Tüm alıştırma soruları

Soru 1541Soru →

In isosceles triangle $ABC$ , side $AB$ is equal in length to side $AC$ . The perimeter of triangle $ABC$ is $36$ , and the length of the altitude from vertex $A$ to base $BC$ is $12$ . What is the area of triangle $ABC$ ?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The area of triangle

ABC

is

60

.

Let

x

be the length of the two equal sides

AB

and

AC

, and

b

be the length of base

BC

. The perimeter is

2x + b = 36

, yielding

x = 18 - \frac{b}{2}

. The altitude from

A

to

BC

has length

12

and bisects

BC

into two segments of length

\frac{b}{2}

. Applying the Pythagorean theorem to one of the right triangles gives

x^2 = 12^2 + \left(\frac{b}{2}\right)^2

. Substituting

x = 18 - \frac{b}{2}

gives

\left(18 - \frac{b}{2}\right)^2 = 144 + \frac{b^2}{4} \implies 324 - 18b + \frac{b^2}{4} = 144 + \frac{b^2}{4} \implies 18b = 180 \implies b = 10

. The area is

\frac{1}{2} \times 10 \times 12 = 60

.

Adım Adım Çözüm

1

Set up an equation for the side lengths using the perimeter.

Let

b

be the length of base

BC

, and

x

be the length of sides

AB

and

AC

. Since the perimeter is

36

,

2x + b = 36

, which gives

x = 18 - \frac{b}{2}

.

An isosceles triangle has two sides of equal length, and perimeter is the sum of all three side lengths.

2

Apply the Pythagorean theorem to the right triangle formed by the altitude.

The altitude of length

12

drops perpendicularly to base

BC

, bisecting it into two equal segments of length

\frac{b}{2}

. Thus,

x^2 = 12^2 + \left(\frac{b}{2}\right)^2 = 144 + \frac{b^2}{4}

.

In an isosceles triangle, the altitude to the base bisects the base and creates two congruent right-angled triangles.

3

Solve for the base length

b

.

Substitute

x = 18 - \frac{b}{2}

into the equation:

\left(18 - \frac{b}{2}\right)^2 = 144 + \frac{b^2}{4} \implies 324 - 18b + \frac{b^2}{4} = 144 + \frac{b^2}{4} \implies 18b = 180 \implies b = 10

.

Expanding the squared binomial allows the

\frac{b^2}{4}

terms to cancel out, resulting in a linear equation for

b

.

4

Calculate the area of the triangle.

\text{Area} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 10 \times 12 = 60.

The area of a triangle is evaluated using half the product of its base and corresponding altitude.

Anahtar Kavram

Isosceles triangle properties, altitude-to-base bisector property, Pythagorean theorem, and triangle area calculation.

Soru 1542Soru →

Dataset $S$ consists of 11 distinct positive integers arranged in increasing order, with a median of 50, an interquartile range of 20, and a standard deviation of $\sigma$ . A new dataset $S'$ is created by adding 10 to each of the 5 integers in $S$ that are strictly greater than 50, while leaving the remaining 6 integers unchanged. Which of the following statements about dataset $S'$ must be true? Select all such statements.

Geçerli olan tümünü seçin

The interquartile range of dataset

S'

is 30.

The range of dataset

S'

is 10 greater than the range of dataset

S

.

The standard deviation of dataset

S'

is strictly greater than

\sigma

.

The median of dataset

S'

is 60.

The interquartile range of dataset

S'

remains 20.

Cevabı ve açıklamayı göster

Cevap: The interquartile range of dataset $S'$ is 30.; The range of dataset $S'$ is 10 greater than the range of dataset $S$ .; The standard deviation of dataset $S'$ is strictly greater than $\sigma$ .

Cevap

The statements asserting that the interquartile range of dataset

S'

is 30, that the range of dataset

S'

is 10 greater than the range of dataset

S

, and that the standard deviation of dataset

S'

is strictly greater than

\sigma

are all correct.

In an ordered dataset of 11 elements, the median is the 6th element,

Q_1

is the 3rd element, and

Q_3

is the 9th element. When 10 is added only to the elements strictly above the median (elements 7 through 11):

Q_3

increases by 10 while

Q_1

is unchanged, making the new IQR equal to

20 + 10 = 30

. The maximum value increases by 10 while the minimum value remains unchanged, increasing the range by 10. Shifting values in the upper tail further right increases the overall distance of data points from the mean, causing the standard deviation to strictly increase.

Adım Adım Çözüm

1

Analyze the position of quartiles and median in a dataset of 11 ordered values.

For 11 ordered values

x_1 < x_2 < \dots < x_{11}

, the median is

x_6 = 50

,

Q_1 = x_3

, and

Q_3 = x_9

. The lower 6 elements (

x_1

through

x_6

) are unchanged. The upper 5 elements (

x_7

through

x_{11}

) each increase by 10.

Determining which specific data positions change allows us to evaluate median, IQR, and range.

2

Calculate the new interquartile range and range.

Q_{1,\text{new}} = x_3

,

Q_{3,\text{new}} = x_9 + 10

. Thus

\text{IQR}_{\text{new}} = (x_9 + 10) - x_3 = \text{IQR}_{\text{old}} + 10 = 20 + 10 = 30

. The maximum element

x_{11}

increases by 10 while

x_1

is unchanged, so

\text{Range}_{\text{new}} = (x_{11} + 10) - x_1 = \text{Range}_{\text{old}} + 10

.

Interquartile range is

Q_3 - Q_1

and range is

\text{maximum} - \text{minimum}

.

3

Evaluate the effect on the median and standard deviation.

The median remains

x_6 = 50

. Moving the upper values further away from the center increases the overall spread around the mean, which strictly increases the standard deviation beyond

\sigma

.

