All practice questions

Question 2081Question →

A line with a positive slope passes through the point $(0, -5)$ and is tangent to the parabola $y = x^2 - 6x + 11$ . What is the slope of this line?

Show answer & explanation

Answer: 2

Answer

The slope of the line is 2.

The correct slope is 2. Representing the line as

y = mx - 5

and setting it equal to the parabola

y = x^2 - 6x + 11

results in the quadratic equation

x^2 - (6 + m)x + 16 = 0

. For the line to be tangent, this equation must have exactly one real solution, meaning its discriminant must equal zero:

(6 + m)^2 - 4(1)(16) = 0

. Solving this gives

6 + m = 8

or

6 + m = -8

, which results in

m = 2

or

m = -14

. Since the problem specifies that the slope is positive, the value of

m

must be 2.

Step-by-Step Solution

1

Write the equation of the line in slope-intercept form.

y = mx - 5

, where

m > 0

.

The line passes through

(0, -5)

, which represents the y-intercept of the line.

2

Equate the line and the parabola to set up the system of equations.

x^2 - 6x + 11 = mx - 5

Setting the two expressions equal allows us to find the x-coordinates of any intersection points.

3

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - (6 + m)x + 16 = 0

Writing the equation in standard form identifies the coefficients

a = 1

,

b = -(6 + m)

, and

c = 16

needed for the discriminant.

4

Apply the condition for tangency by setting the discriminant to zero.

b^2 - 4ac = (-(6 + m))^2 - 4(1)(16) = 0 \implies (6 + m)^2 - 64 = 0

A line is tangent to a parabola if they touch at exactly one point, meaning the quadratic equation has exactly one real root (discriminant equals zero).

5

Solve the quadratic equation for the slope

m

and filter for the positive value.

(6+m)^2 = 64 \implies 6+m = \pm 8

, yielding

m = 2

or

m = -14

. Since the slope is positive,

m = 2

.

Solving the equation gives two possible slope values for tangent lines, and we select the positive slope as specified in the problem statement.

Key Concept

Determining tangency between a linear and quadratic equation by setting the discriminant of the intersection equation to zero.

Question 2082Question →

For all non-zero real numbers

p

and

q

, which of the following expressions is equivalent to

\frac{(p^2 + q^{-1})^2 - p^4}{q^{-2}}?

1

2p^2 q^2 + 1

2p^2 q + 1

2p^2 q^{-1} + 1

2p^2 + 1

Show answer & explanation

Answer: $2p^2 q + 1$

Answer

2p^2 q + 1

Expanding the binomial in the numerator yields

p^4 + 2p^2 q^{-1} + q^{-2}

. After subtracting

p^4

, the numerator is left as

2p^2 q^{-1} + q^{-2}

. Dividing this numerator term-by-term by the denominator

q^{-2}

gives

\frac{2p^2 q^{-1}}{q^{-2}} + \frac{q^{-2}}{q^{-2}}

. Applying the quotient property of exponents to each term yields

2p^2 q^{(-1) - (-2)} + 1 = 2p^2 q^1 + 1 = 2p^2 q + 1

.

Step-by-Step Solution

1

Expand the binomial in the numerator using the perfect square identity:

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

.

(p^2 + q^{-1})^2 = (p^2)^2 + 2(p^2)(q^{-1}) + (q^{-1})^2 = p^4 + 2p^2 q^{-1} + q^{-2}

To begin simplifying the numerator, we must resolve the exponent outside the parentheses.

2

Subtract

p^4

from the expanded numerator expression.

(p^4 + 2p^2 q^{-1} + q^{-2}) - p^4 = 2p^2 q^{-1} + q^{-2}

This simplifies the numerator by combining the

p^4

terms.

3

Divide each term in the simplified numerator by the denominator

q^{-2}

.

\frac{2p^2 q^{-1}}{q^{-2}} + \frac{q^{-2}}{q^{-2}} = 2p^2 q^{-1 - (-2)} + q^{-2 - (-2)}

To divide a polynomial by a monomial, we distribute the division to each term of the polynomial.

4

Apply the quotient rule of exponents,

\frac{x^a}{x^b} = x^{a-b}

, and simplify the terms.

2p^2 q^1 + q^0 = 2p^2 q + 1

Subtracting the exponents of

q

in each term simplifies the division to its final form.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 2083Question →

For all positive real numbers $x$ , which of the following expressions is equivalent to $\log_3(27x^4) - \log_3(3x^2)$ ?

2 + 6\log_3(x)

\log_3(24x^2)

2 + 2\log_3(x)

4\log_3(x)

3 + 2\log_3(x)

Show answer & explanation

Answer: $2 + 2\log_3(x)$

Answer

2 + 2\log_3(x)

The correct answer is found by applying the quotient property of logarithms to combine the terms, yielding

\log_3(9x^2)

. Then, applying the product property splits this into

\log_3(9) + \log_3(x^2)

. Finally, evaluating

\log_3(9) = 2

and using the power property to rewrite

\log_3(x^2)

as

2\log_3(x)

yields the simplified expression

2 + 2\log_3(x)

.

Step-by-Step Solution

1

Apply the quotient property of logarithms:

\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)

.

\log_3\left(\frac{27x^4}{3x^2}\right)

To combine the two logarithmic terms into a single logarithm.

2

Simplify the algebraic expression inside the logarithm.

