All practice questions

Question 2061Question →

The first, third, and eleventh terms of a non-constant arithmetic sequence are the first, second, and third terms, respectively, of a geometric sequence. If the first term of the arithmetic sequence is $6$ , what is the sum of the first $4$ terms of the geometric sequence?

78

384

510

1,554

4,920

Show answer & explanation

Answer: 510

Answer

The sum of the first 4 terms of the geometric sequence is 510.

By writing the first, third, and eleventh terms of the arithmetic sequence as

6

,

6+2d

, and

6+10d

, we can set up the geometric sequence relation

(6+2d)^2 = 6(6+10d)

. Solving this quadratic equation for

d

yields

d=9

(since the sequence is non-constant,

d \neq 0

). Substituting

d=9

back gives the first three geometric terms as

6

,

24

, and

96

, which means the common ratio

r

is

4

. The sum of the first

4

terms of this geometric sequence is calculated as

6(4^4-1)/(4-1) = 510

.

Step-by-Step Solution

1

Define the terms of the arithmetic and geometric sequences.

Let the arithmetic sequence have first term

a_1 = 6

and common difference

d

. The first, third, and eleventh terms are

a_1 = 6

,

a_3 = 6 + 2d

, and

a_{11} = 6 + 10d

. These are the first three terms of the geometric sequence:

g_1 = 6

,

g_2 = 6 + 2d

, and

g_3 = 6 + 10d

.

This establishes algebraic expressions for the terms based on their positions in the sequences.

2

Set up a relation using the constant ratio of the geometric sequence and solve for

d

.

Since

g_1

,

g_2

, and

g_3

form a geometric sequence,

(g_2)^2 = g_1 \cdot g_3

. Substituting the expressions gives

(6 + 2d)^2 = 6(6 + 10d) \implies 36 + 24d + 4d^2 = 36 + 60d \implies 4d^2 - 36d = 0

. Since the sequence is non-constant (

d \neq 0

), we divide by

4d

to get

d = 9

.

Solving the equation yields the common difference of the arithmetic sequence.

3

Determine the terms and common ratio of the geometric sequence.

Using

d = 9

, the first two terms of the geometric sequence are

g_1 = 6

and

g_2 = 6 + 2(9) = 24

. The common ratio is

r = \frac{24}{6} = 4

.

Finding the common ratio allows the use of the geometric series sum formula.

4

Compute the sum of the first 4 terms of the geometric sequence.

S_4 = g_1 \frac{r^4 - 1}{r - 1} = 6 \frac{4^4 - 1}{4 - 1} = 6 \frac{256 - 1}{3} = 2(255) = 510

.

This calculates the final required sum.

Key Concept

Arithmetic and Geometric Sequences and Series

Question 2062Question →

If $\log_3(2x - 1) = 2$ , what is the value of $x$ ?

Show answer & explanation

Answer: 5

Answer

The value of

x

is

5

.

Converting the logarithmic equation

\log_3(2x - 1) = 2

to its equivalent exponential form yields

3^2 = 2x - 1

. Simplifying the exponent gives

9 = 2x - 1

. Adding

1

to both sides results in

10 = 2x

, and dividing by

2

gives

x = 5

.

Step-by-Step Solution

1

Rewrite the logarithmic equation in exponential form.

2x - 1 = 3^2

By definition,

\log_b(y) = z

is equivalent to

b^z = y

.

2

Evaluate the exponent

3^2

.

2x - 1 = 9

Calculating

3

squared yields

9

.

3

Solve the linear equation for

x

.

2x = 10

, which simplifies to

x = 5

Adding

1

to both sides and then dividing by

2

isolates

x

.

Key Concept

Converting logarithmic equations to exponential equations

Estimated Time:45s

Question 2063Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = 2x - 3$ and $g(x) = x^2 - 5$ . If $g(f(x)) = 11$ and $x < 0$ , what is the value of $x$ ?

\frac{7}{2}

-\sqrt{12}

-\frac{\sqrt{7}}{2}

-\frac{1}{2}

-\frac{7}{2}

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

-\frac{1}{2}

The correct answer is

-\frac{1}{2}

. First, the composite function

g(f(x))

is found by substituting

f(x) = 2x - 3

into

g(x) = x^2 - 5

, giving

g(f(x)) = (2x - 3)^2 - 5

. Setting this expression equal to

11

yields the equation

(2x - 3)^2 - 5 = 11

. Adding

5

to both sides results in

(2x - 3)^2 = 16

. Taking the square root of both sides gives

2x - 3 = \pm 4

. Since the problem specifies that

x < 0

, the expression

2x - 3

must be negative because

2x < 0

and subtracting

3

makes the result less than

-3

. Thus, we set

2x - 3 = -4

. Adding

3

to both sides yields

2x = -1

, and dividing by

2

gives

x = -\frac{1}{2}

.

