All practice questions

Question 2101Question →

If $x$ and $y$ are positive real numbers such that $\log_x(y) = 2$ and $\log_4(x) + \log_2(y) = 5$ , what is the value of $y$ ?

Show answer & explanation

Answer: 16

Answer

16

Applying the definition of logarithms to

\log_x(y) = 2

yields

y = x^2

. Substituting this relationship into the second equation gives

\log_4(x) + \log_2(x^2) = 5

. Using the change of base formula, we rewrite

\log_4(x)

as

\frac{\log_2(x)}{\log_2(4)} = \frac{1}{2}\log_2(x)

, and using the power property, we rewrite

\log_2(x^2)

as

2\log_2(x)

. Combining the terms results in

\frac{5}{2}\log_2(x) = 5

, which simplifies to

\log_2(x) = 2

. Converting this back to exponential form gives

x = 2^2 = 4

. Substituting this value back into the relation

y = x^2

yields

y = 4^2 = 16

.

Step-by-Step Solution

1

Apply the definition of a logarithm to the first equation.

y = x^2

By definition,

\log_b(a) = c

is equivalent to

b^c = a

.

2

Substitute

y = x^2

into the second equation.

\log_4(x) + \log_2(x^2) = 5

Substitution reduces the system to a single equation in terms of

x

.

3

Express both logarithmic terms using base 2 properties.

\frac{1}{2}\log_2(x) + 2\log_2(x) = 5

The change of base formula gives

\log_4(x) = \frac{\log_2(x)}{\log_2(4)} = \frac{1}{2}\log_2(x)

, and the power property gives

\log_2(x^2) = 2\log_2(x)

.

4

Combine the coefficients and solve for

x

.

\log_2(x) = 2 \Rightarrow x = 2^2 = 4

Adding the coefficients gives

\frac{5}{2}\log_2(x) = 5

. Multiplying by

\frac{2}{5}

isolates

\log_2(x)

to solve for

x

.

5

Calculate the value of

y

using the relation from Step 1.

y = 4^2 = 16

Since

y = x^2

and

x = 4

, evaluating the square gives the final answer.

Key Concept

Solving systems of logarithmic equations using base conversion and definition of logarithms

Estimated Time:2m 0s

Question 2102Question →

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

Show answer & explanation

Answer: 3

Answer

The

x

-coordinate of the focus is 3.

The equation

y^2 = 12x

represents a parabola with vertex at the origin opening to the right. The standard form for such a parabola is

y^2 = 4px

, where the focus is located at

(p, 0)

. By setting

4p = 12

, we find

p = 3

. Thus, the focus is

(3, 0)

, and its

x

-coordinate is 3.

Step-by-Step Solution

1

Identify the standard form of the parabola's equation.

The equation

y^2 = 12x

fits the standard form of a horizontal parabola with its vertex at the origin,

y^2 = 4px

.

This allows us to relate the given equation to the coordinate of the focus, which is located at

(p, 0)

.

2

Solve for the parameter

p

.

4p = 12

, which gives

p = 3

.

By equating the coefficients of

x

from the given equation and the standard form, we can find the value of

p

.

3

Determine the focus coordinate.

The focus is at

(3, 0)

, so the

x

-coordinate is 3.

The focus of a parabola of the form

y^2 = 4px

has coordinates

(p, 0)

.

Key Concept

Focus of a Parabola

Question 2103Question →

In the standard $(x,y)$ coordinate plane, two opposite vertices of a square are $(1, 2)$ and $(4, 6)$ . If all four vertices of the square lie in the first quadrant, what is the $x$ -coordinate of the vertex that is closest to the $y$ -axis?

Show answer & explanation

Answer: 0.5

Answer

The correct answer is

0.5

. The vertex closest to the

y

-axis is

(0.5, 5.5)

, which has an

x

-coordinate of

0.5

.

The diagonals of a square are perpendicular, equal in length, and bisect each other. Using the given opposite vertices

(1, 2)

and

(4, 6)

, we find the midpoint to be

(2.5, 4)

. The vector between them is

(3, 4)

with length

5

. A perpendicular vector of length

5

is

(-4, 3)

. Adding and subtracting half of this vector,

(2, -1.5)

, from the midpoint yields the other two vertices:

(0.5, 5.5)

and

(4.5, 2.5)

. Since all four vertices are in the first quadrant, we compare their

x

-coordinates:

1

,

4

,

0.5

, and

4.5

. The smallest

x

-coordinate is

0.5

, which represents the vertex closest to the

y

-axis.

Step-by-Step Solution

1

Find the midpoint of the given diagonal.

The midpoint is

M(2.5, 4)

.

The diagonals of a square bisect each other at their common midpoint.

2

Determine the vector representing the given diagonal

AB

and its length.

\vec{AB} = (3, 4)

and its length is

5

.