\log_3(9x^2)

Dividing the coefficients (

27 \div 3 = 9

) and subtracting the exponents of the variable

x

(

4 - 2 = 2

).

3

Apply the product property of logarithms:

\log_b(CD) = \log_b(C) + \log_b(D)

.

\log_3(9) + \log_3(x^2)

To separate the constant and variable parts of the logarithmic argument.

4

Evaluate the numerical logarithm and apply the power property of logarithms:

\log_b(y^k) = k\log_b(y)

.

2 + 2\log_3(x)

Since

3^2 = 9

,

\log_3(9) = 2

, and the exponent of

x

can be moved in front of the logarithm as a multiplier.

Key Concept

Logarithmic and Exponential Expressions and Equations

Alternative Method

Alternatively, expand each logarithm first using the product and power properties of logarithms:
1.

\log_3(27x^4) = \log_3(27) + \log_3(x^4) = 3 + 4\log_3(x)

2.

\log_3(3x^2) = \log_3(3) + \log_3(x^2) = 1 + 2\log_3(x)

Subtracting the second expanded expression from the first yields:

(3 + 4\log_3(x)) - (1 + 2\log_3(x)) = 3 - 1 + 4\log_3(x) - 2\log_3(x) = 2 + 2\log_3(x)

.

Estimated Time:1m 30s

Question 2084Question →

A linear function $f(x) = mx + b$ satisfies the inequality $f(x + 3) - f(x) \geq 6$ for all real numbers $x$ , and its graph passes through the point $(1, -2)$ . If the region bounded by the graph of $f$ , the line $x = -2$ , the $x$ -axis, and the $y$ -axis has an area of $20$ square units, what is the value of $b$ ?

-10

-8

-6

-4

4

Show answer & explanation

Answer: -6

Answer

The value of

b

is

-6

.

To find the value of

b

, we first determine the constraints on the slope

m

. Since

f(x) = mx + b

, the difference

f(x+3) - f(x) = 3m

. Given

3m \geq 6

, we find

m \geq 2

. The graph of

f

passes through

(1, -2)

, which gives

-2 = m + b

, or

b = -2 - m

. Since

m \geq 2

, the

y

-intercept

b

must be less than or equal to

-4

, meaning the function is negative for the entire interval

[-2, 0]

. The region bounded by the graph of

f

,

x = -2

, the

x

-axis, and the

y

-axis forms a trapezoid below the

x

-axis. The vertical parallel sides of this trapezoid have lengths equal to the absolute values of the

y

-coordinates at

x = -2

and

x = 0

, which are

2m - b

and

-b

, respectively. The width is

2

. The area of the trapezoid is

\frac{(2m - b) + (-b)}{2} \times 2 = 2m - 2b

. Setting the area to

20

square units gives

2m - 2b = 20

, which simplifies to

m - b = 10

. Solving the system of equations

m + b = -2

and

m - b = 10

yields

m = 4

and

b = -6

.

Step-by-Step Solution

1

Find the constraint on the slope

m

using the given inequality

f(x+3) - f(x) \geq 6

.

3m \geq 6 \implies m \geq 2

.

The difference in function values over an interval of

3

for a linear function is

3

times the slope

m

.

2

Substitute the point

(1, -2)

into the equation

f(x) = mx + b

to find a relationship between

m

and

b

.

-2 = m(1) + b \implies b = -2 - m

.

The graph of the function must pass through the given coordinate point.

3

Determine the shape and boundaries of the region bounded by the graph of

f

,

x = -2

, the

x

-axis, and the

y

-axis.

A trapezoid below the

x

-axis with parallel vertical sides of lengths

2m - b

(at

x = -2

) and

-b

(at

x = 0

), and a horizontal width of

2

.

Since

m \geq 2

, the

y

-intercept

b

is at most

-4

, so the function is strictly negative on the interval

[-2, 0]

.

4

Set up the area formula for the trapezoid and set it equal to

20

to find another relation between

m

and

b

.

\frac{(2m-b) + (-b)}{2} \times 2 = 20 \implies 2m - 2b = 20 \implies m - b = 10

.

The area of a trapezoid is the average of the parallel side lengths multiplied by the width.

5

Solve the system of equations:

m + b = -2

and

m - b = 10

.

Adding the equations gives

2m = 8 \implies m = 4

. Substituting

m = 4

gives

b = -6

.

To find the specific value of

b

that satisfies both the point condition and the area condition.

Key Concept

Graphing linear equations, calculating area of bounded regions on the coordinate plane, and using linear slope and point-intercept forms.

Alternative Method

Instead of solving the system of equations algebraically, we can express the line in point-slope form as

y + 2 = m(x - 1)

. At

x = 0

,

y = -m - 2

, and at

x = -2

,

y = -3m - 2

. The heights of the trapezoid are

m + 2

and

3m + 2

(since they are below the

x

-axis and

m \geq 2

). The area of the trapezoid is

\frac{(m+2) + (3m+2)}{2} \times 2 = 4m + 4

. Setting

4m + 4 = 20

yields

4m = 16 \implies m = 4

, which gives

b = -m - 2 = -6

.

Estimated Time:3m 0s

Question 2085Question →

If $y$ is a real number such that $|3 - \frac{1}{2}y| = 5$ , what is the sum of all possible values of $y$ ?