Step-by-Step Solution

1

Substitute the expression for

f(x)

into the function

g(x)

to obtain the composite function

g(f(x))

.

g(f(x)) = g(2x - 3) = (2x - 3)^2 - 5

To find

g(f(x))

, we substitute the entire function

f(x)

in place of the input variable in

g(x)

.

2

Set the composite function equal to

11

and isolate the squared term.

(2x - 3)^2 - 5 = 11 \Rightarrow (2x - 3)^2 = 16

We are given that

g(f(x)) = 11

, and adding

5

to both sides isolates the squared binomial.

3

Solve for

2x - 3

by taking the square root of both sides, applying the constraint

x < 0

.

2x - 3 = -4

Since

x < 0

, it follows that

2x < 0

, which means

2x - 3 < -3

. Because

2x-3

must be negative, we take the negative square root of

16

.

4

Solve the linear equation

2x - 3 = -4

for

x

.

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

Adding

3

to both sides gives

2x = -1

, and dividing by

2

yields the final value of

x

.

Key Concept

Function Composition and Solving Quadratic/Linear Equations

Estimated Time:1m 30s

Question 2064Question →

A manufacturing company produces sheet metal. The area of a large rectangular sheet is represented by the expression $2x^2(3x^3 - 4x + 5)$ square inches. A square piece with a side length of $3x - 4$ inches is cut out from the sheet. Which of the following expressions represents the area of the remaining metal sheet, in square inches, for all $x > 2$ ?

6x^6 - 8x^3 + x^2 + 24x - 16

6x^5 - 8x^3 + x^2 - 16

6x^5 - 8x^3 + x^2 - 24x + 16

6x^5 - 8x^3 + x^2 + 24x - 16

6x^5 - 8x^3 + x^2 + 24x + 16

Show answer & explanation

Answer: $6x^5 - 8x^3 + x^2 + 24x - 16$

Answer

The remaining area of the metal sheet is represented by

6x^5 - 8x^3 + x^2 + 24x - 16

.

To find the remaining area, the area of the square piece must be subtracted from the area of the large rectangular sheet. Expanding the large rectangular sheet's area yields

2x^2(3x^3 - 4x + 5) = 6x^5 - 8x^3 + 10x^2

. Expanding the area of the square piece yields

(3x - 4)^2 = 9x^2 - 24x + 16

. Subtracting the second polynomial from the first requires distributing the negative sign to all terms:

(6x^5 - 8x^3 + 10x^2) - (9x^2 - 24x + 16) = 6x^5 - 8x^3 + 10x^2 - 9x^2 + 24x - 16

. Combining the quadratic terms

10x^2 - 9x^2 = x^2

yields the correct simplified expression

6x^5 - 8x^3 + x^2 + 24x - 16

.

Step-by-Step Solution

1

Expand the expression for the area of the large rectangular sheet.

2x^2(3x^3 - 4x + 5) = 6x^5 - 8x^3 + 10x^2

Distribute the term

2x^2

to each term in the trinomial, using the exponent rule

x^a \cdot x^b = x^{a+b}

to combine the variable factors.

2

Expand the expression for the area of the square piece.

(3x - 4)^2 = 9x^2 - 24x + 16

Apply the binomial squaring formula

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

, where

a = 3x

and

b = 4

.

3

Subtract the area of the square piece from the area of the large rectangular sheet.

(6x^5 - 8x^3 + 10x^2) - (9x^2 - 24x + 16) = 6x^5 - 8x^3 + 10x^2 - 9x^2 + 24x - 16

Distribute the negative sign to each term of the subtracted polynomial, changing the sign of each term inside the second parenthesis.

4

Combine like terms and write the final expression in descending order.

6x^5 - 8x^3 + x^2 + 24x - 16

Combine the quadratic terms:

10x^2 - 9x^2 = x^2

. Arrange the remaining terms in descending order of their exponents.

Key Concept

Polynomial operations involving monomial distribution with exponent rules, binomial squaring, and polynomial subtraction with sign distribution.

Estimated Time:2m 0s

Question 2065Question →

A smart home heating system has two operating modes: Eco mode and Comfort mode. In a certain week, the system was active in one of these two modes for a total of $168$ hours. The number of hours the system was in Eco mode was $24$ hours less than three times the number of hours it was in Comfort mode. For how many hours was the heating system in Comfort mode during that week?

Show answer & explanation

Answer: 48

Answer

The heating system was in Comfort mode for 48 hours.

The correct answer is obtained by representing the hours in Comfort mode as

c

and Eco mode as

3c - 24

. Since their sum is

168

, we write the equation

c + 3c - 24 = 168

. Solving this linear equation gives

4c = 192

, which yields

c = 48

.

Step-by-Step Solution

1

Define variables for the Comfort and Eco modes.

Let

c

represent the hours in Comfort mode and

e

represent the hours in Eco mode.