The vector is found by subtracting coordinates:

(4 - 1, 6 - 2) = (3, 4)

, and its length is

\sqrt{3^2 + 4^2} = 5

.

3

Find a perpendicular vector of the same length to represent the other diagonal.

A perpendicular vector is

(-4, 3)

.

The dot product of

(3, 4)

and

(-4, 3)

is

3(-4) + 4(3) = 0

, and its length is

\sqrt{(-4)^2 + 3^2} = 5

.

4

Calculate the coordinates of the other two vertices of the square.

The vertices are

C(0.5, 5.5)

and

D(4.5, 2.5)

.

The vertices are located at

M \pm \frac{1}{2}\vec{CD}

, which gives

(2.5, 4) \pm ( -2, 1.5 )

.

5

Determine which of the four vertices is closest to the

y

-axis and identify its

x

-coordinate.

The vertex closest to the

y

-axis is

C(0.5, 5.5)

, and its

x

-coordinate is

0.5

.

The distance to the

y

-axis is the

x

-coordinate of the point. Comparing the

x

-coordinates

1

,

4

,

0.5

, and

4.5

, the smallest value is

0.5

.

Key Concept

Properties of diagonals of a square on a coordinate plane, including midpoint and perpendicularity.

Alternative Method

Instead of using vectors, one can set up a system of equations. Let

(x, y)

be one of the unknown vertices. Since it forms a right isosceles triangle with the midpoint

(2.5, 4)

and has distance

2.5

from it along a line with slope

-3/4

, we can write the equation of the line as

y - 4 = -0.75(x - 2.5)

and use the distance formula

(x - 2.5)^2 + (y - 4)^2 = 2.5^2

to solve for

x

and

y

.

Estimated Time:3m 0s

Question 2104Question →

In the coordinate plane, the line representing the linear equation $3x - 4y = 8$ contains the point $(-4, b)$ . What is the value of $b$ ?

Show answer & explanation

Answer: -5

Answer

The value of

b

is

-5

.

Substituting the coordinates

(-4, b)

into the equation

3x - 4y = 8

yields

3(-4) - 4b = 8

. Simplifying gives

-12 - 4b = 8

. Adding

12

to both sides results in

-4b = 20

. Finally, dividing by

-4

gives

b = -5

.

Step-by-Step Solution

1

Substitute the point

(-4, b)

into the equation

3x - 4y = 8

.

3(-4) - 4(b) = 8

Since the point lies on the line, its coordinates must satisfy the line's equation.

2

Simplify the constant term.

-12 - 4b = 8

Multiplying

3

by

-4

yields

-12

.

3

Isolate the variable term by adding

12

to both sides.

-4b = 20

Adding

12

to

8

gives

20

.

4

Solve for

b

by dividing both sides by

-4

.

b = -5

Dividing

20

by

-4

yields

-5

.

Key Concept

Determining an unknown coordinate of a point on a line by substitution into the linear equation.

Question 2105Question →

In the standard $(x, y)$ coordinate plane, a circle passes through the points $A(-1, -2)$ and $B(3, 6)$ . The center of the circle, $C$ , lies on the line with the equation $y = 2x - 5$ . What is the radius of this circle?

Show answer & explanation

Answer: 5

Answer

The radius of the circle is 5.

The perpendicular bisector of the segment connecting

A(-1, -2)

and

B(3, 6)

passes through their midpoint

(1, 2)

with a slope of

-\frac{1}{2}

, giving the equation

x + 2y = 5

. Solving the system of equations with

y = 2x - 5

yields the center at

C(3, 1)

. The distance from

C(3, 1)

to

A(-1, -2)

is

\sqrt{(3 - (-1))^2 + (1 - (-2))^2} = \sqrt{4^2 + 3^2} = 5

.

Step-by-Step Solution

1

Find the midpoint and slope of the segment

AB

connecting

A(-1, -2)

and

B(3, 6)

.

Midpoint

M = (1, 2)

and slope

m = 2

.

The center of any circle passing through

A

and

B

must lie on the perpendicular bisector of segment

AB

.

2

Determine the equation of the perpendicular bisector of

AB

.

x + 2y = 5

The perpendicular bisector passes through the midpoint

M(1, 2)

and has a slope that is the negative reciprocal of the slope of

AB

, which is

-\frac{1}{2}

.

3

Find the intersection point of the perpendicular bisector

x + 2y = 5

and the given line

y = 2x - 5

.

Center

C(3, 1)

The center of the circle lies on both the perpendicular bisector of

AB

and the line

y = 2x - 5

.

4

Calculate the distance from the center

C(3, 1)

to point

A(-1, -2)

using the distance formula.

Radius

r = 5

The radius is the distance from the center of the circle to any point on its circumference.

