x + 7$ and the function $g$ is defined by $g(x) = 2x + 1$ . If $f(g(k)) = 19$ and $k < 0$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: -1.5

Cevap

The value of

k

is

-1.5

.

To solve for

k

, substitute

g(k) = 2k + 1

into

f(x)

to get

f(g(k)) = (2k + 1)^2 - 4(2k + 1) + 7

. Expanding and combining like terms yields

4k^2 - 4k + 4

. Setting this equal to

19

gives

4k^2 - 4k - 15 = 0

. Factoring the quadratic expression yields

(2k - 5)(2k + 3) = 0

, which gives

k = \frac{5}{2} = 2.5

or

k = -\frac{3}{2} = -1.5

. Since the problem specifies that

k < 0

, the value of

k

must be

-1.5

.

Adım Adım Çözüm

1

Express the nested composite function

f(g(k))

in terms of

k

f(g(k)) = (2k + 1)^2 - 4(2k + 1) + 7

Substituting

g(k) = 2k + 1

for every occurrence of

x

in

f(x)

sets up the composite function evaluation.

2

Expand terms and simplify the algebraic expression

f(g(k)) = 4k^2 - 4k + 4

Expanding

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

and distributing

-4(2k + 1) = -8k - 4

, then combining like terms yields

4k^2 - 4k + 4

.

3

Set the composite function expression equal to 19 and form a standard quadratic equation

4k^2 - 4k - 15 = 0

Subtracting

19

from both sides produces a standard quadratic equation

ax^2 + bx + c = 0

.

4

Solve the quadratic equation and apply the condition

k < 0

k = -1.5

Factoring

(2k - 5)(2k + 3) = 0

gives

k = 2.5

or

k = -1.5

. The constraint

k < 0

selects

k = -1.5

.

Anahtar Kavram

Composite function evaluation

f(g(x))

combined with quadratic equation solving under domain constraints.

Soru 1597Soru →

A total of 120 software engineers at a firm were surveyed about their attendance at three professional development workshops: Machine Learning ( $M$ ), Cloud Computing ( $C$ ), and Cybersecurity ( $S$ ). The survey results revealed the following:

- 65 engineers attended Machine Learning
- 55 engineers attended Cloud Computing
- 48 engineers attended Cybersecurity
- 25 engineers attended both Machine Learning and Cloud Computing
- 18 engineers attended both Cloud Computing and Cybersecurity
- 20 engineers attended both Machine Learning and Cybersecurity
- 10 engineers attended all three workshops

How many of the surveyed software engineers attended exactly one of the three workshops?

42

48

72

105

115

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

72 engineers attended exactly one of the three workshops.

To find the number of engineers attending exactly one workshop, calculate the exclusive single-set counts for each category using a 3-set Venn diagram model. Subtracting the overlapping regions gives 30 for Machine Learning only, 22 for Cloud Computing only, and 20 for Cybersecurity only. Summing these exclusive regions yields

30 + 22 + 20 = 72

. Alternatively, using the algebraic formula

|\text{Exactly 1}| = (|M| + |C| + |S|) - 2(|M \cap C| + |C \cap S| + |M \cap S|) + 3|M \cap C \cap S| = (65 + 55 + 48) - 2(25 + 18 + 20) + 3(10) = 168 - 126 + 30 = 72

.

Adım Adım Çözüm

1

Find the number of engineers in the regions corresponding to exactly two workshops.

Machine Learning and Cloud Computing only =

25 - 10 = 15

; Cloud Computing and Cybersecurity only =

18 - 10 = 8

; Machine Learning and Cybersecurity only =

20 - 10 = 10

.

The given pairwise intersection values include engineers who attended all three workshops.

2

Calculate the number of engineers who attended only one specific workshop.

Machine Learning only =

65 - (15 + 10 + 10) = 30

; Cloud Computing only =

55 - (15 + 8 + 10) = 22

; Cybersecurity only =

48 - (10 + 8 + 10) = 20

.

Subtract all overlapping regions (two-workshop-only and three-workshop) from each set's total.

3

Sum the single-workshop regions to find the total attending exactly one workshop.

30 + 22 + 20 = 72

.