To represent the unknown quantities algebraically.

2

Translate the given information into a system of equations.

c + e = 168

and

e = 3c - 24

The sum of the hours in both modes is

168

, and the relationship between Eco and Comfort hours is described.

3

Substitute the expression for

e

into the total hours equation.

c + (3c - 24) = 168

To create a single linear equation with one variable.

4

Solve for the variable

c

.

4c - 24 = 168 \Rightarrow 4c = 192 \Rightarrow c = 48

Simplifying the equation gives the value of

c

, which is the hours spent in Comfort mode.

Key Concept

Translating and Solving Algebraic Word Problems

Question 2066Question →

If $\log_4 x = \frac{3}{2}$ , what is the value of $x$ ?

6

8

12

64

\frac{81}{16}

Show answer & explanation

Answer: $8$

Answer

The correct answer is

8

.

To solve the equation

\log_4 x = \frac{3}{2}

, we apply the definition of a logarithm to rewrite it in exponential form:

x = 4^{3/2}

. We then evaluate the exponent by first taking the square root of

4

, which is

2

, and then cubing it to get

2^3 = 8

. Thus, the value of

x

is

8

.

Step-by-Step Solution

1

Rewrite the logarithmic equation in its equivalent exponential form.

x = 4^{3/2}

By definition, a logarithmic equation of the form

\log_b a = c

is equivalent to the exponential equation

b^c = a

.

2

Evaluate the exponential expression

4^{3/2}

.

x = 8

The fractional exponent can be simplified by taking the square root of the base first, which is

\sqrt{4} = 2

, and then raising the result to the power of the numerator, giving

2^3 = 8

.

Key Concept

Converting logarithmic equations to exponential form and evaluating fractional exponents

Alternative Method

You can solve this by substituting the answer choices back into the equation. For example, testing the value

8

gives

\log_4 8 = \log_4 (2^3) = 3 \log_4 2

. Since

2

is the square root of

4

,

\log_4 2 = \frac{1}{2}

. Therefore,

3 \times \frac{1}{2} = \frac{3}{2}

, confirming that

8

is the correct solution.

Estimated Time:45s

Question 2067Question →

In a chemistry laboratory, a beaker contains a mixture of water and acid. The volume of water in the beaker is $3$ liters more than twice the volume of acid. After $5$ liters of water are added to the beaker, the ratio of the volume of water to the volume of acid is $5$ to $2$ . If no acid was added or removed, what was the initial volume of water, in liters, in the beaker?

5

16

19

35

50

Show answer & explanation

Answer: 35

Answer

35

The correct answer of 35 liters is found by setting up the linear equation where the initial volume of acid is

a

and the initial volume of water is

2a + 3

. Adding 5 liters of water gives

2a + 8

liters of water. Setting the ratio of water to acid to

\frac{5}{2}

and solving gives

a = 16

. Finally, substituting

a = 16

into the expression for the initial volume of water (

2a + 3

) yields 35 liters.

Step-by-Step Solution

1

Define variables for the initial volumes of acid and water based on the given relationships.

Let

a

be the initial volume of acid in liters. The initial volume of water is

2a + 3

liters.

This translates the statement 'the volume of water is 3 liters more than twice the volume of acid' into algebraic expressions.

2

Set up a linear equation representing the ratio after adding 5 liters of water.

The new volume of water is

(2a + 3) + 5 = 2a + 8

liters. The ratio of water to acid is

\frac{2a + 8}{a} = \frac{5}{2}

.

This uses the new state of the mixture to form an equation that can be solved for

a

.

3

Solve the linear equation for

a

by cross-multiplying and isolating the variable.

2(2a + 8) = 5a \implies 4a + 16 = 5a \implies a = 16

.

Cross-multiplication removes the fractions and allows for standard term isolation.

4

Calculate the initial volume of water.

Initial water volume =

2a + 3 = 2(16) + 3 = 35

liters.

The question asks for the initial volume of water, which is represented by

2a + 3

, not the volume of acid

a

.

Key Concept

Solving linear equations in one variable derived from word problems.

Alternative Method

Instead of solving algebraically, one could test the answer choices. For example, testing 35 liters of water means the initial acid is

(35 - 3) / 2 = 16

liters. Adding 5 liters of water gives 40 liters of water. The ratio of water to acid is

40 : 16

, which simplifies to

5 : 2

. This confirms 35 is correct.

Estimated Time:1m 30s

Question 2068Question →

In the standard $(x,y)$ coordinate plane, the graph of the linear equation $y - 7 = -2(x + 3)$ crosses the $y$ -axis. What is the $y$ -coordinate of this intersection point?

Show answer & explanation

Answer: 1

Answer

The

y

-coordinate of the intersection point is

1

.