Key Concept

The perpendicular bisector of a chord of a circle passes through the center of that circle.

Question 2106Question →

What is the complete solution set for the inequality $|4 - 3v| \ge 7$ ?

v \ge -1

or

v \le \frac{11}{3}

-1 \le v \le \frac{11}{3}

v \le -1

or

v \ge \frac{11}{3}

v \le -1

v \le -7

or

v \ge 7

Show answer & explanation

Answer: $v \le -1$ or $v \ge \frac{11}{3}$

Answer

The solution set is the union of two open-ended intervals, representing all values of

v

such that

v

is less than or equal to

-1

or

v

is greater than or equal to

\frac{11}{3}

.

To solve the absolute value inequality

|4 - 3v| \ge 7

, we rewrite it as two separate inequalities:

4 - 3v \ge 7

or

4 - 3v \le -7

. For the first case, subtracting 4 from both sides of

4 - 3v \ge 7

results in

-3v \ge 3

. Dividing by

-3

and reversing the inequality sign gives

v \le -1

. For the second case, subtracting 4 from both sides of

4 - 3v \le -7

results in

-3v \le -11

. Dividing by

-3

and reversing the inequality sign gives

v \ge \frac{11}{3}

. Combining these gives the correct solution set:

v \le -1

or

v \ge \frac{11}{3}

.

Step-by-Step Solution

1

Set up two separate linear inequalities based on the absolute value inequality template.

4 - 3v \ge 7

or

4 - 3v \le -7

An absolute value inequality of the form

|A| \ge B

where

B > 0

is equivalent to the compound statement

A \ge B

or

A \le -B

.

2

Solve the first inequality:

4 - 3v \ge 7

. Subtract 4 from both sides, then divide by

-3

and reverse the inequality sign.

v \le -1

Subtracting 4 yields

-3v \ge 3

. Dividing by the negative number

-3

requires reversing the direction of the inequality sign.

3

Solve the second inequality:

4 - 3v \le -7

. Subtract 4 from both sides, then divide by

-3

and reverse the inequality sign.

v \ge \frac{11}{3}

Subtracting 4 yields

-3v \le -11

. Dividing by the negative number

-3

requires reversing the direction of the inequality sign.

4

Combine the individual solutions into a single compound statement.

v \le -1

or

v \ge \frac{11}{3}

The solution to a 'greater than or equal to' absolute value inequality is the union of the individual solutions.

Key Concept

Solving absolute value inequalities by splitting them into two linear cases and correctly reversing the inequality sign when dividing by a negative number.

Alternative Method

Alternatively, test test-values from each interval. For instance, choosing

v = 0

(which is in the middle interval) yields

|4 - 0| = 4 \ge 7

, which is false. Choosing

v = -2

(which is in the left interval) yields

|4 - 3(-2)| = |10| = 10 \ge 7

, which is true. Choosing

v = 4

(which is in the right interval since

4 > \frac{11}{3}

) yields

|4 - 12| = |-8| = 8 \ge 7

, which is true. This confirms the solution set must cover

v \le -1

and

v \ge \frac{11}{3}

.

Estimated Time:1m 30s

Question 2107Question →

A landscaping company purchases bags of grass seed for $30$ each and bags of fertilizer for $18$ each. For a large project, the company purchased $2$ fewer than $1.5$ times as many bags of fertilizer as bags of grass seed. If the total cost of the grass seed and fertilizer was $420$ , how many bags of fertilizer did the company purchase?

Show answer & explanation

Answer: 10

Answer

The company purchased 10 bags of fertilizer.

The correct answer is 10. By writing the number of fertilizer bags as

f = 1.5g - 2

and substituting this expression into the total cost equation

30g + 18f = 420

, we get

30g + 18(1.5g - 2) = 420

. Simplifying the expression leads to

57g - 36 = 420

, which yields

g = 8

. Substituting

g = 8

back into the relationship for

f

gives

f = 1.5(8) - 2 = 10

.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

g

represent the number of grass seed bags and

f

represent the number of fertilizer bags.

Setting up variables is the first step in translating the word problem into solvable algebraic equations.

2

Translate the relationship between the quantities of bags into an equation.

f = 1.5g - 2

The problem states that the number of fertilizer bags purchased is 2 fewer than 1.5 times the number of grass seed bags purchased.

3

Write the total cost equation using the prices and variables.

30g + 18f = 420

The total cost of

420

is the sum of the cost of the grass seed (

30

per bag) and the fertilizer (

18

per bag).

4

Substitute the equation from Step 2 into the cost equation from Step 3.

30g + 18(1.5g - 2) = 420

Substituting allows us to solve a single linear equation with one variable.

5

Distribute the 18 through the parentheses and combine like terms.

57g - 36 = 420

Distributing gives

18 \times 1.5g = 27g

and

18 \times -2 = -36

. Combining

30g

and

27g

yields

57g

.