-4

12

16

-12

20

Show answer & explanation

Answer: 12

Answer

12

Solving the equation

|3 - \frac{1}{2}y| = 5

requires setting up two cases:

3 - \frac{1}{2}y = 5

and

3 - \frac{1}{2}y = -5

. Solving the first case gives

y = -4

, and solving the second case gives

y = 16

. The sum of these two solutions is

-4 + 16 = 12

.

Step-by-Step Solution

1

Set up the two equations represented by the absolute value expression.

3 - \frac{1}{2}y = 5

and

3 - \frac{1}{2}y = -5

An absolute value equation

|u| = C

(where

C \ge 0

) splits into two cases:

u = C

and

u = -C

.

2

Solve the first equation

3 - \frac{1}{2}y = 5

.

y = -4

Subtracting 3 from both sides yields

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

. Multiplying both sides by

-2

isolates

y

.

3

Solve the second equation

3 - \frac{1}{2}y = -5

.

y = 16

Subtracting 3 from both sides yields

-\frac{1}{2}y = -8

. Multiplying both sides by

-2

isolates

y

.

4

Calculate the sum of all possible values of

y

.

12

Add the two solutions together:

-4 + 16 = 12

.

Key Concept

Absolute Value Equations

Estimated Time:1m 30s

Question 2086Question →

A line graphed on the $(x, y)$ coordinate plane has a $y$ -intercept of $-1$ and passes through the point $(3, 8)$ . If the point $(a, 14)$ also lies on this line, what is the value of $a$ ?

Show answer & explanation

Answer: 5

Answer

5

The line has a y-intercept of

-1

, which corresponds to the point

(0, -1)

. The slope is calculated as

m = \frac{8 - (-1)}{3 - 0} = 3

. The equation of the line is

y = 3x - 1

. Substituting

(a, 14)

into this equation gives

14 = 3a - 1

, which simplifies to

15 = 3a

, and solving for

a

yields

5

.

Step-by-Step Solution

1

Identify the coordinate representation of the y-intercept.

The point is

(0, -1)

.

A y-intercept of

-1

means the line crosses the y-axis at the point where

x = 0

, which is

(0, -1)

.

2

Calculate the slope of the line.

m = 3

Using the slope formula

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

with the points

(0, -1)

and

(3, 8)

, we get

m = \frac{8 - (-1)}{3 - 0} = 3

.

3

Write the equation of the line and solve for

a

.

a = 5

The equation in slope-intercept form is

y = 3x - 1

. Substituting the point

(a, 14)

yields

14 = 3a - 1

. Adding 1 to both sides gives

15 = 3a

, so

a = 5

.

Key Concept

Finding the equation of a line from two points and solving for a missing coordinate.

Estimated Time:1m 30s

Question 2087Question →

Matrices $C$ and $D$ are defined as follows:

C = \begin{bmatrix} 4 & 1 \\ -2 & k \end{bmatrix}, \quad D = \begin{bmatrix} 3 & -1 \\ 5 & 2 \end{bmatrix}

If the element in the second row and first column of the product matrix $CD$ is $-11$ , what is the value of $k$ ?

Show answer & explanation

Answer: -1

Answer

The value of

k

is

-1

.

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

CD

, we take the dot product of the second row of

C

, which is

\begin{bmatrix} -2 & k \end{bmatrix}

, and the first column of

D

, which is

\begin{bmatrix} 3 \\ 5 \end{bmatrix}

. This gives

(-2)(3) + (k)(5) = -6 + 5k

. Setting this equal to the given value of

-11

yields the equation

-6 + 5k = -11

. Adding

6

to both sides gives

5k = -5

, and dividing by

5

results in

k = -1

.

Step-by-Step Solution

1

Identify the second row of matrix

C

and the first column of matrix

D

.

Row 2 of

C

is

\begin{bmatrix} -2 & k \end{bmatrix}

and Column 1 of

D

is

\begin{bmatrix} 3 \\ 5 \end{bmatrix}

.

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

CD

, we must compute the dot product of the second row of the first matrix (

C

) and the first column of the second matrix (

D

).

2

Multiply the corresponding elements of the row and column and add the products.

The element at row 2, column 1 of

CD

is

(-2)(3) + (k)(5) = -6 + 5k

.

This defines the matrix multiplication rule for that specific position in the resulting matrix.

3

Set the expression

-6 + 5k

equal to the given value of

-11

and solve the linear equation for

k

.

-6 + 5k = -11 \implies 5k = -5 \implies k = -1

.

Solving the equation yields the value of the unknown variable

k

.

Key Concept

2x2 Matrix Multiplication

Question 2088Question →

If $\log_3(x + 4) = 4$ , what is the value of $x$ ?

Show answer & explanation

Answer: 77

Answer

The value of

x

is 77.

The correct answer is 77. To solve the equation

\log_3(x + 4) = 4

, convert the equation from its logarithmic form to its exponential form. Since

\log_b(a) = c

means

b^c = a

, the equation becomes

3^4 = x + 4

. Calculating

3^4

yields 81. Thus,

81 = x + 4

. Subtracting 4 from both sides gives

x = 77

.

Step-by-Step Solution

1

Rewrite the logarithmic equation in exponential form.

x + 4 = 3^4

By definition,

\log_b(a) = c

is equivalent to

b^c = a

.

2

Calculate the value of the exponential expression.

3^4 = 81

3^4 = 3 \times 3 \times 3 \times 3 = 81

.

3

Solve for

x

by isolating the variable.

x = 77

Subtract 4 from both sides of the equation:

81 - 4 = 77

.