To find where the graph of the linear equation crosses the

y

-axis, substitute

x = 0

into the equation

y - 7 = -2(x + 3)

. This gives

y - 7 = -2(0 + 3)

, which simplifies to

y - 7 = -6

. Adding

7

to both sides results in

y = 1

. Thus, the

y

-coordinate of the intersection point is

1

.

Step-by-Step Solution

1

Substitute

x = 0

into the linear equation.

y - 7 = -2(0 + 3)

The graph of an equation crosses the

y

-axis (which defines the

y

-intercept) when the

x

-coordinate is equal to

0

.

2

Simplify the expression on the right side of the equation.

y - 7 = -6

Evaluating

-2(0 + 3)

yields

-2(3) = -6

.

3

Solve the simplified linear equation for

y

.

y = 1

Adding

7

to both sides isolates

y

, giving

y = -6 + 7 = 1

.

Key Concept

Finding the

y

-intercept of a line from its point-slope form equation

Question 2069Question →

A manufacturing company produces two types of metal alloys. The production cost, in dollars per ton, of Alloy A is modeled by the expression $800 - 15(x - 5)$ , where $x$ is the amount of stabilizer added in kilograms. The production cost of Alloy B, in dollars per ton, is modeled by the expression $520 + 5(x + 3)$ . If the company requires the production cost of Alloy A to be strictly less than the production cost of Alloy B, what is the minimum integer amount of stabilizer $x$ , in kilograms, that must be added?

Show answer & explanation

Answer: 18

Answer

The minimum integer amount of stabilizer that must be added is

18

kg.

The inequality representing the condition is

800 - 15(x - 5) < 520 + 5(x + 3)

. Expanding both sides gives

875 - 15x < 535 + 5x

. Isolating

x

yields

-20x < -340

. Dividing by

-20

and flipping the inequality sign results in

x > 17

. The minimum integer value that is strictly greater than

17

is

18

.

Step-by-Step Solution

1

Set up the linear inequality using the cost expressions for Alloy A and Alloy B.

800 - 15(x - 5) < 520 + 5(x + 3)

The cost of Alloy A must be strictly less than the cost of Alloy B.

2

Distribute and combine like terms to simplify both sides of the inequality.

875 - 15x < 535 + 5x

Simplifying the expressions makes it easier to isolate the variable.

3

Isolate the variable term on one side of the inequality.

-20x < -340

Subtracting

5x

and

875

from both sides moves all variable terms to the left and constant terms to the right.

4

Divide both sides by the negative coefficient

-20

and reverse the inequality sign.

x > 17

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

5

Identify the smallest integer that satisfies the inequality.

18

The solution requires a strict inequality

x > 17

, so the smallest integer value that is strictly greater than

17

is

18

.

Key Concept

Solving multi-step linear inequalities with variables on both sides, including reversing the inequality sign when dividing by a negative number.

Question 2070Question →

If $x$ and $y$ are real numbers such that $2^x \cdot 3^y = 24$ and $3^x \cdot 2^y = 54$ , what is the value of $x^2 + y^2$ ?

Show answer & explanation

Answer: 10

Answer

10

Multiplying the two equations gives

(2^x \cdot 3^y)(3^x \cdot 2^y) = 24 \cdot 54 \implies 6^{x+y} = 1296 = 6^4

, which yields

x + y = 4

. Dividing the first equation by the second equation gives \frac{2^x \cdot 3^y}{3^x \cdot 2^y} = \frac{24}{54} \implies (\frac{2}{3})^{x-y} = \frac{4}{9} = (\frac{2}{3})^2

, which yields

x - y = 2

. Solving the system of equations

x + y = 4

and

x - y = 2

gives

x = 3

and

y = 1

. The value of

x^2 + y^2

is therefore

3^2 + 1^2 = 10$.

Step-by-Step Solution

1

Multiply the two equations together.

6^{x+y} = 1296

, which simplifies to

x + y = 4

.

Multiplying the equations groups bases of 2 and 3 together to form base 6, allowing us to find the sum of the variables.

2

Divide the first equation by the second equation.

(\frac{2}{3})^{x-y} = \frac{4}{9}

, which simplifies to

x - y = 2

.

Dividing the equations groups bases of 2 and 3 to form base

2/3

, allowing us to find the difference of the variables.

3

Solve the system of equations for

x

and

y

.

x = 3

and

y = 1

.

Solving the linear system of equations

x + y = 4

and

x - y = 2

gives the individual values of

x

and

y

.

4

Calculate

x^2 + y^2

.

10

Substitute the values of

x

and

y

into the target expression.

Key Concept

Solving systems of exponential equations using properties of exponents and bases

Alternative Method

Take the logarithm of both sides of each equation to convert them into a system of linear equations in terms of

x

and

y

:

x \log 2 + y \log 3 = \log 24

and

x \log 3 + y \log 2 = \log 54

. Solving this system using elimination or substitution yields

x = 3

and

y = 1

, so

x^2 + y^2 = 10

.