6

Isolate the variable term by adding 36 to both sides of the equation.

57g = 456

To solve for

g

, we must isolate the term containing the variable.

7

Divide both sides by 57 to find the value of

g

.

g = 8

This division yields the number of grass seed bags purchased.

8

Substitute the value of

g

back into the relation for

f

to find the final answer.

f = 1.5(8) - 2 = 10

The question asks for the number of fertilizer bags (

f

), not grass seed bags (

g

).

Key Concept

Solving a linear equation by substitution and distributing across linear terms.

Estimated Time:1m 30s

Question 2108Question →

If $a$ , $b$ , and $c$ are positive real numbers greater than $1$ such that $\log_b(a) = \frac{3}{2}$ and $\log_c(b) = \frac{4}{3}$ , what is the value of $\log_a(abc)$ ?

\frac{1}{3}

\frac{13}{6}

\frac{23}{9}

3

\frac{9}{2}

Show answer & explanation

Answer: $\frac{13}{6}$

Answer

\frac{13}{6}

The correct answer is obtained by expanding

\log_a(abc)

into

\log_a(a) + \log_a(b) + \log_a(c)

. Since

\log_a(a) = 1

and

\log_a(b)

is the reciprocal of

\log_b(a)

, we find

\log_a(b) = \frac{2}{3}

. Using the chain rule of change of base, we calculate

\log_a(c) = \log_a(b) \cdot \log_b(c) = \frac{2}{3} \cdot \frac{3}{4} = \frac{1}{2}

. Adding these three parts together yields

1 + \frac{2}{3} + \frac{1}{2} = \frac{13}{6}

.

Step-by-Step Solution

1

Apply the logarithmic product rule to expand the target expression.

\log_a(abc) = \log_a(a) + \log_a(b) + \log_a(c)

The logarithm of a product is equal to the sum of the logarithms of its individual factors.

2

Evaluate the first two terms of the expanded expression.

\log_a(a) = 1

, and since

\log_b(a) = \frac{3}{2}

, we have

\log_a(b) = \frac{1}{\log_b(a)} = \frac{2}{3}

.

The logarithm of a base to itself is always

1

, and the base reciprocal property states that

\log_x(y) = \frac{1}{\log_y(x)}

.

3

Evaluate the third term

\log_a(c)

using base properties and the change of base formula.

Since

\log_c(b) = \frac{4}{3}

, we have

\log_b(c) = \frac{3}{4}

. Then,

\log_a(c) = \log_a(b) \cdot \log_b(c) = \frac{2}{3} \cdot \frac{3}{4} = \frac{1}{2}

.

By applying the change of base formula, we can rewrite

\log_a(c)

in terms of base

b

as

\frac{\log_b(c)}{\log_b(a)} = \log_a(b) \cdot \log_b(c)

.

4

Sum the three evaluated logarithmic terms together.

1 + \frac{2}{3} + \frac{1}{2} = \frac{6}{6} + \frac{4}{6} + \frac{3}{6} = \frac{13}{6}

Combining the values of the individual terms gives the final value of the expanded expression.

Key Concept

Properties of Logarithms and Change of Base Formula

Estimated Time:2m 0s

Question 2109Question →

A hyperbola in the standard coordinate plane is represented by the equation $9x^2 - 16y^2 - 36x - 32y - 124 = 0$ . Which of the following equations represents one of the asymptotes of this hyperbola?

y = \frac{3}{4}x - \frac{1}{2}

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

y = \frac{3}{4}x - \frac{5}{2}

y = \frac{3}{4}x + \frac{5}{2}

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

Show answer & explanation

Answer: $y = \frac{3}{4}x - \frac{5}{2}$

Answer

y = \frac{3}{4}x - \frac{5}{2}

The correct equation is found by rewriting the hyperbola equation in standard form

\frac{(x - 2)^2}{16} - \frac{(y + 1)^2}{9} = 1

through completing the square. This indicates a horizontal hyperbola centered at

(2, -1)

with

a = 4

and

b = 3

. The asymptotes are given by

y - k = \pm \frac{b}{a}(x - h)

. Substituting the values gives

y + 1 = \pm \frac{3}{4}(x - 2)

. Simplifying the positive case results in the correct equation.

Step-by-Step Solution

1

Group the

x

-terms and

y

-terms and factor out the coefficients of the squared terms.

9(x^2 - 4x) - 16(y^2 + 2y) = 124

Grouping prepares the algebraic expression for completing the square for both variables.

2

Complete the square for both the

x

and

y

expressions by adding the balanced constants to the right side.

9(x - 2)^2 - 16(y + 1)^2 = 144

Adding

9(4) = 36

and subtracting

16(1) = 16

to the right side balances the equation after completing the square.