Standard deviation measures the average spread of values from the mean.

Anahtar Kavram

Effect of asymmetric data shifts on measures of central tendency and dispersion

Soru 1543Soru →

In the coordinate plane, segment $AB$ lies along the positive $x$ -axis with point $A$ at the origin $(0,0)$ and point $B$ at $(6\sqrt{3}, 0)$ . Point $C$ is located in the first quadrant such that $\triangle ABC$ is a right triangle with $\angle ACB = 90^\circ$ and $\angle CAB = 30^\circ$ . Point $D$ is also located in the first quadrant such that $\triangle ABD$ is an isosceles right triangle with hypotenuse $AB$ and $\angle ADB = 90^\circ$ .

Which of the following statements regarding this figure are true? Select all that apply.

Geçerli olan tümünü seçin

The length of segment

AC

is

9

.

The length of segment

AD

is

3\sqrt{6}

.

The area of triangle

ABD

is

27

.

The length of segment

BC

is

9

.

The length of segment

AD

is

6\sqrt{6}

.

Cevabı ve açıklamayı göster

Cevap: The length of segment $AC$ is $9$ .; The length of segment $AD$ is $3\sqrt{6}$ .; The area of triangle $ABD$ is $27$ .

Cevap

The correct statements are: the length of segment

AC

is

9

, the length of segment

AD

is

3\sqrt{6}

, and the area of triangle

ABD

is

27

.

The statement that

AC = 9

is correct because in

\triangle ABC

,

AC = AB \cos(30^\circ) = 6\sqrt{3} \cdot \frac{\sqrt{3}}{2} = 9

. The statement that

AD = 3\sqrt{6}

is correct because in isosceles right

\triangle ABD

,

AD = \frac{AB}{\sqrt{2}} = \frac{6\sqrt{3}}{\sqrt{2}} = 3\sqrt{6}

. The statement that the area of

\triangle ABD

is

27

is correct because

\frac{1}{2}(3\sqrt{6})(3\sqrt{6}) = 27

.

Adım Adım Çözüm

1

Determine the side lengths of 30-60-90 triangle ABC

Hypotenuse

AB = 6\sqrt{3}

. Leg

BC

opposite

30^\circ

is

\frac{1}{2}(6\sqrt{3}) = 3\sqrt{3}

. Leg

AC

opposite

60^\circ

is

3\sqrt{3} \cdot \sqrt{3} = 9

.

In a

30^\circ-60^\circ-90^\circ

triangle, side ratios are

1 : \sqrt{3} : 2

relative to angles

30^\circ : 60^\circ : 90^\circ

.

2

Determine the side lengths of 45-45-90 triangle ABD

Hypotenuse

AB = 6\sqrt{3}

. Legs

AD = BD = \frac{6\sqrt{3}}{\sqrt{2}} = 3\sqrt{6}

.

In a

45^\circ-45^\circ-90^\circ

isosceles right triangle, side ratios are

1 : 1 : \sqrt{2}

, so leg length equals hypotenuse divided by

\sqrt{2}

.

3

Calculate the area of right triangle ABD

Area

(\triangle ABD) = \frac{1}{2} \cdot AD \cdot BD = \frac{1}{2} (3\sqrt{6})(3\sqrt{6}) = \frac{1}{2} (54) = 27

.

The area of a right triangle is half the product of its perpendicular legs.

4

Evaluate each given option against computed values

Segment

AC = 9

is true. Segment

AD = 3\sqrt{6}

is true. Area of

\triangle ABD = 27

is true. Segment

BC = 9

is false (

BC = 3\sqrt{3}

). Segment

AD = 6\sqrt{6}

is false (

AD = 3\sqrt{6}

).

Direct comparison with calculated geometric dimensions.

Anahtar Kavram

Side ratios of 30-60-90 (

1:\sqrt{3}:2

) and 45-45-90 (

1:1:\sqrt{2}

) special right triangles

Soru 1544Soru →

Dataset $X$ consists of $50$ distinct real numbers with range $R$ , interquartile range $Q$ , and standard deviation $s$ , where $R > Q > s > 0$ . A new dataset, Dataset $Y$ , is formed by applying the transformation $y = -3x - 7$ to each data point $x$ in Dataset $X$ . If $R_Y$ , $Q_Y$ , and $s_Y$ represent the range, interquartile range, and standard deviation of Dataset $Y$ , respectively, which of the following expressions represents the sum $R_Y + Q_Y + s_Y$ ?

3(R + Q + s)

3(R + Q + s) - 21

3(R + Q + s) - 7

-3(R + Q + s) - 7

-3(R + Q + s)

Cevabı ve açıklamayı göster

Cevap: $3(R + Q + s)$

Cevap

The sum of the dispersion metrics for Dataset

Y

is

3(R + Q + s)

.

Under a linear transformation

y = ax + b

, all measures of dispersion (range, interquartile range, standard deviation) scale by

|a|

and are unaffected by

b

. With

a = -3

and

b = -7

, each measure of dispersion is multiplied by

|-3| = 3

. Therefore,

R_Y = 3R

,

Q_Y = 3Q

, and

s_Y = 3s

, making their sum

3(R + Q + s)

.

Adım Adım Çözüm

1

Identify the effect of linear transformations on measures of dispersion.

For any linear transformation of data

y = ax + b

, measures of dispersion (Range, Interquartile Range, Standard Deviation) scale by the absolute value of the multiplicative constant,

|a|

, and are completely unaffected by the constant addition or subtraction

b

.

Measures of dispersion quantify spread and distances between data points, which shift uniformly when a constant is added but stretch by

|a|

when scaled.

2

Calculate individual dispersion measures for Dataset

Y

.