Estimated Time:2m 30s

Question 2071Question →

A line in the standard $(x,y)$ coordinate plane passes through the points shown in the table below:

$x$	$y$
$0$	$5$
$2$	$9$
$4$	$13$

Which of the following equations represents this line?

y = 5x + 2

y = 2x - 5

y = 2x + 5

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

y = 4x + 5

Show answer & explanation

Answer: $y = 2x + 5$

Answer

y = 2x + 5

The correct equation is

y = 2x + 5

because the slope of the line is

2

, calculated from the points

(0, 5)

and

(2, 9)

as

\frac{9 - 5}{2 - 0} = 2

, and the

y

-intercept is

5

, corresponding to the point

(0, 5)

.

Step-by-Step Solution

1

Identify the

y

-intercept from the table.

The point

(0, 5)

indicates that the

y

-intercept is

5

, which means

b = 5

in the slope-intercept form

y = mx + b

.

The

y

-intercept is the point where the line crosses the

y

-axis, which occurs when

x = 0

.

2

Calculate the slope using two points from the table.

Using

(0, 5)

and

(2, 9)

, the slope

m = \frac{9 - 5}{2 - 0} = \frac{4}{2} = 2

.

The slope formula is the change in

y

divided by the change in

x

.

3

Substitute the slope and

y

-intercept into the slope-intercept equation.

Substituting

m = 2

and

b = 5

into

y = mx + b

yields

y = 2x + 5

.

This represents the linear relationship of the points.

Key Concept

Writing a linear equation in slope-intercept form from a table of values.

Estimated Time:45s

Question 2072Question →

A taxi company charges a flat fee plus a constant rate per mile driven. The total cost, $C$ (in dollars), for a ride of $d$ miles is given by a linear relationship. If a 6-mile ride costs $18.50 and a 10-mile ride costs$ 26.50, what is the flat fee, in dollars, charged by the taxi company?

15.50

2.00

6.50

12.50

25.00

Show answer & explanation

Answer: 6.50

Answer

The flat fee charged by the taxi company is 6.50 dollars.

The correct flat fee is 6.50 dollars. The linear relationship is of the form

C = md + b

, where

m

is the cost per mile and

b

is the flat fee. The slope is calculated as

m = \frac{26.50 - 18.50}{10 - 6} = 2.00

. Substituting the point

(6, 18.50)

into the equation gives

18.50 = 2(6) + b

, which simplifies to

18.50 = 12.00 + b

, yielding

b = 6.50

.

Step-by-Step Solution

1

Identify the two coordinates from the word problem and calculate the slope (rate per mile).

The points are

(6, 18.50)

and

(10, 26.50)

. The slope

m = \frac{26.50 - 18.50}{10 - 6} = \frac{8.00}{4} = 2.00

.

In a linear relationship, the constant rate of change is represented by the slope of the line.

2

Use the slope-intercept form

C = md + b

and substitute one of the points to solve for the vertical intercept

b

(flat fee).

Using

(6, 18.50)

:

18.50 = 2.00(6) + b \implies 18.50 = 12.00 + b \implies b = 6.50

.

The flat fee corresponds to the cost when the distance is zero miles, which is the vertical intercept

b

.

Key Concept

Determining the equation of a line from two points in a real-world linear application.

Question 2073Question →

For a real number $a$ , the quadratic equation $x^2 + ax + (a - 3)^2 = 0$ has complex roots. One of these roots is $2 + i\sqrt{b}$ , where $b$ is a positive real number and $i = \sqrt{-1}$ . What is the value of $a + b$ ?

17

-21

41

-49

1

Show answer & explanation

Answer: 41

Answer

The value of

a + b

is

41

.

The correct answer is

41

. Since the quadratic equation has real coefficients, its complex roots must occur in conjugate pairs. Thus, the roots are

2 + i\sqrt{b}

and

2 - i\sqrt{b}

. Their sum is

4

, which by Vieta's formulas equals

-a

, giving

a = -4

. The product of the roots is

4 + b

, which equals the constant term

(a - 3)^2 = (-4 - 3)^2 = 49

. Solving for

b

gives

b = 45

. Therefore,

a + b = -4 + 45 = 41

.

Step-by-Step Solution

1

Identify the second root using the complex conjugate root theorem.

The conjugate root is

2 - i\sqrt{b}

.

Since the coefficients of the quadratic equation are real, complex roots must occur in conjugate pairs.

2

Find the value of

a

using the sum of the roots.

a = -4

By Vieta's formulas, the sum of the roots is equal to

-a

. The sum of the conjugate roots is

(2 + i\sqrt{b}) + (2 - i\sqrt{b}) = 4

, so

-a = 4

.

3

Calculate the constant term of the quadratic equation.

The constant term is

49

.

Substitute

a = -4

into the constant term

(a - 3)^2

to get

(-4 - 3)^2 = (-7)^2 = 49

.