3

Divide both sides by

144

to express the equation in the standard form of a hyperbola.

\frac{(x - 2)^2}{16} - \frac{(y + 1)^2}{9} = 1

The standard form equation

\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1

reveals the center

(h, k)

and the semi-axes values

a

and

b

.

4

Identify the key parameters of the hyperbola and state the general formula for its asymptotes.

Center

(h, k) = (2, -1)

,

a = 4

,

b = 3

. The asymptotes are given by

y - k = \pm \frac{b}{a}(x - h)

, which simplifies to

y + 1 = \pm \frac{3}{4}(x - 2)

.

For a horizontal hyperbola, the rise-over-run slope of the asymptotes is governed by the ratio

\frac{b}{a}

.

5

Simplify the positive slope case to find the matching slope-intercept equation.

y = \frac{3}{4}x - \frac{5}{2}

Distributing the slope gives

y + 1 = \frac{3}{4}x - \frac{3}{2}

, and subtracting

1

from both sides yields the final equation.

Key Concept

Rewriting a hyperbola equation using completing the square to find its asymptotes.

Alternative Method

Instead of completing the square entirely, find the center of the hyperbola by taking partial derivatives. The derivative with respect to

x

is

18x - 36 = 0 \implies x = 2

. The derivative with respect to

y

is

-32y - 32 = 0 \implies y = -1

. Thus, the center is

(2, -1)

. The slope of the asymptotes can be found from the ratio of the square roots of the coefficients of the quadratic terms:

m = \pm \sqrt{\frac{9}{16}} = \pm \frac{3}{4}

. Using the point-slope form with the center

(2, -1)

and slope

\frac{3}{4}

gives

y + 1 = \frac{3}{4}(x - 2)

, which simplifies directly to

y = \frac{3}{4}x - \frac{5}{2}

.

Estimated Time:2m 30s

Question 2110Question →

For all real numbers $x$ such that $x \neq 1$ , the function $f$ is defined by $f(x) = \frac{12}{x - 1}$ . For all real numbers $x$ , the function $g$ is defined by $g(x) = x^2 + 2$ . What is the value of the composite function $f(g(3))$ ?

Show answer & explanation

Answer: 1.2

Answer

1.2

To evaluate the composite function

f(g(3))

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

g(3) = 3^2 + 2 = 11

. Next, substitute this result into the outer function

f(x)

to get

f(11) = \frac{12}{11 - 1} = \frac{12}{10} = 1.2

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = 3

.

g(3) = 11

To evaluate a composite function of the form

f(g(x))

at a given value, we must first find the output of the inner function,

g(x)

, at that value.

2

Substitute the output from Step 1 as the input for the outer function

f(x)

and evaluate.

f(g(3)) = 1.2

Using the result

g(3) = 11

as the input for

f(x)

gives

f(11) = \frac{12}{11 - 1} = 1.2

.

Key Concept

Function Composition and Evaluation

Question 2111Question →

In the standard $(x,y)$ coordinate plane, a line with a negative slope passes through the point $(3, 4)$ . If the sum of the line's $x$ -intercept and $y$ -intercept is $14$ , which of the following could be the slope of this line?

-\frac{3}{4}

-\frac{4}{3}

-\frac{2}{3}

-4

\frac{4}{3}

Show answer & explanation

Answer: $-\frac{4}{3}$

Answer

-\frac{4}{3}

The correct answer is

-\frac{4}{3}

. By representing the line in point-slope form as

y - 4 = m(x - 3)

, we determine that the

y

-intercept is at

(0, -3m + 4)

and the

x

-intercept is at

\left(3 - \frac{4}{m}, 0\right)

. Adding these intercepts together and setting the sum equal to

14

gives the equation

-3m + 7 - \frac{4}{m} = 14

. Simplifying this equation leads to the quadratic expression

3m^2 + 7m + 4 = 0

, which factors as

(3m + 4)(m + 1) = 0

. This yields two possible negative slopes:

-\frac{4}{3}

and

-1

. Among the choices,

-\frac{4}{3}

is the only matching option.

Step-by-Step Solution

1

Write the general equation of a line passing through

(3, 4)

with slope

m

.

y - 4 = m(x - 3)

Using the point-slope form

y - y_1 = m(x - x_1)

is a direct way to parameterize the line in terms of its slope.

2

Find the

y

-intercept of the line by setting

x = 0

.

y = -3m + 4

The

y

-intercept is the point where the line crosses the

y

-axis, which occurs when

x = 0

.

3

Find the

x

-intercept of the line by setting

y = 0

and solving for

x

.

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

The

x

-intercept is the point where the line crosses the

x

-axis, which occurs when

y = 0

.

4

Set the sum of the

x

-intercept and

y

-intercept equal to

14

.