R_Y = |-3| R = 3R

,

Q_Y = |-3| Q = 3Q

, and

s_Y = |-3| s = 3s

.

The multiplicative factor is

a = -3

, so

|a| = |-3| = 3

. The constant shift

b = -7

has zero effect on spread.

3

Sum the three dispersion measures for Dataset

Y

.

R_Y + Q_Y + s_Y = 3R + 3Q + 3s = 3(R + Q + s)

.

Factoring out

3

yields the simplified combined expression.

Anahtar Kavram

Linear Transformations on Dispersion Metrics

Tahmini Süre:2m 0s

Soru 1545Soru →

Fill in each blank in the passage below with the word that best completes the text in a logically coherent manner.

Aşağıdaki boşlukları doldurun

Although the curator initially dismissed the artifact as a modern , a rigorous chemical analysis of the patina revealed trace elements unique to antiquity, thereby its historical authenticity.

Cevabı ve açıklamayı göster

Cevap

Blank 1: forgery (or fabrication/counterfeit); Blank 2: substantiating (or confirming/verifying)

The sentence relies on a concession-reversal structure signaled by 'Although'. The initial dismissal of an ancient artifact as a 'modern' object implies it was viewed as an inauthentic creation, making 'forgery' or 'fabrication' the logical fit for the first blank. The subsequent discovery of genuine ancient trace elements provides concrete evidence that refutes the initial skepticism, thereby 'substantiating' or 'confirming' the artifact's historical authenticity for the second blank.

Adım Adım Çözüm

1

Analyze structural contrast and directional clues in the sentence.

The opening transition 'Although' establishes a concession and contrast between the curator's initial reaction and the subsequent analytical findings.

The relationship between the initial view and the chemical analysis determines the polarity of both blanks.

2

Determine the meaning of the first blank based on context.

The curator 'dismissed' the object as a 'modern' attempt to imitate an ancient piece, indicating it was considered a false creation.

The word for Blank 1 must denote a fake or artificial creation, such as 'forgery' or 'fabrication'.

3

Determine the meaning of the second blank based on the evidence provided.

Finding trace elements 'unique to antiquity' refutes the initial dismissal and provides positive proof of genuine age.

The phrase 'thereby {{blank_2}} its historical authenticity' requires a participle meaning supporting or proving, such as 'substantiating' or 'confirming'.

Anahtar Kavram

Multi-Blank Dependency Tracking via Concessive and Resultant Signals

Tahmini Süre:1m 15s

Soru 1546Soru →

Dataset $X$ consists of $80$ numerical observations with a standard deviation of $s$ ( $s > 0$ ) and an interquartile range of $I$ ( $I > 0$ ). A new dataset, Dataset $Y$ , is created by transforming each observation $x$ in Dataset $X$ using the linear formula $y = -4x + 25$ . Which of the following correctly gives the standard deviation and the interquartile range of Dataset $Y$ in terms of $s$ and $I$ ?

Standard deviation:

4s

; Interquartile range:

4I

Standard deviation:

-4s + 25

; Interquartile range:

-4I + 25

Standard deviation:

4s + 25

; Interquartile range:

4I + 25

Standard deviation:

-4s

; Interquartile range:

-4I

Standard deviation:

16s

; Interquartile range:

16I

Cevabı ve açıklamayı göster

Cevap: Standard deviation: $4s$ ; Interquartile range: $4I$

Cevap

The standard deviation of Dataset

Y

is

4s

and the interquartile range is

4I

.

For any linear transformation of the form

y = ax + b

, the measures of dispersion (such as standard deviation, interquartile range, and range) scale by

|a|

and are unaffected by the additive constant

b

. Here

a = -4

and

b = 25

, so both standard deviation and IQR scale by

|-4| = 4

, resulting in

4s

and

4I

.

Adım Adım Çözüm

1

Analyze the impact of adding a constant to data values.

Adding

+25

to each data value shifts the position of the data points along the number line, but the relative distances between data points remain constant. Thus, constant addition has zero effect on standard deviation or interquartile range.

Measures of dispersion measure the spread of data around a central value, which is invariant under pure horizontal translations.

2

Analyze the impact of multiplying data values by a scalar factor.

Multiplying each value by

k = -4

expands the distances between points by a factor of

|k| = |-4| = 4

.

Standard deviation and IQR are defined as non-negative distance quantities; scaling data by

k

scales dispersion by

|k|

.

3

Combine the scale and shift transformations.

New Standard Deviation =

|-4| \times s = 4s

, and New IQR =

|-4| \times I = 4I

.

Applying the transformation

y = ax + b

transforms standard deviation

\sigma_y = |a|\sigma_x

and

\text{IQR}_y = |a|\text{IQR}_x

.

Anahtar Kavram

Effect of Linear Transformations on Measures of Dispersion

Soru 1547Soru →

If

x

and

y

are real numbers such that

x > y > 0

and they satisfy the following system of equations:

\frac{3}{x+y} + \frac{4}{x-y} = \frac{11}{4}

\frac{5}{x+y} - \frac{2}{x-y} = \frac{1}{4}

what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

x

is 3.

Substituting

u = \frac{1}{x+y}

and

v = \frac{1}{x-y}

transforms the non-linear looking equations into the linear system

3u + 4v = \frac{11}{4}

and

5u - 2v = \frac{1}{4}

. Solving this system yields

u = \frac{1}{4}

and

v = \frac{1}{2}

. Consequently,

x + y = 4

and

x - y = 2

. Adding these two equations gives

2x = 6

, so

x = 3

.

Adım Adım Çözüm

1

Introduce auxiliary variables to linearize the system.