4

Find the value of

b

using the product of the roots.

b = 45

By Vieta's formulas, the product of the roots is the constant term. The product is

(2 + i\sqrt{b})(2 - i\sqrt{b}) = 4 - i^2 b = 4 + b

(since

i^2 = -1

). Equating this to the constant term gives

4 + b = 49

, so

b = 45

.

5

Calculate the final value of

a + b

.

41

Substitute the values of

a

and

b

:

a + b = -4 + 45 = 41

.

Key Concept

Vieta's formulas and complex conjugate roots of quadratic equations with real coefficients.

Question 2074Question →

If the equation $\frac{2}{5}(x - 4) + \frac{1}{3}(2x + k) = 7$ is true when $x = 9$ , what is the value of $k$ ?

-17

1

-3

3

8

Show answer & explanation

Answer: $-3$

Answer

-3

Substituting

x = 9

into the equation gives

\frac{2}{5}(9 - 4) + \frac{1}{3}(2(9) + k) = 7

. Simplifying the terms yields

2 + \frac{18 + k}{3} = 7

. Subtracting 2 from both sides results in

\frac{18 + k}{3} = 5

. Multiplying by 3 gives

18 + k = 15

. Subtracting 18 from both sides gives

k = -3

.

Step-by-Step Solution

1

Substitute

x = 9

into the given equation.

\frac{2}{5}(9 - 4) + \frac{1}{3}(2(9) + k) = 7

We are given that the equation is true when

x = 9

.

2

Simplify the operations inside the parentheses.

2 + \frac{18 + k}{3} = 7

Simplifying

\frac{2}{5}(5)

gives 2, and

2(9)

gives 18.

3

Subtract 2 from both sides of the equation.

\frac{18 + k}{3} = 5

Isolating the fraction term simplifies the equation.

4

Multiply both sides by 3.

18 + k = 15

Clearing the denominator allows us to isolate the variable

k

.

5

Subtract 18 from both sides of the equation.

k = -3

This isolates

k

to find its value.

Key Concept

Solving linear equations in one variable by substitution and simplification

Question 2075Question →

If $x$ is a positive real number such that $\log_2(x) \cdot \log_3(x) - \log_2(x) - \log_3(x) = 1$ , what is the product of all possible real values of $x$ ?

\frac{1}{6}

5

6

\frac{1}{5}

12

Show answer & explanation

Answer: $6$

Answer

6

The correct answer is 6. By converting the equation using the change-of-base formula into natural logarithms, we obtain the quadratic equation

(\ln x)^2 - (\ln 2 + \ln 3)\ln x - \ln 2 \cdot \ln 3 = 0

. By setting

w = \ln x

, we have a quadratic in terms of

w

. The product of the two solutions

x_1

and

x_2

is

x_1 x_2 = e^{w_1} e^{w_2} = e^{w_1 + w_2}

. Using Vieta's formulas, the sum of the roots

w_1 + w_2 = \ln 2 + \ln 3 = \ln 6

. Therefore, the product of the solutions is

e^{\ln 6} = 6

.

Step-by-Step Solution

1

Rewrite the logarithmic equation using a common base.

Using the change-of-base formula

\log_b(a) = \frac{\ln a}{\ln b}

, the equation becomes:

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

This allows all logarithmic terms to be expressed in terms of the natural logarithm, making it easier to solve.

2

Eliminate denominators by multiplying the equation by

\ln 2 \cdot \ln 3

.

(\ln x)^2 - (\ln 3)\ln x - (\ln 2)\ln x = \ln 2 \cdot \ln 3

Multiplying through by the common denominator simplifies the rational equation into a standard polynomial form.

3

Rearrange the equation into a standard quadratic form.

(\ln x)^2 - (\ln 2 + \ln 3)\ln x - \ln 2 \cdot \ln 3 = 0

Grouping the coefficients of

\ln x

reveals a quadratic equation of the form

Aw^2 + Bw + C = 0

, where

w = \ln x

.

4

Define the relationship between the roots of the quadratic equation and the values of

x

.

Let the two real roots of the quadratic equation be

w_1

and

w_2

. These correspond to the solutions for

x

, namely

x_1 = e^{w_1}

and

x_2 = e^{w_2}

. The product of the solutions is

x_1 \cdot x_2 = e^{w_1} \cdot e^{w_2} = e^{w_1 + w_2}

.

To find the product of the solutions for

x

, we must compute the exponential of the sum of the roots of the quadratic equation.

5

Apply Vieta's formulas to find the sum of the roots

w_1 + w_2

.

The sum of the roots is

w_1 + w_2 = -\frac{B}{A} = -\frac{-(\ln 2 + \ln 3)}{1} = \ln 2 + \ln 3

. Using the logarithmic product rule, this simplifies to

\ln(2 \cdot 3) = \ln 6

.