(-3m + 4) + \left(3 - \frac{4}{m}\right) = 14

This translates the given condition that the sum of the intercepts is

14

.

5

Simplify the equation and clear the fraction by multiplying by

-m

.

3m^2 + 7m + 4 = 0

Grouping like terms yields

-3m - \frac{4}{m} - 7 = 0

. Multiplying by

-m

transforms it into a standard quadratic equation.

6

Factor the quadratic equation to solve for

m

.

(3m + 4)(m + 1) = 0 \implies m = -\frac{4}{3}

or

m = -1

Factoring helps find the values of

m

that satisfy the equation. Both values represent lines with negative slopes.

Key Concept

Using the point-slope form of a linear equation to find intercepts and solving the resulting quadratic equation to determine the slope.

Alternative Method

An alternative approach is using the intercept form of a line,

\frac{x}{a} + \frac{y}{b} = 1

, where

a

and

b

are the

x

- and

y

-intercepts. Since the point

(3, 4)

lies on the line, we have

\frac{3}{a} + \frac{4}{b} = 1

. We are also given

a + b = 14 \implies b = 14 - a

. Substituting

b

into the equation gives

\frac{3}{a} + \frac{4}{14-a} = 1

. Solving this equation by finding a common denominator results in

3(14-a) + 4a = a(14-a) \implies 42 + a = 14a - a^2 \implies a^2 - 13a + 42 = 0

. Factoring gives

(a-6)(a-7) = 0

, so

a = 6

or

a = 7

. If

a = 6

, then

b = 8

, and the slope

m = -\frac{b}{a} = -\frac{8}{6} = -\frac{4}{3}

. If

a = 7

, then

b = 7

, and the slope

m = -\frac{7}{7} = -1

. This confirms the possible slopes.

Estimated Time:2m 30s

Question 2112Question →

In the standard $(x, y)$ coordinate plane, a hyperbola is defined by the equation $9x^2 - 16y^2 - 54x - 64y - 127 = 0$ . What is the shortest distance from the focus of the hyperbola with the larger $x$ -coordinate to the asymptote with the positive slope?

\frac{15}{7}

3

4

5

\frac{17}{5}

Show answer & explanation

Answer: 3

Answer

3

The correct answer is 3. Completing the square for

9x^2 - 16y^2 - 54x - 64y - 127 = 0

yields the standard form equation

\frac{(x-3)^2}{16} - \frac{(y+2)^2}{9} = 1

. The focus with the larger

x

-coordinate is at

(8, -2)

and the asymptote with the positive slope is

3x - 4y - 17 = 0

. Applying the point-to-line distance formula yields a distance of 3.

Step-by-Step Solution

1

Group the

x

-terms and

y

-terms and move the constant to the right side of the equation.

9(x^2 - 6x) - 16(y^2 + 4y) = 127

Grouping the terms allows us to complete the square for the

x

and

y

variables separately.

2

Complete the square for both the

x

and

y

expressions, adjusting the right side of the equation by adding the weighted constants.

9(x^2 - 6x + 9) - 16(y^2 + 4y + 4) = 127 + 9(9) - 16(4) \implies 9(x-3)^2 - 16(y+2)^2 = 144

Completing the square allows us to write the quadratic expressions as perfect squares to put the equation in standard form.

3

Divide both sides of the equation by 144 to obtain the standard form of the hyperbola.

\frac{(x-3)^2}{16} - \frac{(y+2)^2}{9} = 1

The standard form of a horizontal hyperbola is

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

, which reveals the center

(h, k)

and the semi-axes

a

and

b

.

4

Identify the key parameters of the hyperbola: center,

a

,

b

, and calculate the focal distance

c

.

Center is

(3, -2)

,

a = 4

,

b = 3

, and

c = \sqrt{a^2 + b^2} = \sqrt{16 + 9} = 5

.

These parameters are required to find the coordinates of the focus and the equation of the asymptote.

5

Find the coordinates of the focus with the larger

x

-coordinate.

Focus is

(3 + 5, -2) = (8, -2)

.

For a horizontal hyperbola, the foci are located at

(h \pm c, k)

. The focus with the larger

x

-coordinate is at

(h+c, k)

.

6

Determine the equation of the asymptote with the positive slope.

The asymptote equation is

y + 2 = \frac{3}{4}(x - 3) \implies 3x - 4y - 17 = 0

.

The asymptotes of a horizontal hyperbola are given by

y - k = \pm \frac{b}{a}(x - h)

. The one with the positive slope uses

+\frac{b}{a}

.

7

Use the point-to-line distance formula

d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}

to find the distance from the focus

(8, -2)

to the asymptote line

3x - 4y - 17 = 0

.

d = \frac{|3(8) - 4(-2) - 17|}{\sqrt{3^2 + (-4)^2}} = \frac{|24 + 8 - 17|}{\sqrt{25}} = \frac{15}{5} = 3

.