Let

u = \frac{1}{x+y}

and

v = \frac{1}{x-y}

. The system becomes

3u + 4v = \frac{11}{4}

and

5u - 2v = \frac{1}{4}

.

Replacing non-linear reciprocal terms with simple variables allows elimination or substitution methods for linear systems.

2

Solve the system of linear equations for

u

and

v

using elimination.

Multiply

5u - 2v = \frac{1}{4}

by 2 to get

10u - 4v = \frac{1}{2}

. Add this to

3u + 4v = \frac{11}{4}

:

13u = \frac{11}{4} + \frac{2}{4} = \frac{13}{4} \implies u = \frac{1}{4}

. Then

4v = \frac{11}{4} - 3\left(\frac{1}{4}\right) = 2 \implies v = \frac{1}{2}

.

Eliminating

v

yields a single equation in

u

, which provides the values of both auxiliary variables.

3

Convert auxiliary values back to equations in

x

and

y

.

Since

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

, we get

x + y = 4

. Since

v = \frac{1}{x-y} = \frac{1}{2}

, we get

x - y = 2

.

Inverting the fractions restores the original variables in a standard 2x2 linear system.

4

Solve for

x

by adding the two linear equations.

(x + y) + (x - y) = 4 + 2 \implies 2x = 6 \implies x = 3

.

Adding the equations eliminates

y

directly, isolating

x

.

Anahtar Kavram

Solving systems of linear equations using substitution variables for algebraic simplification

Soru 1548Soru →

For all real numbers $x$ , what is the numerical value of the expression $\frac{3^{x+2} - 3^x}{3^{x-1} + 3^{x-2}}$ ?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The numerical value of the expression is 18.

Factoring

3^x

from both terms in the numerator gives

3^x(3^2 - 1) = 8 \cdot 3^x

. Factoring

3^x

from both terms in the denominator gives

3^x(3^{-1} + 3^{-2}) = 3^x\left(\frac{1}{3} + \frac{1}{9}\right) = \frac{4}{9} \cdot 3^x

. Dividing the two expressions cancels

3^x

entirely, resulting in

\frac{8}{\frac{4}{9}} = 8 \times \frac{9}{4} = 18

.

Adım Adım Çözüm

1

Factor out

3^x

from the numerator.

3^{x+2} - 3^x = 3^x(3^2 - 1) = 3^x(9 - 1) = 8 \cdot 3^x

Applying the exponent rule

a^{m+n} = a^m \cdot a^n

allows factoring out the common exponential factor

3^x

.

2

Factor out

3^x

from the denominator.

3^{x-1} + 3^{x-2} = 3^x(3^{-1} + 3^{-2}) = 3^x\left(\frac{1}{3} + \frac{1}{9}\right) = 3^x\left(\frac{4}{9}\right)

Applying negative exponent rules

a^{-n} = \frac{1}{a^n}

allows evaluating the remaining numerical terms inside the parentheses.

3

Simplify the overall fraction by dividing the factored numerator by the factored denominator.

\frac{8 \cdot 3^x}{\frac{4}{9} \cdot 3^x} = \frac{8}{\frac{4}{9}} = 8 \times \frac{9}{4} = 18

The non-zero common term

3^x

cancels from both numerator and denominator, leaving a constant integer.

Anahtar Kavram

Factoring and simplifying exponential expressions with variable exponents.

Soru 1549Soru →

For all real numbers $a$ and $b$ , the custom operation $\diamondsuit$ is defined by $a \diamondsuit b = a^2 - b^2 + 2ab$ . The functions $f$ and $g$ are defined by $f(x) = x \diamondsuit 2$ and $g(x) = 2 \diamondsuit x$ . If $k$ is a positive real number such that $f(k) = g(k)$ , what is the value of $f(g(-1))$ ?

-15

-7

-1

1

9

Cevabı ve açıklamayı göster

Cevap: $-7$

Cevap

-7

Applying the custom symbol rule gives

f(x) = x^2 + 4x - 4

and

g(x) = -x^2 + 4x + 4

. Equating

f(k) = g(k)

yields

2k^2 = 8

, so the positive solution is

k = 2

. Evaluating the inner function gives

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

. Substituting this value into

f

gives

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

.

Adım Adım Çözüm

1

Express

f(x)

and

g(x)

using the definition of the custom operation

\diamondsuit

.

f(x) = x \diamondsuit 2 = x^2 - 2^2 + 2(x)(2) = x^2 + 4x - 4

and

g(x) = 2 \diamondsuit x = 2^2 - x^2 + 2(2)(x) = -x^2 + 4x + 4

.

Applying

a \diamondsuit b = a^2 - b^2 + 2ab

with

(a, b) = (x, 2)

and

(a, b) = (2, x)

separately.

2

Set

f(k) = g(k)

to solve for the positive constant

k

.

k^2 + 4k - 4 = -k^2 + 4k + 4 \implies 2k^2 = 8 \implies k^2 = 4 \implies k = 2

(since

k > 0

).

Equating the two algebraic function expressions and solving the resulting quadratic equation.

3

Evaluate the inner function expression

g(-1)

.

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

.

Substituting

x = -1

into the formula for

g(x)

.

4

Evaluate the outer function

f(g(-1)) = f(-1)

.

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

.

Substituting the result from Step 3 into the formula for

f(x)

.

Anahtar Kavram

Evaluating algebraic custom operations, solving functional equalities, and applying nested function compositions.

Soru 1550Soru →

Consider the following system of linear equations in variables

x

,

y

, and

z

, where

a

is a real constant:

\begin{aligned} x + y + z &= 6 \\ x + 2y + 3z &= 10 \\ x + 2y + (a^2 - 1)z &= a + 8 \end{aligned}

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

Geçerli olan tümünü seçin

If

a = 2

, the system has infinitely many solutions.