Key Concept

Converting logarithmic equations to exponential form

Question 2089Question →

For a real constant $k$ , the quadratic equation $x^2 - 2(k - 2)x + (k^2 - 3k + 2) = 0$ has two real roots, $r_1$ and $r_2$ . What is the value of $k$ that minimizes the sum of the squares of these roots, $r_1^2 + r_2^2$ ?

Show answer & explanation

Answer: 2

Answer

The value of

k

that minimizes the sum of the squares of the roots is 2.

Applying Vieta's formulas and algebraic identities, the sum of the squares of the roots is expressed as

2k^2 - 10k + 12

. The discriminant condition for the roots to be real requires

k \leq 2

. Since the vertex of the upward-opening parabola

2k^2 - 10k + 12

is at

k = 2.5

, the function is strictly decreasing for all

k \leq 2

. Thus, the minimum value on the interval

k \leq 2

occurs at the boundary

k = 2

.

Step-by-Step Solution

1

Use Vieta's formulas to find the sum and product of the roots.

r_1 + r_2 = 2(k - 2)

and

r_1 r_2 = k^2 - 3k + 2

Vieta's formulas relate the roots of a quadratic equation

Ax^2 + Bx + C = 0

to its coefficients by

r_1 + r_2 = -B/A

and

r_1 r_2 = C/A

.

2

Express the sum of the squares of the roots,

r_1^2 + r_2^2

, in terms of

k

.

r_1^2 + r_2^2 = 2k^2 - 10k + 12

Since

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

, substituting the Vieta relations gives

[2(k-2)]^2 - 2(k^2 - 3k + 2) = 4(k^2 - 4k + 4) - 2k^2 + 6k - 4 = 2k^2 - 10k + 12

.

3

Determine the condition for the quadratic equation to have real roots using the discriminant.

\Delta = -4k + 8 \geq 0 \implies k \leq 2

For the roots

r_1

and

r_2

to be real, the discriminant

\Delta = B^2 - 4AC

must be greater than or equal to 0.

4

Minimize the quadratic function

f(k) = 2k^2 - 10k + 12

subject to the constraint

k \leq 2

.

k = 2

The parabola

f(k)

opens upwards with its vertex at

k = 2.5

. For

k \leq 2

, the function is strictly decreasing, meaning its minimum value on this interval occurs at the upper boundary,

k = 2

.

Key Concept

Quadratic Equations and the Quadratic Formula

Question 2090Question →

A commercial printing press charges a business a setup fee of $100$ plus $0.15$ per brochure printed. To encourage green practices, the press offers a recycling discount equal to $30$ less than $0.05$ per brochure printed. If the business has a budget of at most $380$ for their brochure order, what is the maximum number of brochures they can print?

1,250

1,550

2,500

3,100

3,500

Show answer & explanation

Answer: 2,500

Answer

2,500

The correct answer is 2,500 brochures. The total cost is the setup fee of

100 plus the printing cost of

0.15 per brochure, minus the discount of

30 less than

0.05 per brochure. Setting

b

as the number of brochures, the discount is

0.05b - 30

. The total cost equation is

100 + 0.15b - (0.05b - 30) = 130 + 0.10b

. Setting this less than or equal to

380

yields

130 + 0.10b \leq 380 \Rightarrow 0.10b \leq 250 \Rightarrow b \leq 2,500

.

Step-by-Step Solution

1

Define the variable and write the algebraic expressions for the cost components.

Let

b

be the number of brochures printed. The total cost before the discount is

100 + 0.15b

. The recycling discount is represented as

0.05b - 30

.

Establishing clear variables and translating the textual relationships into mathematical expressions is the first step in solving word problems.

2

Set up the inequality representing the budget constraint.

100 + 0.15b - (0.05b - 30) \leq 380

The total cost (original cost minus the discount) must be at most the budget of

380

.

3

Simplify the left side of the inequality.

100 + 0.15b - 0.05b + 30 \leq 380 \Rightarrow 130 + 0.10b \leq 380

Distributing the subtraction sign across the discount expression and combining like terms simplifies the inequality.

4

Solve for the variable

b

.

0.10b \leq 250 \Rightarrow b \leq 2500

Isolating

b

gives the maximum number of brochures that can be printed.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 2091Question →

A circle in the standard $(x, y)$ coordinate plane has its center at $(5, -2)$ and a radius of $7$ units. Which of the following is an equation of this circle?

x^2 + y^2 - 10x + 4y = 20

x^2 + y^2 + 10x - 4y = 20

x^2 + y^2 - 10x + 4y = -22

x^2 + y^2 = 70

x^2 + y^2 - 10x + 4y = 7

Show answer & explanation

Answer: $x^2 + y^2 - 10x + 4y = 20$

Answer

The equation

x^2 + y^2 - 10x + 4y = 20

The standard form equation of a circle is

(x - h)^2 + (y - k)^2 = r^2

, where

(h, k)

is the center and

r

is the radius. Substituting

h = 5

,

k = -2

, and

r = 7

yields

(x - 5)^2 + (y + 2)^2 = 49

. Expanding both binomials gives

x^2 - 10x + 25 + y^2 + 4y + 4 = 49

. Combining the constant terms on the left side gives

x^2 + y^2 - 10x + 4y + 29 = 49

. Subtracting

29

from both sides results in the equation

x^2 + y^2 - 10x + 4y = 20

.

Step-by-Step Solution

1

Write down the standard equation of a circle.