The subsets representing single workshop attendance are disjoint.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets and Venn Diagram Region Partitioning

Tahmini Süre:2m 0s

Soru 1598Soru →

An industrial facility has three supply pipes—Pipe A, Pipe B, and Pipe C—that can fill a storage reservoir. Pipe A operating alone can fill the empty reservoir in $a$ hours. Pipe B operating alone takes $50\%$ longer than Pipe A to fill the empty reservoir. Pipe C operating alone fills the empty reservoir at a rate equal to the combined filling rate of Pipe A and Pipe B.

Initially, the reservoir is empty. Pipe A and Pipe B are opened simultaneously. After $2$ hours, Pipe B is closed and Pipe C is opened, while Pipe A remains open. The reservoir becomes completely full exactly $4$ hours after Pipe A and Pipe B were initially opened.

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

Geçerli olan tümünü seçin

Pipe A operating alone would fill the empty reservoir in

8

hours and

40

minutes.

During the first

2

hours of the process, exactly

\frac{5}{13}

of the total capacity of the reservoir is filled.

Pipe C operating alone would fill the empty reservoir in

5

hours and

12

minutes.

Pipe B operating alone would fill the empty reservoir in

11

hours and

30

minutes.

If all three pipes were opened simultaneously from the beginning, the empty reservoir would be filled in

3

hours and

15

minutes.

Cevabı ve açıklamayı göster

Cevap: Pipe A operating alone would fill the empty reservoir in $8$ hours and $40$ minutes.; During the first $2$ hours of the process, exactly $\frac{5}{13}$ of the total capacity of the reservoir is filled.; Pipe C operating alone would fill the empty reservoir in $5$ hours and $12$ minutes.

Cevap

The statements asserting that Pipe A operating alone takes 8 hours and 40 minutes, that 5/13 of the capacity is filled in the first 2 hours, and that Pipe C operating alone takes 5 hours and 12 minutes are all true.

Using the rate relationships

R_A = \frac{1}{a}

,

R_B = \frac{2}{3a}

, and

R_C = \frac{5}{3a}

, the equation

2(R_A + R_B) + 2(R_A + R_C) = 1

simplifies to

\frac{26}{3a} = 1

, giving

a = \frac{26}{3}

hours. This verifies that Pipe A takes 8 hours 40 minutes alone, Phase 1 fills 5/13 of the reservoir capacity, and Pipe C takes 5.2 hours (5 hours 12 minutes) alone.

Adım Adım Çözüm

1

Express the individual work rates of Pipe A, Pipe B, and Pipe C in terms of parameter

a

.

R_A = \frac{1}{a}

,

R_B = \frac{1}{1.5a} = \frac{2}{3a}

, and

R_C = R_A + R_B = \frac{1}{a} + \frac{2}{3a} = \frac{5}{3a}

.

Work rate is defined as the reciprocal of the time required to complete one unit of work.

2

Formulate an equation for total work completed over the two 2-hour phases.

2(R_A + R_B) + 2(R_A + R_C) = 1 \implies 2\left(\frac{5}{3a}\right) + 2\left(\frac{8}{3a}\right) = 1 \implies \frac{10}{3a} + \frac{16}{3a} = 1

.

Pipes A and B operate for the first 2 hours, followed by Pipes A and C operating for the next 2 hours to complete 1 full reservoir.

3

Solve the work equation for

a

and determine Pipe A's solo time.

\frac{26}{3a} = 1 \implies a = \frac{26}{3} = 8\frac{2}{3} \text{ hours} = 8 \text{ hours } 40 \text{ minutes}.

Fractional hours are converted to minutes by multiplying

\frac{2}{3}

by

60

.

4

Evaluate the volume filled in Phase 1 and the solo filling time for Pipe C.

\text{Phase 1 volume} = \frac{10}{3(26/3)} = \frac{5}{13}. \text{ Pipe C solo time} = \frac{1}{R_C} = \frac{3a}{5} = \frac{26}{5} = 5.2 \text{ hours} = 5 \text{ hours } 12 \text{ minutes}.

Substituting

a = \frac{26}{3}

into the respective rate expressions yields the exact time and volume parameters.

Anahtar Kavram

Formulating combined rate models and solving sequential multi-stage work problems.

Soru 1599Soru →

A health club surveyed $100$ members regarding their use of cardio machines and weight equipment. Of the members surveyed, $60$ use cardio machines, $50$ use weight equipment, and $15$ use neither cardio machines nor weight equipment. Which of the following statements must be true? Indicate all such statements.