If

a = -2

, the system has no solution.

If

a = 2

, every solution to the system satisfies

2x + y = 8

.

If

a = 3

, the unique solution has

y = 3

.

The system has a unique solution for all real values of

a

except

a = 2

.

Cevabı ve açıklamayı göster

Cevap: If $a = 2$ , the system has infinitely many solutions.; If $a = -2$ , the system has no solution.; If $a = 2$ , every solution to the system satisfies $2x + y = 8$ .

Cevap

The correct statements are that

a = 2

yields infinitely many solutions,

a = -2

results in no solution, and for

a = 2

every solution satisfies

2x + y = 8

.

The reduced equation

(a^2 - 4)z = a - 2

determines the behavior of the system. Setting

a = 2

gives

0 = 0

, leading to infinitely many solutions where

x = z + 2

and

y = 4 - 2z

, which identically satisfies

2x + y = 8

. Setting

a = -2

gives

0 = -4

, an inconsistency yielding no solutions.

Adım Adım Çözüm

1

Eliminate

x

and

y

using elimination between the second and third equations.

(x + 2y + (a^2 - 1)z) - (x + 2y + 3z) = (a + 8) - 10 \implies (a^2 - 4)z = a - 2

Isolating the parameter dependence onto a single variable

z

reveals existence and uniqueness conditions.

2

Analyze the equation

(a - 2)(a + 2)z = a - 2

for key parameter values.

If

a = 2

,

0 \cdot z = 0

(infinitely many solutions). If

a = -2

,

0 \cdot z = -4

(no solution). If

a \neq \pm 2

,

z = \frac{1}{a + 2}

(unique solution).

Determining system consistency depends on whether the leading coefficient and right-hand side evaluate to zero.

3

Express

x

and

y

in terms of

z

for the consistent case

a = 2

.

Subtracting the first equation from the second gives

y + 2z = 4 \implies y = 4 - 2z

. Substituting into the first gives

x = z + 2

.

Parameterizing the solution set allows verification of linear combinations.

4

Evaluate the linear combination

2x + y

when

a = 2

.

2x + y = 2(z + 2) + (4 - 2z) = 2z + 4 + 4 - 2z = 8

.

Verifies that

2x + y = 8

is an invariant across all parametric solutions.

Anahtar Kavram

Parametric Analysis of 3x3 Systems of Linear Equations

Soru 1551Soru →

A dataset consists of $9$ distinct positive integers $a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9$ listed in strictly increasing order. The mean of the $9$ integers is $28$ , and the median is $24$ . A $10\text{th}$ positive integer $x$ , where $x > a_9$ , is added to the dataset, causing the new mean of the $10$ integers to become $31$ . If $L$ represents the minimum possible value of $a_9$ , what is the value of $x - L$ ?

2

6

21

37

58

Cevabı ve açıklamayı göster

Cevap: $21$

Cevap

The value of

x - L

is

21

.

The sum of the original

9

elements is

9 \times 28 = 252

, and the sum of the

10

elements is

10 \times 31 = 310

, giving

x = 58

. To minimize

a_9

(

L

), the sum of all other elements must be maximized. The median

a_5 = 24

. The maximum possible values for the first four distinct elements below

24

are

20, 21, 22, 23

(summing to

86

). To minimize

a_9

, the upper four elements must be consecutive integers

(a_9 - 3, a_9 - 2, a_9 - 1, a_9)

. Setting the total sum equation

86 + 24 + (4a_9 - 6) = 252

yields

4a_9 = 148

, so

L = 37

. Consequently,

x - L = 58 - 37 = 21

.

Adım Adım Çözüm

1

Calculate the sum of the original 9 integers and determine the value of the 10th integer

x

.

The sum of the original

9

integers is

9 \times 28 = 252

. The sum of the

10

integers after adding

x

is

10 \times 31 = 310

. Therefore,

x = 310 - 252 = 58

.

The sum of a set of numbers equals the number of elements multiplied by the mean.

2

Identify the median of the ordered 9-element set.

In an ordered set of

9

elements, the median is the

5\text{th}

element, so

a_5 = 24

.

For an odd number of ordered elements, the median is the exact middle element.

3

Maximize the sum of the first 4 elements

a_1, a_2, a_3, a_4

to minimize

a_9

.

Since all integers are distinct and strictly increasing,

a_4 < 24

. To maximize

a_1 + a_2 + a_3 + a_4

, choose

a_4 = 23, a_3 = 22, a_2 = 21, a_1 = 20

. Their sum is

20 + 21 + 22 + 23 = 86

.

Maximizing lower elements leaves the smallest possible remainder of the total sum for the upper elements.

4

Express

a_6, a_7, a_8

in terms of

a_9

to minimize

a_9

.

To make

a_9

as small as possible,

a_6, a_7, a_8

should be as large as possible relative to

a_9

, meaning they are consecutive integers:

a_8 = a_9 - 1

,

a_7 = a_9 - 2

,

a_6 = a_9 - 3

.

Making elements above the median consecutive integers directly below

a_9

minimizes

a_9

for a fixed sum.

5

Set up the sum equation for the 9 elements to solve for

L = \text{min}(a_9)

.

(a_1 + a_2 + a_3 + a_4) + a_5 + (a_6 + a_7 + a_8 + a_9) = 252 \implies 86 + 24 + (a_9 - 3 + a_9 - 2 + a_9 - 1 + a_9) = 252 \implies 104 + 4a_9 = 252 \implies 4a_9 = 148 \implies a_9 = 37

. Thus,

L = 37

.

Solving the algebraic equation derived from the total sum yields the minimum integer value for

a_9

while satisfying

a_6 = 34 > 24

.