(x - h)^2 + (y - k)^2 = r^2

, where

(h, k)

is the center and

r

is the radius.

This is the fundamental formula relating the geometric properties of a circle to its algebraic representation.

2

Substitute the given values into the standard equation.

(x - 5)^2 + (y - (-2))^2 = 7^2

, which simplifies to

(x - 5)^2 + (y + 2)^2 = 49

.

Substituting the center

(5, -2)

and radius

7

into the standard equation sets up the expression for expansion.

3

Expand the squared binomials.

x^2 - 10x + 25 + y^2 + 4y + 4 = 49

.

Expanding

(x - 5)^2

into

x^2 - 10x + 25

and

(y + 2)^2

into

y^2 + 4y + 4

allows conversion to the general form.

4

Simplify and rearrange the equation to match the form of the options.

x^2 + y^2 - 10x + 4y + 29 = 49

, which simplifies to

x^2 + y^2 - 10x + 4y = 20

.

Combining the constants and subtracting

29

from both sides of the equation yields the final simplified general form.

Key Concept

Equation of a Circle in General Form

Estimated Time:1m 0s

Question 2092Question →

An ellipse in the standard coordinate plane is defined by the equation $9x^2 + 25y^2 - 36x + 50y - 164 = 0$ . What is the distance between the two foci of this ellipse?

8

4

16

2\sqrt{34}

12

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. Rearranging the given equation

9x^2 + 25y^2 - 36x + 50y - 164 = 0

by completing the square gives the standard form

\frac{(x-2)^2}{25} + \frac{(y+1)^2}{9} = 1

. In this standard horizontal ellipse equation, the semi-major axis squared is

a^2 = 25

and the semi-minor axis squared is

b^2 = 9

. The focal distance

c

from the center to each focus is found using the relation

c^2 = a^2 - b^2

, which yields

c = \sqrt{25 - 9} = 4

. Since the distance between the two foci is

2c

, the final distance is

2(4) = 8

.

Step-by-Step Solution

1

Group the

x

and

y

terms and move the constant to the right-hand side.

(9x^2 - 36x) + (25y^2 + 50y) = 164

Grouping like variables allows us to factor out coefficients before completing the square.

2

Factor out the leading coefficients of the quadratic terms.

9(x^2 - 4x) + 25(y^2 + 2y) = 164

Completing the square requires the quadratic terms inside the parentheses to have a coefficient of 1.

3

Complete the square for both variables by adding the square of half the linear coefficients inside the parentheses, and balance the equation by adding the distributed values to the right side.

9(x^2 - 4x + 4) + 25(y^2 + 2y + 1) = 164 + 9(4) + 25(1)

which simplifies to

9(x-2)^2 + 25(y+1)^2 = 225

.

This rewrites the quadratic expressions into perfect square binomials.

4

Divide both sides of the equation by 225 to write the equation in standard form.

\frac{(x-2)^2}{25} + \frac{(y+1)^2}{9} = 1

The standard form of an ellipse equation is equal to 1.

5

Identify the values of

a^2

and

b^2

to calculate the distance

c

from the center to each focus.

a^2 = 25

and

b^2 = 9

. Using

c^2 = a^2 - b^2

, we get

c^2 = 25 - 9 = 16

, so

c = 4

.

For a horizontal ellipse, the larger denominator is

a^2

and the smaller is

b^2

, and the focal distance satisfies

c^2 = a^2 - b^2

.

6

Multiply the focal distance from the center by 2 to find the total distance between the two foci.

\text{Distance} = 2c = 2(4) = 8

.

The distance between the two foci is the length of the segment connecting them, which is centered at

(2, -1)

and extends

c

units in both horizontal directions.

Key Concept

Rewriting an ellipse equation in standard form to determine its key geometric features including its foci.

Estimated Time:1m 30s

Question 2093Question →

For all real values of $q$ that satisfy the inequality $8 - 2|3q - 1| \ge -6$ , which of the following inequality expressions represents the complete solution set for $q$ ?

$

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

q \le -2\text{ or }q \ge \frac{8}{3}

q \le \frac{8}{3}

q \ge \frac{8}{3}

q \ge 0

Show answer & explanation

Answer: $ $-2 \le q \le \frac{8}{3}$

Answer

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

The correct answer is the interval containing all real numbers between

-2

and

\frac{8}{3}

inclusive. This is determined by subtracting 8 from both sides of the inequality, dividing by

-2

(which reverses the inequality sign to yield

|3q - 1| \le 7

), expressing this as the compound inequality

-7 \le 3q - 1 \le 7

, and isolating

q

.

Step-by-Step Solution

1

Subtract 8 from both sides of the inequality to isolate the absolute value term.

-2|3q - 1| \ge -14

To solve for the variable, we must first isolate the term containing the absolute value by performing inverse operations.

2

Divide both sides by

-2

and reverse the inequality sign.

|3q - 1| \le 7

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

3

Rewrite the absolute value inequality as a compound inequality.

-7 \le 3q - 1 \le 7

An inequality of the form

|x| \le d

(where

d \ge 0

) is equivalent to the compound inequality

-d \le x \le d

.

4

Add 1 to all three parts of the compound inequality.

-6 \le 3q \le 8

Adding 1 eliminates the constant term from the middle section of the inequality.

5

Divide all three parts by 3 to solve for

q

.

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

Dividing by 3 isolates the variable

q

, yielding the complete solution set.