Vieta's formulas state that the sum of the roots of

Aw^2 + Bw + C = 0

is

-\frac{B}{A}

.

6

Calculate the product of the real values of

x

.

x_1 \cdot x_2 = e^{\ln 6} = 6

Substituting the sum of the roots back into the exponent gives the final product.

Key Concept

Solving equations involving logarithmic properties, change of base, and relating quadratic roots to exponential functions

Question 2076Question →

A cleaning service company offers two types of packages: Standard Clean and Deep Clean. The number of packages sold during the first two weeks of the month is represented by matrix $Q$ :

Q = \begin{bmatrix} 20 & 15 \\ 25 & 18 \end{bmatrix}

where row 1 represents Week 1, row 2 represents Week 2, column 1 represents Standard Clean packages, and column 2 represents Deep Clean packages.

The price charged and the cost to perform each package (in dollars) are represented by matrix $P$ :

P = \begin{bmatrix} 80 & 30 \\ 150 & 60 \end{bmatrix}

where row 1 represents Standard Clean packages, row 2 represents Deep Clean packages, column 1 represents the price charged, and column 2 represents the cost.

Which of the following matrices represents the total revenue and total cost for Week 1 and Week 2, respectively?

\begin{bmatrix} 1,600 & 450 \\ 3,750 & 1,080 \end{bmatrix}

\begin{bmatrix} 2,350 & 1,740 \\ 4,500 & 3,330 \end{bmatrix}

\begin{bmatrix} 3,850 & 1,500 \\ 4,700 & 1,830 \end{bmatrix}

\begin{bmatrix} 3,800 & 1,500 \\ 4,000 & 1,830 \end{bmatrix}

\begin{bmatrix} 650 & 300 \\ 700 & 330 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 3,850 & 1,500 \\ 4,700 & 1,830 \end{bmatrix}$

Answer

The matrix with elements 3,850 and 1,500 in the first row, and 4,700 and 1,830 in the second row.

The matrix representing the total revenue and total cost for Week 1 and Week 2 is found by multiplying matrix

Q

by matrix

P

. Since

Q

is a

2 \times 2

matrix (weeks by package types) and

P

is a

2 \times 2

matrix (package types by financial categories), the product

Q \times P

yields a

2 \times 2

matrix representing weeks by financial categories (revenue and cost).

Calculating each element:
- Row 1, Column 1 (Week 1 Revenue):

(20 \times 80) + (15 \times 150) = 1,600 + 2,250 = 3,850

- Row 1, Column 2 (Week 1 Cost):

(20 \times 30) + (15 \times 60) = 600 + 900 = 1,500

- Row 2, Column 1 (Week 2 Revenue):

(25 \times 80) + (18 \times 150) = 2,000 + 2,700 = 4,700

- Row 2, Column 2 (Week 2 Cost):

(25 \times 30) + (18 \times 60) = 750 + 1,080 = 1,830

This results in the matrix:

\begin{bmatrix} 3,850 & 1,500 \\ 4,700 & 1,830 \end{bmatrix}

Step-by-Step Solution

1

Identify the appropriate matrix operation to combine the weekly package sales with their respective price and cost.

We need to compute the product of matrix

Q

and matrix

P

(

Q \times P

), since multiplying a

2 \times 2

matrix of (weeks

\times

packages) by a

2 \times 2

matrix of (packages

\times

financial category) results in a

2 \times 2

matrix of (weeks

\times

financial category).

This alignment ensures that the package types cancel out, leaving the desired weekly revenue and cost breakdown.

2

Calculate the elements of the first row of the product matrix.

Row 1, Column 1 (Week 1 Revenue):

20(80) + 15(150) = 1,600 + 2,250 = 3,850

. Row 1, Column 2 (Week 1 Cost):

20(30) + 15(60) = 600 + 900 = 1,500

.

The first row represents Week 1 revenue and cost.

3

Calculate the elements of the second row of the product matrix.

Row 2, Column 1 (Week 2 Revenue):

25(80) + 18(150) = 2,000 + 2,700 = 4,700

. Row 2, Column 2 (Week 2 Cost):

25(30) + 18(60) = 750 + 1,080 = 1,830

.

The second row represents Week 2 revenue and cost.

Key Concept

Matrix multiplication represents linear combinations of variables, useful for calculating total cost and revenue in multi-variable business scenarios.

Question 2077Question →

Let matrix $A = \begin{bmatrix} 3 & -2 \\ 5 & 1 \end{bmatrix}$ and matrix $B = \begin{bmatrix} -2 & 3 \\ 2 & -3 \end{bmatrix}$ . If matrix $C$ is defined by the equation $C = 3A - 2B$ , what is the value of the element in the second row and first column of $C$ ?

Show answer & explanation

Answer: 11

Answer

The element in the second row and first column of matrix

C

is 11.