6

Calculate

x - L

.

x - L = 58 - 37 = 21

.

Subtracting the minimum bound

L

from

x

fulfills the target question requirement.

Anahtar Kavram

Optimization of Dataset Values using Central Tendency Constraints

Tahmini Süre:2m 30s

Soru 1552Soru →

In the $xy$ -plane, line $m$ passes through the point $(4, -2)$ and has a $y$ -intercept of $6$ . What is the slope of line $m$ ?

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

The slope of line

m

is

-2

.

The

y

-intercept of

6

indicates that the line passes through

(0, 6)

. Substituting

(0, 6)

and

(4, -2)

into the slope formula yields

m = \frac{6 - (-2)}{0 - 4} = \frac{8}{-4} = -2

.

Adım Adım Çözüm

1

Identify the coordinates of two points on line

m

.

The line passes through

(4, -2)

and the

y

-intercept point

(0, 6)

.

The

y

-intercept is the point where

x = 0

.

2

Apply the slope formula

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

.

Substitute

(x_1, y_1) = (4, -2)

and

(x_2, y_2) = (0, 6)

into the formula:

m = \frac{6 - (-2)}{0 - 4}

.

Slope represents the ratio of vertical change to horizontal change between two points on a line.

3

Simplify the fractional expression.

m = \frac{8}{-4} = -2

.

Dividing positive

8

by negative

4

gives

-2

.

Anahtar Kavram

Slope of a line given two points or a point and intercept

Soru 1553Soru →

A department consisting of $5$ employees has a mean monthly sales total of $\$12{}000$ . If a new employee with a monthly sales total of $\$18{}000$ joins the department, what is the new mean monthly sales total, in dollars, for the $6$ employees?

Cevabı ve açıklamayı göster

Cevap: 13000

Cevap

13000

To find the new mean, multiply the initial mean by the initial number of employees to get total sales:

5 \times 12{}000 = 60{}000

. Add the new employee's sales:

60{}000 + 18{}000 = 78{}000

. Finally, divide by the new total number of employees (

6

) to get

\frac{78{}000}{6} = 13{}000

.

Adım Adım Çözüm

1

Find the total sales of the original 5 employees.

5 \times 12{}000 = 60{}000

The sum of values is equal to the mean multiplied by the number of observations.

2

Calculate the total sales for all 6 employees.

60{}000 + 18{}000 = 78{}000

Add the new employee's sales to the initial total.

3

Calculate the new mean sales per employee.

\frac{78{}000}{6} = 13{}000

Divide the combined total sales by the new total number of employees (6).

Anahtar Kavram

Mean of Combined Data Sets

Tahmini Süre:1m 0s

Soru 1554Soru →

A dataset consists of $25$ distinct integers arranged in increasing order. The mean of all $25$ integers is $52$ . The mean of the smallest $12$ integers is $30$ , and the mean of the largest $12$ integers is $70$ .

If $5$ additional numbers, each equal to the median of the original dataset, are added to the dataset, what is the mean of the new set of $30$ numbers?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The mean of the new set of 30 numbers is 60.

The sum of all 25 numbers is

25 \times 52 = 1300

. Since the numbers are ordered, the median is the 13th value. The 12 smallest numbers sum to

12 \times 30 = 360

and the 12 largest sum to

12 \times 70 = 840

. The sum of these 24 numbers is

360 + 840 = 1200

, so the 13th number (the median) must be

1300 - 1200 = 100

. Adding 5 copies of 100 increases the sum to

1300 + 500 = 1800

across

30

numbers. The new mean is

\frac{1800}{30} = 60

.

Adım Adım Çözüm

1

Calculate the total sum of the original dataset of 25 numbers.

Sum =

25 \times 52 = 1300

.

The mean multiplied by the number of elements gives the total sum.

2

Calculate the combined sum of the 12 smallest and 12 largest integers.

Sum of 12 smallest =

12 \times 30 = 360

; Sum of 12 largest =

12 \times 70 = 840

; Total =

360 + 840 = 1200

.

The 25 numbers consist of the 12 smallest, the 1 median (13th element), and the 12 largest.

3

Determine the value of the median.

Median =

1300 - 1200 = 100

.

Subtracting the sum of the 24 non-median values from the total sum yields the 13th element, which is the median.

4

Find the sum and count of the modified dataset.

New Sum =

1300 + 5(100) = 1800

; New Count =

25 + 5 = 30

.

Adding 5 numbers each equal to 100 increases the sum by 500 and the count by 5.

5

Calculate the mean of the new dataset.

New Mean =

\frac{1800}{30} = 60

.

Divide the new total sum by the new total count of numbers.

Anahtar Kavram

Relationship between Mean, Median, and Total Sum in Partitioned Datasets

Tahmini Süre:2m 30s

Soru 1555Soru →

Three water pumps, A, B, and C, are used to drain a large industrial reservoir. Operating alone at its constant rate, Pump A can drain the reservoir in $x$ hours, where $x > 0$ . Pump B operating alone takes $x + 4$ hours to drain the reservoir. When operating together for 2 hours, Pump A and Pump B complete the exact same fraction of the total job that Pump C completes operating alone in 3 hours. If all three pumps working simultaneously at their respective constant rates can drain the entire reservoir in $\frac{9}{4}$ hours, what is the value of $x$ ?

4

6

8

10

12

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

x

is 6.

The correct answer 6 is derived by properly formulating the work rates

R_A = \frac{1}{x}

,

R_B = \frac{1}{x+4}

, and

R_C = \frac{4(x+2)}{3x(x+4)}

. Setting their sum

\frac{10(x+2)}{3x(x+4)}

equal to the combined rate

\frac{4}{9}

forms the quadratic

2x^2 - 7x - 30 = 0

, which yields the unique positive solution

x = 6

.