Key Concept

Solving multi-step absolute value inequalities, including reversing the inequality sign when dividing by a negative number and setting up a compound inequality to represent both positive and negative cases.

Question 2094Question →

A system of equations consists of the line $2x - y = 5$ and the parabola $y = x^2 - 4x + c$ , where $c$ is a constant. If the line and the parabola intersect at two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ such that the positive difference between their $x$ -coordinates is 2, what is the value of $c$ ?

0

3

5

8

11

Show answer & explanation

Answer: 3

Answer

3

To find the constant

c

, we equate the line and the parabola equations:

x^2 - 4x + c = 2x - 5

. Bringing all terms to one side gives the quadratic equation

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

. Applying the quadratic formula, the

x

-coordinates of the intersection points are

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

. The positive difference between these coordinates is

(3 + \sqrt{4 - c}) - (3 - \sqrt{4 - c}) = 2\sqrt{4 - c}

. Setting this difference equal to the given value of 2 gives

2\sqrt{4 - c} = 2

, which simplifies to

\sqrt{4 - c} = 1

. Squaring both sides yields

4 - c = 1

, which gives

c = 3

. This corresponds to the correct option.

Step-by-Step Solution

1

Equate the equations for the line and the parabola to set up the equation for their intersection points.

x^2 - 4x + c = 2x - 5

The intersection points of the system of equations occur where the

y

-values are equal.

2

Rearrange the equation into standard quadratic form:

ax^2 + bx + c = 0

.

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

Grouping the terms allows us to identify the coefficients:

a = 1

,

b = -6

, and the constant term is

c + 5

.

3

Apply the quadratic formula to find the

x

-coordinates of the intersection points in terms of

c

.

x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(c + 5)}}{2} = 3 \pm \sqrt{4 - c}

The quadratic formula yields the roots

x_1 = 3 + \sqrt{4 - c}

and

x_2 = 3 - \sqrt{4 - c}

.

4

Set up an equation representing the positive difference between the

x

-coordinates and solve for

c

.

(3 + \sqrt{4 - c}) - (3 - \sqrt{4 - c}) = 2 \implies 2\sqrt{4 - c} = 2 \implies \sqrt{4 - c} = 1 \implies 4 - c = 1 \implies c = 3

The difference between the two coordinates is given as 2, which allows us to isolate and solve for

c

.

Key Concept

Solving systems of linear and non-linear equations by finding the intersection of a line and a parabola and using root properties to determine unknown constants

Alternative Method

Instead of using the quadratic formula, you can apply Vieta's formulas. Let the roots of

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

be

x_1

and

x_2

. Vieta's formulas state that

x_1 + x_2 = 6

and

x_1 x_2 = c + 5

. We are given that

|x_1 - x_2| = 2

. Squaring this equation gives

(x_1 - x_2)^2 = 4

. Since

(x_1 - x_2)^2 = (x_1 + x_2)^2 - 4x_1 x_2

, we substitute the known values:

6^2 - 4(c + 5) = 4 \implies 36 - 4c - 20 = 4 \implies 16 - 4c = 4 \implies 4c = 12 \implies c = 3

.

Estimated Time:2m 0s

Question 2095Question →

An arithmetic sequence $a_1, a_2, a_3, \dots$ has a first term $a_1 = a$ and a non-zero common difference $d$ . A geometric sequence $g_1, g_2, g_3, \dots$ has a first term $g_1 = a$ and a common ratio $r > 1$ . The third term of the arithmetic sequence is equal to the second term of the geometric sequence ( $a_3 = g_2$ ), and the seventh term of the arithmetic sequence is equal to the third term of the geometric sequence ( $a_7 = g_3$ ). If the sum of the first five terms of the arithmetic sequence is 150, what is the value of the fifth term of the geometric sequence, $g_5$ ?

240

480

800

96

810,000

Show answer & explanation

Answer: 240

Answer

The fifth term of the geometric sequence is 240.

The correct answer is 240. The term relationships

a_3 = g_2

and

a_7 = g_3

translate to

a + 2d = ar

and

a + 6d = ar^2

. Expressing the first equation as

2d = a(r - 1)

and substituting it into the second equation yields

a + 3a(r - 1) = ar^2

. Dividing by

a

gives the quadratic equation

r^2 - 3r + 2 = 0

. Since

r > 1

, we find

r = 2

, which implies

d = a/2

. The sum of the first five terms of the arithmetic sequence is

5(a + 2d) = 150

, which simplifies to

a + 2d = 30

. Substituting

d = a/2

gives

2a = 30

, so

a = 15

. The fifth term of the geometric sequence is then

g_5 = a \cdot r^4 = 15 \cdot 2^4 = 240

.

Step-by-Step Solution

1

Express the given sequence term relationships in terms of

a

,

d

, and

r

.

a_3 = a + 2d

,

a_7 = a + 6d

,

g_2 = ar

, and

g_3 = ar^2

.

To represent the equality of terms algebraically.

2

Solve the system of equations

a + 2d = ar

and

a + 6d = ar^2

for the common ratio

r

.

From the first equation,

2d = a(r - 1) \Rightarrow d = \frac{a(r - 1)}{2}

. Substituting this into the second equation gives

a + 3a(r - 1) = ar^2

. Dividing by

a

(since

a \neq 0

) yields

r^2 - 3r + 2 = 0

, which factors as

(r-1)(r-2) = 0

. Since

r > 1

, we must have

r = 2

. Thus,

d = a/2

.