Calculating this gives the shortest distance from the focus to the asymptote.

Key Concept

Rewriting the general equation of a hyperbola into standard form by completing the square, identifying its center, foci, and asymptotes, and applying the distance formula from a point to a line.

Question 2113Question →

For a certain quadratic equation $0.25x^2 - kx + 4.5 = 0$ , where $k$ is a constant, the difference between the two real solutions is exactly $3$ . What is the positive value of $k$ ?

Show answer & explanation

Answer: 2.25

Answer

The positive value of

k

is

2.25

.

By applying the root difference formula

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

with

a = 0.25

,

b = -k

, and

c = 4.5

, we get the equation

\frac{\sqrt{k^2 - 4.5}}{0.25} = 3

. This simplifies to

4\sqrt{k^2 - 4.5} = 3

. Dividing by

4

and squaring both sides gives

k^2 - 4.5 = 0.5625

, which means

k^2 = 5.0625

. Taking the positive square root yields

k = 2.25

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation and express the formula for the difference of the roots.

The coefficients are

a = 0.25

,

b = -k

, and

c = 4.5

. The difference between the roots is

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

.

This sets up the algebraic relationship between the root difference and the coefficients of the quadratic equation.

2

Substitute the coefficients and the given root difference of

3

into the difference formula.

\frac{\sqrt{(-k)^2 - 4(0.25)(4.5)}}{0.25} = 3

This creates an equation in terms of the variable

k

using the given problem constraints.

3

Simplify the equation and isolate the radical term.

4\sqrt{k^2 - 4.5} = 3 \implies \sqrt{k^2 - 4.5} = 0.75

Simplifying the fraction makes it easier to isolate the radical prior to squaring.

4

Square both sides of the equation to eliminate the radical and solve for

k^2

.

k^2 - 4.5 = 0.5625 \implies k^2 = 5.0625

Squaring is the inverse operation of the square root, allowing us to solve for

k^2

.

5

Calculate the positive square root of

5.0625

to find the value of

k

.

k = 2.25

Since the question asks for the positive value of

k

, we choose the positive square root.

Key Concept

Relating the difference of the roots of a quadratic equation to its coefficients using the discriminant and the quadratic formula.

Question 2114Question →

A boutique chocolatier packages two types of gift boxes: Standard and Deluxe. Each Standard box contains $4$ dark chocolates, and each Deluxe box contains $8$ dark chocolates. The chocolatier plans to prepare a batch of boxes such that the number of Standard boxes is exactly twice the number of Deluxe boxes. If the chocolatier has a total of $160$ dark chocolates available and uses all of them for this batch, what is the total number of boxes (both Standard and Deluxe combined) they can make?

Show answer & explanation

Answer: 30

Answer

The total number of boxes they can make is 30.

By defining the number of Deluxe boxes as

x

, the number of Standard boxes is

2x

. The equation for the total number of dark chocolates is

4(2x) + 8x = 160

. Solving this gives

16x = 160

, which simplifies to

x = 10

. Thus, there are

10

Deluxe boxes and

20

Standard boxes, for a combined total of

30

boxes.

Step-by-Step Solution

1

Define variables for the number of boxes of each type.

Let

x

be the number of Deluxe boxes and

2x

be the number of Standard boxes.

We are given that the number of Standard boxes is exactly twice the number of Deluxe boxes.

2

Set up an equation representing the total number of dark chocolates used.

4(2x) + 8x = 160

Each Standard box requires

4

dark chocolates, each Deluxe box requires

8

, and the total used is

160

.

3

Solve the equation for

x

.

16x = 160 \Rightarrow x = 10

Combining like terms gives

16x = 160

, and dividing both sides by

16

yields

x = 10

.

4

Calculate the total number of boxes.

x + 2x = 30

The total number of boxes is the sum of Deluxe boxes (

10

) and Standard boxes (

20

).

Key Concept

Translating verbal descriptions of relationships and totals into linear equations to solve algebraic word problems.

Estimated Time:1m 30s

Question 2115Question →

A circle in the standard $(x,y)$ coordinate plane has its center at $(2, 3)$ and passes through the point $(8, 11)$ . What is the length of the radius of this circle?

10

14

28

70

100

Show answer & explanation

Answer: 10

Answer

10

The correct answer is 10. The radius of the circle is the distance from the center

(2,3)

to the point

(8,11)

on the circle. Substituting these coordinates into the distance formula yields

\sqrt{(8-2)^2 + (11-3)^2} = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10

.

Step-by-Step Solution

1

Identify the formula for the radius of a circle given its center and a point on the circle.

The radius

r

is the distance between the center

(x_1, y_1) = (2, 3)

and the point

(x_2, y_2) = (8, 11)

.