Adım Adım Çözüm

1

Express the individual work rates of Pump A and Pump B in terms of

x

.

R_A = \frac{1}{x}

and

R_B = \frac{1}{x+4}

. Combined rate

R_{A+B} = \frac{1}{x} + \frac{1}{x+4} = \frac{2x+4}{x(x+4)}

.

Work rate is defined as the fraction of the job completed per hour.

2

Determine the work rate of Pump C using the given relationship.

Work done by A and B in 2 hours is

2 \cdot \frac{2x+4}{x(x+4)} = \frac{4x+8}{x(x+4)}

. Since Pump C does this in 3 hours,

R_C = \frac{1}{3} \cdot \frac{4x+8}{x(x+4)} = \frac{4(x+2)}{3x(x+4)}

.

Pump C's hourly rate is one-third of the total work completed by A and B in 2 hours.

3

Sum all three rates to find the total combined rate and set it equal to the given combined rate.

R_{total} = \frac{2x+4}{x(x+4)} + \frac{4x+8}{3x(x+4)} = \frac{3(2x+4) + 4x+8}{3x(x+4)} = \frac{10(x+2)}{3x(x+4)}

. Given total time is

\frac{9}{4}

hours, total rate is

\frac{4}{9}

. Thus,

\frac{10(x+2)}{3x(x+4)} = \frac{4}{9}

.

The sum of individual rates equals the inverse of total time required when working simultaneously.

4

Solve the algebraic equation for

x

.

Multiply both sides by 9 to get

\frac{90(x+2)}{3x(x+4)} = 4 \implies \frac{30(x+2)}{x(x+4)} = 4 \implies 15(x+2) = 2x(x+4) \implies 15x + 30 = 2x^2 + 8x \implies 2x^2 - 7x - 30 = 0

. Factoring gives

(2x + 5)(x - 6) = 0

. Since

x > 0

,

x = 6

.

Solving the quadratic yields the valid positive real root for time

x

.

Anahtar Kavram

Work Rate Modeling and Rational Equation Systems

Soru 1556Soru →

In the $xy$ -plane, line $k$ passes through the points $(2, 5)$ and $(6, 13)$ . What is the slope of line $k$ ?

2

\frac{1}{2}

-2

-\frac{1}{2}

\frac{9}{4}

Cevabı ve açıklamayı göster

Cevap: $2$

Cevap

The slope of line

k

is

2

.

The slope of a straight line passing through points

(x_1, y_1)

and

(x_2, y_2)

is calculated as

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

. Substituting

(2, 5)

and

(6, 13)

gives

m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2

.

Adım Adım Çözüm

1

Identify the given points on the line.

(x_1, y_1) = (2, 5)

and

(x_2, y_2) = (6, 13)

.

These coordinates provide the required values for the slope formula.

2

Apply the slope formula

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

.

m = \frac{13 - 5}{6 - 2} = \frac{8}{4}

.

The slope is defined as the change in vertical position (

y

) divided by the change in horizontal position (

x

).

3

Simplify the fraction.

m = 2

.

Dividing

8

by

4

yields the simplified slope of the line.

Anahtar Kavram

Slope of a Line in Coordinate Geometry

Tahmini Süre:45s

Soru 1557Soru →

A software engineer logged the response time, in milliseconds, for seven independent server requests: $8$ , $16$ , $3$ , $21$ , $14$ , $5$ , and $10$ . What is the median response time, in milliseconds, for these seven requests?

8

10

11

14

21

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

The median response time is

10

milliseconds.

To find the median, first order the seven data points in ascending order:

3, 5, 8, 10, 14, 16, 21

. Since there are

7

values, the median is the middle value at position

\frac{7 + 1}{2} = 4

. The fourth value in this ordered list is

10

.

Adım Adım Çözüm

1

Arrange the given dataset in ascending order.

The sorted dataset is

3, 5, 8, 10, 14, 16, 21

.

Finding the median of a numerical dataset requires ordering the values from smallest to largest first.

2

Determine the position of the median element.

For

n = 7

items, the position is

\frac{7 + 1}{2} = 4

th item.

When the number of observations

n

is odd, the median is the exact middle value located at position

\frac{n+1}{2}

.

3

Identify the value at the 4th position.

The 4th value in the sorted list is

10

.

The 4th element in

3, 5, 8, 10, 14, 16, 21

is

10

.

Anahtar Kavram

Median of a finite numerical dataset with an odd count

Tahmini Süre:45s

Soru 1558Soru →

If $x$ is a real number that satisfies both $|3 - x| \le 5$ and $\frac{2x - 8}{-3} \le -2$ , which of the following could be the value of $x$ ? Indicate all such values.

Geçerli olan tümünü seçin

-1

4

7

7.5

8

Cevabı ve açıklamayı göster

Cevap: $7$ ; $7.5$ ; $8$

Cevap

The values

7

,

7.5

, and

8

satisfy both inequalities.

Solving

|3 - x| \le 5

gives

-2 \le x \le 8

. Solving

\frac{2x - 8}{-3} \le -2

by multiplying by

-3

and reversing the inequality sign gives

2x - 8 \ge 6

, or

x \ge 7

. Taking the intersection of both conditions yields

7 \le x \le 8

. Among the choices, the values

7

,

7.5

, and

8

fall within this range.

Adım Adım Çözüm

1

Solve the absolute value inequality

|3 - x| \le 5

.

-2 \le x \le 8

Rewrite as a compound inequality

-5 \le 3 - x \le 5

. Subtracting

3

gives

-8 \le -x \le 2

. Multiplying by

-1

and reversing inequality signs yields

-2 \le x \le 8

.