To find the relationship between the arithmetic common difference, the geometric common ratio, and the first term.

3

Use the sum of the first five terms of the arithmetic sequence to solve for

a

.

S_5 = 5(a + 2d) = 150 \Rightarrow a + 2d = 30

. Substituting

d = a/2

gives

a + a = 30 \Rightarrow 2a = 30 \Rightarrow a = 15

.

To determine the numerical value of the first term.

4

Calculate the fifth term of the geometric sequence.

g_5 = a \cdot r^4 = 15 \cdot 2^4 = 15 \cdot 16 = 240

.

To find the requested term of the geometric sequence.

Key Concept

Solving systems of linear and exponential relationships using arithmetic and geometric sequence properties.

Question 2096Question →

For the functions $f(x) = (x - 3)^2 - 2$ and $g(x) = |3x - 10|$ , what is the value of the composite function $g(f(1))$ ?

-4

4

24

28

40

Show answer & explanation

Answer: 4

Answer

The correct answer is 4

Evaluating

f(1)

gives

(1 - 3)^2 - 2 = (-2)^2 - 2 = 4 - 2 = 2

. Substituting this result into

g(x)

gives

g(2) = |3(2) - 10| = |6 - 10| = |-4| = 4

. Therefore, the value of the composite function is 4.

Step-by-Step Solution

1

Evaluate the inner function

f(x)

at

x = 1

.

f(1) = (1 - 3)^2 - 2 = (-2)^2 - 2 = 4 - 2 = 2

To evaluate the composite function

g(f(1))

, the output of the inner function

f(1)

must be calculated first.

2

Substitute the output

f(1) = 2

into the outer function

g(x)

.

g(2) = |3(2) - 10| = |6 - 10| = |-4| = 4

The output of the inner function becomes the input for the outer function

g(x)

.

Key Concept

Evaluating a composite function by first calculating the value of the inner function and then using that result as the input for the outer function.

Estimated Time:1m 0s

Question 2097Question →

A linear function contains the points shown in the table below:

$x$	$y$
$k - 2$	$5$
$k + 1$	$14$
$2k + 3$	$35$

What is the $y$ -intercept of the line representing this function?

Show answer & explanation

Answer: -4

Answer

The

y

-intercept of the line representing this function is

-4

.

By using the first two coordinates, the constant slope of the linear function is determined to be

3

. Equating the slope between the second and third coordinates to

3

yields the parameter

k = 5

. Substituting this back into the coordinate expressions gives the points

(3, 5)

and

(6, 14)

. Using the point-slope form, the equation of the line is

y = 3x - 4

, meaning the

y

-intercept is

-4

.

Step-by-Step Solution

1

Calculate the slope of the line using the first two points:

(k-2, 5)

and

(k+1, 14)

.

m = \frac{14 - 5}{(k+1) - (k-2)} = \frac{9}{3} = 3

Since the function is linear, the rate of change (slope) remains constant between any two points.

2

Express the slope using the second and third points,

(k+1, 14)

and

(2k+3, 35)

, set it equal to

3

, and solve for

k

.

\frac{35 - 14}{(2k+3) - (k+1)} = 3 \implies \frac{21}{k+2} = 3 \implies 3(k+2) = 21 \implies k+2 = 7 \implies k = 5

The slope between the second and third points must also equal the constant slope of

3

.

3

Substitute

k = 5

back into the coordinates to determine the actual points on the line.

The points are

(3, 5)

,

(6, 14)

, and

(13, 35)

.

This provides concrete coordinates that can be used to write the equation of the line.

4

Use the point-slope form with the point

(3, 5)

and slope

m = 3

to write the equation of the line, then convert to slope-intercept form.

y - 5 = 3(x - 3) \implies y - 5 = 3x - 9 \implies y = 3x - 4

Converting to slope-intercept form (

y = mx + b

) directly gives the

y

-intercept as the constant term

b

.

Key Concept

Determining the equation and intercepts of a line using the constant slope property of linear functions.

Alternative Method

Once the slope

m = 3

and a point such as

(3, 5)

are established, substitute these values directly into the slope-intercept equation

y = mx + b

to solve for

b

. This gives

5 = 3(3) + b \implies 5 = 9 + b \implies b = -4

.

Estimated Time:2m 30s

Question 2098Question →

A parabola in the standard $(x, y)$ coordinate plane is defined by the equation $y = \frac{1}{12}x^2 - 2x + 13$ . What is the $y$ -coordinate of the focus of this parabola?

Show answer & explanation

Answer: 4

Answer

The y-coordinate of the focus is 4.

The standard form of the parabola is

y - 1 = \frac{1}{12}(x - 12)^2

. Comparing this to

y - k = \frac{1}{4p}(x - h)^2

gives the vertex

(h, k) = (12, 1)

and

4p = 12 \implies p = 3

. Since the parabola opens upward, the focus is at

(12, 1 + 3) = (12, 4)

, making the

y

-coordinate 4.

Step-by-Step Solution

1

Complete the square to rewrite the equation in standard vertex form.

y = \frac{1}{12}(x - 12)^2 + 1

Rewriting the general quadratic equation into standard form allows us to directly identify the vertex and focal parameters.

2

Equate the coefficients to find the focal distance

p

and vertex

(h, k)

.