To find the element in the second row and first column of matrix

C

, we apply the operations defined by

C = 3A - 2B

directly to the corresponding elements of

A

and

B

. The element in row 2, column 1 of matrix

A

is

5

, and of matrix

B

is

2

. Computing

3(5) - 2(2)

yields

15 - 4 = 11

.

Step-by-Step Solution

1

Identify the elements in the second row and first column of matrices

A

and

B

.

A_{21} = 5

and

B_{21} = 2

.

To find a specific element of the resulting matrix

C = 3A - 2B

, we only need to perform the operations on the elements in the corresponding position.

2

Set up the equation for the element in the second row and first column of

C

.

C_{21} = 3A_{21} - 2B_{21}

Matrix addition, subtraction, and scalar multiplication are performed element-wise.

3

Substitute the identified values into the equation and compute the result.

C_{21} = 3(5) - 2(2) = 15 - 4 = 11

Evaluating the expression gives the value of the target element.

Key Concept

Matrix scalar multiplication and element-wise subtraction

Question 2078Question →

Let $P(x) = (3x^2 - 2x + 4)(2x - 5) - (4x^3 - 7x^2 + x - 3)$ . When $P(x)$ is simplified and written in standard form, what is the coefficient of the $x^2$ term?

Show answer & explanation

Answer: -12

Answer

The coefficient of the

x^2

term is -12.

Expanding

(3x^2 - 2x + 4)(2x - 5)

yields

6x^3 - 19x^2 + 18x - 20

. Distributing the negative sign across the second polynomial yields

-4x^3 + 7x^2 - x + 3

. Combining the

x^2

terms gives

-19x^2 + 7x^2 = -12x^2

, so the coefficient of the

x^2

term is -12.

Step-by-Step Solution

1

Expand the product of the trinomial and the binomial:

(3x^2 - 2x + 4)(2x - 5)

6x^3 - 19x^2 + 18x - 20

To find the expanded form of the first polynomial component before subtraction

2

Distribute the negative sign across the second polynomial:

-(4x^3 - 7x^2 + x - 3)

-4x^3 + 7x^2 - x + 3

To prepare the second polynomial for combination of like terms

3

Combine the like terms from the two expanded components

2x^3 - 12x^2 + 17x - 17

To write the entire polynomial in standard form and identify the coefficient of

x^2

Key Concept

Operations on Polynomials

Question 2079Question →

Let the functions $f$ and $g$ be defined by $f(x) = (x - 3)^2$ and $g(x) = 2x + 1$ . What is the value of the composite function $f(g(-2))$ ?

Show answer & explanation

Answer: 36

Answer

36

To evaluate the composite function

f(g(-2))

, evaluate from the inside out. First, evaluate the inner function

g(-2) = 2(-2) + 1 = -3

. Next, substitute this output value into the outer function

f(x)

to get

f(-3) = (-3 - 3)^2 = (-6)^2 = 36

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = -2

g(-2) = -3

Before evaluating the outer function

f

, we must determine the output of the inner function

g

at the given input value.

2

Evaluate the outer function

f(x)

at the result of the inner function

f(-3) = 36

Substitute the inner output

-3

as the input for

f(x) = (x - 3)^2

, giving

(-3 - 3)^2 = (-6)^2 = 36

.

Key Concept

Function composition and evaluation

Estimated Time:1m 0s

Question 2080Question →

If the quadratic equation $x^2 + kx + 25 = 0$ has exactly one real solution, and $k > 0$ , what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The value of

k

is

10

.

For any quadratic equation in standard form,

ax^2 + bx + c = 0

, the equation has exactly one real solution when the discriminant

b^2 - 4ac

is equal to

0

. Here, the equation is

x^2 + kx + 25 = 0

, so

a = 1

,

b = k

, and

c = 25

. Setting the discriminant to

0

gives

k^2 - 4(1)(25) = 0

, which simplifies to

k^2 - 100 = 0

. Solving for

k

yields

k = \pm 10

. Since we are given that

k > 0

, the only valid solution is

10

.

Step-by-Step Solution

1

Set the discriminant of the quadratic equation to zero.

b^2 - 4ac = 0

A quadratic equation has exactly one real solution if and only if its discriminant is zero.

2

Identify coefficients

a

,

b

, and

c

from the equation

x^2 + kx + 25 = 0

and substitute them into the discriminant equation.

k^2 - 4(1)(25) = 0

For

x^2 + kx + 25 = 0

, the coefficients are

a=1

,

b=k

, and

c=25

.

3

Simplify the equation and solve for

k

.

k^2 = 100

, so

k = 10

or

k = -10

Simplifying

k^2 - 100 = 0

gives

k^2 = 100

.

4

Apply the constraint

k > 0

to find the final value.

k = 10

The problem specifies that

k

must be greater than zero.

Key Concept

Using the discriminant (

b^2 - 4ac

) of a quadratic equation to determine when there is exactly one real solution.

Estimated Time:45s

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