2

Solve the linear inequality

\frac{2x - 8}{-3} \le -2

.

x \ge 7

Multiply both sides by

-3

, making sure to flip the inequality sign:

2x - 8 \ge 6

. Adding

8

yields

2x \ge 14

, so

x \ge 7

.

3

Find the intersection of the two solution sets.

7 \le x \le 8

Combining

-2 \le x \le 8

and

x \ge 7

gives the range

7 \le x \le 8

.

4

Test the given options against the combined range

7 \le x \le 8

.

The values

7

,

7.5

, and

8

fall within

[7, 8]

, while

-1

and

4

do not.

Only numbers greater than or equal to

7

and less than or equal to

8

satisfy both conditions.

Anahtar Kavram

Solving compound linear and absolute value inequalities, ensuring inequality signs are reversed when multiplying or dividing by negative numbers.

Soru 1559Soru →

A rectangle has a length of $8$ units and a width of $6$ units. Which of the following statements about this rectangle must be true? Select all that apply.

Geçerli olan tümünü seçin

The area of the rectangle is

48

square units.

The length of each diagonal of the rectangle is

10

units.

The perimeter of the rectangle is

28

units.

The length of each diagonal of the rectangle is

14

units.

The area of the rectangle is

28

square units.

Cevabı ve açıklamayı göster

Cevap: The area of the rectangle is $48$ square units.; The length of each diagonal of the rectangle is $10$ units.; The perimeter of the rectangle is $28$ units.

Cevap

The correct statements are that the area of the rectangle is 48 square units, the length of each diagonal is 10 units, and the perimeter of the rectangle is 28 units.

The statement regarding the area being 48 square units is correct because

8 \times 6 = 48

. The statement regarding the diagonal being 10 units is correct because

\sqrt{8^2 + 6^2} = 10

. The statement regarding the perimeter being 28 units is correct because

2 \times (8 + 6) = 28

.

Adım Adım Çözüm

1

Calculate the area of the rectangle

\text{Area} = 8 \times 6 = 48

square units

The area formula for a rectangle is length multiplied by width.

2

Calculate the length of the diagonal using the Pythagorean theorem

\text{Diagonal} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = 10

units

The sides and diagonal of a rectangle form a right triangle where the diagonal is the hypotenuse.

3

Calculate the perimeter of the rectangle

\text{Perimeter} = 2 \times (8 + 6) = 28

units

The perimeter formula for a rectangle is twice the sum of its length and width.

Anahtar Kavram

Basic geometric properties of rectangles including area, perimeter, and diagonal calculation via the Pythagorean theorem.

Soru 1560Soru →

If $k$ is a real constant such that the inequality $|2x - 3| + |x + 4| \le k$ has no real solutions for $x$ , which of the following inequality statements expresses all possible values of $k$ ?

k < \frac{11}{2}

k \le \frac{11}{2}

k < 7

k < 11

k > \frac{11}{2}

Cevabı ve açıklamayı göster

Cevap: $k < \frac{11}{2}$

Cevap

The statement expressing all possible values of

k

is

k < \frac{11}{2}

.

The function

f(x) = |2x - 3| + |x + 4|

represents a continuous piecewise linear curve. Evaluating

f(x)

at its critical points

x = -4

and

x = \frac{3}{2}

yields

f(-4) = 11

and

f\left(\frac{3}{2}\right) = 5.5 = \frac{11}{2}

. Since the slope is

-3

for

x < -4

,

-1

for

-4 < x < \frac{3}{2}

, and

+3

for

x > \frac{3}{2}

, the global minimum value of

f(x)

across all real numbers is

\frac{11}{2}

. Consequently, the inequality

f(x) \le k

has no real solutions if and only if

k

is strictly less than this minimum value, leading to

k < \frac{11}{2}

.

Adım Adım Çözüm

1

Identify the critical points of the absolute value terms.

The terms

|2x - 3|

and

|x + 4|

change behavior at

x = \frac{3}{2}

and

x = -4

, respectively.

Absolute value functions

f(x) = |ax + b|

reach zero and change slope at their roots.

2

Evaluate

f(x) = |2x - 3| + |x + 4|

at the critical points and analyze its piecewise behavior.

At

x = -4

,

f(-4) = |-11| + |0| = 11

. At

x = \frac{3}{2}

,

f\left(\frac{3}{2}\right) = |0| + |\frac{11}{2}| = \frac{11}{2}

. For

x < -4

,

f(x) = (3 - 2x) - (x + 4) = -3x - 1

. For

-4 \le x \le \frac{3}{2}

,

f(x) = (3 - 2x) + (x + 4) = -x + 7

. For

x > \frac{3}{2}

,

f(x) = (2x - 3) + (x + 4) = 3x + 1

.

Because

f(x)

is a convex piecewise linear function that grows to

\infty

as

x \to \pm\infty

, its global minimum must occur at one of its critical points.

3

Determine the global minimum value of

f(x)

.

Comparing values,

f\left(\frac{3}{2}\right) = \frac{11}{2}

is smaller than

f(-4) = 11

, so the minimum value of

|2x - 3| + |x + 4|

for all real

x

is

\frac{11}{2}

.

The function output is always greater than or equal to

\frac{11}{2}

for any real number

x

.

4

Apply the condition for no real solutions.

For

|2x - 3| + |x + 4| \le k

to have no solutions,

k

must be strictly less than the absolute minimum value of the expression, so

k < \frac{11}{2}

.

If

k \ge \frac{11}{2}

, there is at least one

x

value (such as

x = \frac{3}{2}

) satisfying the inequality.

Anahtar Kavram

Minimizing Sums of Absolute Values and Boundary Conditions of Inequalities

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