Vertex

(h, k) = (12, 1)

and

p = 3

The standard vertex form of a vertical parabola is

y - k = \frac{1}{4p}(x - h)^2

. Setting

\frac{1}{4p} = \frac{1}{12}

gives

p = 3

.

3

Calculate the focus coordinates

(h, k + p)

.

Focus

= (12, 4)

, so the

y

-coordinate is

4

For an upward-opening parabola, the focus is located

p

units directly above the vertex.

Key Concept

Rewriting a quadratic equation into standard vertex form to find the properties of a parabola, including its vertex and focus.

Question 2099Question →

If the expression

\frac{(x^3 y^2)^3 \cdot (x^{-2} y^{-1})^2}{x^k y^3}

is equivalent to $x^2 y$ for all non-zero real numbers $x$ and $y$ , what is the value of the exponent $k$ ?

Show answer & explanation

Answer: 3

Answer

The value of the exponent

k

is 3.

Applying exponent rules simplifies the numerator of the expression to

x^5 y^4

. Dividing this by the denominator

x^k y^3

yields

x^{5-k} y

. Equating the exponent of

x

to 2 in the target expression

x^2 y

gives

5 - k = 2

, which solves to

k = 3

.

Step-by-Step Solution

1

Apply the power of a power rule

(a^m)^n = a^{m \cdot n}

to simplify each factor in the numerator.

(x^3 y^2)^3 = x^9 y^6

and

(x^{-2} y^{-1})^2 = x^{-4} y^{-2}

To raise a product to a power, raise each factor to that power by multiplying the exponents.

2

Multiply the two simplified factors in the numerator together by adding the exponents of like bases.

x^9 y^6 \cdot x^{-4} y^{-2} = x^5 y^4

When multiplying exponential expressions with the same base, add their exponents:

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

.

3

Divide the numerator by the denominator by subtracting the exponents of like bases.

\frac{x^5 y^4}{x^k y^3} = x^{5-k} y

When dividing exponential expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator:

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

.

4

Set the exponent of

x

in the simplified expression equal to the exponent of

x

in the target expression

x^2 y

, and solve for

k

.

5 - k = 2 \implies k = 3

For the expressions to be equivalent for all non-zero real numbers, the corresponding exponents of like bases must be equal.

Key Concept

Properties of exponents (power of a power, product of powers, and quotient of powers rules)

Estimated Time:1m 30s

Question 2100Question →

A manufacturer calculates the net monthly operating profit $P$ (in thousands of dollars) using the expression $\frac{8 - 3x}{2}$ , where $x$ is the number of maintenance hours scheduled. The auxiliary support costs are modeled by the expression $\frac{2x - 1}{3}$ thousand dollars. For the plant to be considered efficient, the net operating profit minus the auxiliary support costs must be at least $3$ thousand dollars. Which of the following inequalities represents the range of maintenance hours, $x$ , that satisfy this efficiency requirement?

x \leq \frac{4}{13}

x \geq \frac{8}{13}

x \leq \frac{8}{13}

x \leq \frac{4}{5}

x \geq \frac{12}{5}

Show answer & explanation

Answer: $x \leq \frac{8}{13}$

Answer

The range of maintenance hours that satisfy the efficiency requirement is

x \leq \frac{8}{13}

.

First, translate the word problem into a mathematical inequality. The net monthly operating profit minus the auxiliary support costs must be at least 3, which translates to:

\frac{8 - 3x}{2} - \frac{2x - 1}{3} \geq 3

To solve this inequality, find the least common denominator (LCD) of 2 and 3, which is 6. Multiply every term in the inequality by 6:

3(8 - 3x) - 2(2x - 1) \geq 18

Distribute the constants on the left side, paying careful attention to distribute the negative sign:

24 - 9x - 4x + 2 \geq 18

Combine like terms:

26 - 13x \geq 18

Subtract 26 from both sides:

-13x \geq -8

Divide both sides by -13. Because we are dividing by a negative number, the direction of the inequality sign must be reversed:

x \leq \frac{8}{13}

Thus, the option stating

x \leq \frac{8}{13}

is correct.

Step-by-Step Solution

1

Translate the word problem into a mathematical inequality.

\frac{8 - 3x}{2} - \frac{2x - 1}{3} \geq 3

The net operating profit minus the auxiliary support costs must be at least 3, which translates to a subtraction operation set greater than or equal to 3.

2

Multiply all terms in the inequality by the least common denominator (LCD) of 2 and 3, which is 6.

3(8 - 3x) - 2(2x - 1) \geq 18

Multiplying by the LCD eliminates the denominators and simplifies the equation for algebraic manipulation.

3

Distribute the constants on the left side of the inequality.

24 - 9x - 4x + 2 \geq 18

Removing the parentheses is required to combine like terms. Be careful to distribute the negative sign:

-2 \times -1 = +2

.

4

Combine like terms on the left side of the inequality.

26 - 13x \geq 18

Simplifying the left-hand side reduces the inequality to a standard two-step linear inequality.

5

Subtract 26 from both sides of the inequality to isolate the variable term.

-13x \geq -8

This isolates the variable term

-13x

on the left side.

6

Divide both sides by -13 and reverse the direction of the inequality sign.

x \leq \frac{8}{13}

Dividing both sides of an inequality by a negative number requires reversing the inequality sign to maintain equivalence.

Key Concept

Solving multi-step linear inequalities, including fraction clearance and reversing the inequality sign when dividing by a negative number.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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