Geçerli olan tümünü seçin

Exactly

25

members use both cardio machines and weight equipment.

Exactly

35

members use cardio machines but do not use weight equipment.

Exactly

10

members use both cardio machines and weight equipment.

Exactly

110

members use at least one of the two types of equipment.

Exactly

85

members use at least one of the two types of equipment.

Cevabı ve açıklamayı göster

Cevap: Exactly $25$ members use both cardio machines and weight equipment.; Exactly $35$ members use cardio machines but do not use weight equipment.; Exactly $85$ members use at least one of the two types of equipment.

Cevap

The true statements are those stating that exactly 25 members use both types of equipment, exactly 35 members use cardio machines only, and exactly 85 members use at least one type of equipment.

The total surveyed population of

100

consists of members using at least one type of equipment and members using neither (

15

). Thus,

100 - 15 = 85

members use at least one type of equipment. Using the inclusion-exclusion formula

\text{Union} = \text{Cardio} + \text{Weights} - \text{Intersection}

, we have

85 = 60 + 50 - \text{Intersection}

, giving an intersection of

25

members using both types of equipment. Subtracting

25

from the

60

total cardio users shows that

35

members use cardio machines exclusively.

Adım Adım Çözüm

1

Calculate the total number of members using at least one type of equipment.

Total using at least one type

= 100 - 15 = 85

.

Members using at least one type of equipment represent the total surveyed population minus those who use neither.

2

Calculate the number of members using both cardio machines and weight equipment using the principle of inclusion-exclusion.

\text{Both} = \text{Cardio} + \text{Weights} - \text{At least one} = 60 + 50 - 85 = 25

.

Adding the individual set counts double-counts the overlapping region, so subtracting the total union gives the intersection size.

3

Calculate the number of members using cardio machines only.

\text{Cardio Only} = \text{Cardio} - \text{Both} = 60 - 25 = 35

.

Subtracting the intersection from the total cardio group yields the exclusive cardio users.

Anahtar Kavram

Principle of Inclusion-Exclusion for Two Sets

Tahmini Süre:1m 15s

Soru 1600Soru →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . Point $D$ lies on segment $AC$ such that $AD = 11$ and $BD = 13$ . If $DC = 5$ , what is the perimeter of triangle $ABD$ ?

Cevabı ve açıklamayı göster

Cevap: 44

Cevap

The perimeter of triangle

ABD

is

44

.

To find the perimeter of triangle

ABD

, we need the lengths of its three sides:

AD

,

BD

, and

AB

. We are given

AD = 11

and

BD = 13

. To find

AB

, we first analyze right triangle

BCD

with right angle at

C

, hypotenuse

BD = 13

, and leg

DC = 5

. Using the Pythagorean theorem, leg

BC = \sqrt{13^2 - 5^2} = 12

. Next, the full leg

AC = AD + DC = 11 + 5 = 16

. In right triangle

ABC

, legs are

AC = 16

and

BC = 12

, making hypotenuse

AB = \sqrt{16^2 + 12^2} = 20

. Finally, the perimeter of triangle

ABD

is

11 + 13 + 20 = 44

.

Adım Adım Çözüm

1

Find the length of side

BC

using right triangle

BCD

BC = 12

In right triangle

BCD

, angle

C = 90^\circ

, hypotenuse

BD = 13

, and leg

DC = 5

. By the Pythagorean theorem,

BC = \sqrt{13^2 - 5^2} = \sqrt{144} = 12

.

2

Determine the total length of side

AC

AC = 16

Since point

D

lies on segment

AC

,

AC = AD + DC = 11 + 5 = 16

.

3

Calculate the hypotenuse

AB

of the main right triangle

ABC

AB = 20

In right triangle

ABC

, legs are

AC = 16

and

BC = 12

. By the Pythagorean theorem,

AB = \sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20

.

4

Calculate the perimeter of triangle

ABD

Perimeter

= 44

The sides of triangle

ABD

are

AD = 11

,

BD = 13

, and

AB = 20

. Adding these gives

11 + 13 + 20 = 44

.

Anahtar Kavram

Pythagorean Theorem and Multi-Step Triangle Properties

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