By definition, the radius of a circle is the straight-line distance from the center to any point on its outer boundary.

2

Substitute the coordinates into the distance formula

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

.

r = \sqrt{(8 - 2)^2 + (11 - 3)^2}

This sets up the calculation for the horizontal and vertical changes between the center and the point on the circle.

3

Simplify the terms inside the parentheses and evaluate the exponents.

r = \sqrt{6^2 + 8^2} = \sqrt{36 + 64}

Following the order of operations, we first perform the subtraction within the parentheses, and then evaluate the squares.

4

Add the values inside the radical and take the square root.

r = \sqrt{100} = 10

We must sum the terms under the square root before taking the square root of the total sum.

Key Concept

Using the distance formula to find the radius of a circle given its center and a point on the circle.

Estimated Time:45s

Question 2116Question →

For what value of the constant $k$ does the quadratic equation $0.5x^2 - 1.5x + k = 0$ have two complex roots of the form $a \pm 2i$ , where $a$ is a real number?

0.875

1.5625

2.125

3.125

6.25

Show answer & explanation

Answer: 3.125

Answer

3.125

Applying the quadratic formula to

0.5x^2 - 1.5x + k = 0

yields

x = 1.5 \pm \sqrt{2.25 - 2k}

. For the roots to be of the form

a \pm 2i

, the imaginary component must be

2i

, which requires the radicand

2.25 - 2k

to equal

(2i)^2 = -4

. Solving the equation

2.25 - 2k = -4

yields

k = 3.125

. This corresponds to the option containing

3.125

.

Step-by-Step Solution

1

Identify the coefficients

a

,

b

, and

c

of the given quadratic equation

0.5x^2 - 1.5x + k = 0

.

a = 0.5

,

b = -1.5

, and

c = k

.

These coefficients are required to substitute into the quadratic formula.

2

Apply the quadratic formula,

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

, to express the roots of the equation.

x = \frac{-(-1.5) \pm \sqrt{(-1.5)^2 - 4(0.5)k}}{2(0.5)} = \frac{1.5 \pm \sqrt{2.25 - 2k}}{1} = 1.5 \pm \sqrt{2.25 - 2k}

.

Substituting the coefficients simplifies the expression for the roots of the quadratic equation.

3

Set the radical term equal to the imaginary component

2i

to find the value of

k

that yields complex roots of the form

a \pm 2i

.

\sqrt{2.25 - 2k} = 2i \Rightarrow 2.25 - 2k = (2i)^2 \Rightarrow 2.25 - 2k = -4

.

For the roots to have an imaginary part of

2i

, the term under the square root must equal

(2i)^2

, which is

-4

.

4

Solve the linear equation

2.25 - 2k = -4

for

k

.

-2k = -4 - 2.25 \Rightarrow -2k = -6.25 \Rightarrow k = 3.125

.

Isolating

k

gives the constant value that satisfies the root condition.

Key Concept

Using the quadratic formula and discriminant to determine the value of a parameter for a specified type of complex roots.

Question 2117Question →

A line is defined by the equation $y = 3x + k$ , where $k$ is a constant. This line intersects the parabola $y = x^2 - x + 2$ at two distinct points, $P$ and $Q$ . If the midpoint of the line segment $PQ$ lies on the line $y = 2x + 7$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of

k

is

5

.

Equating the equations of the line and the parabola gives a quadratic equation

x^2 - 4x + (2 - k) = 0

. The average of the roots of this quadratic equation gives the

x

-coordinate of the midpoint,

x_m = 2

. Substituting this into the first line's equation gives the

y

-coordinate of the midpoint,

y_m = 6 + k

. Since the midpoint

(2, 6+k)

lies on the line

y = 2x + 7

, we substitute these coordinates to get

6 + k = 11

, which simplifies to

k = 5

.

Step-by-Step Solution

1

Equate the equations of the line and the parabola.

x^2 - 4x + (2 - k) = 0

To find the

x

-coordinates of the intersection points

P

and

Q

.

2

Determine the sum of the

x

-coordinates and find the midpoint's

x

-coordinate.

x_m = 2

By Vieta's formulas, the sum of the roots of the quadratic equation is

4

. The

x

-coordinate of the midpoint is the average of these roots:

4 / 2 = 2

.

3

Find the

y

-coordinate of the midpoint.

y_m = 6 + k

Because the midpoint lies on the line segment

PQ

, its coordinates must satisfy the equation of the line passing through

P

and

Q

, which is

y = 3x + k

.

4

Substitute the midpoint coordinates into the second line's equation and solve for

k

.

k = 5

We are given that the midpoint lies on the line

y = 2x + 7

.

Key Concept

Systems of Linear and Quadratic Equations and Midpoint Properties

Question 2118Question →

In the standard $(x,y)$ coordinate plane, a line has an $x$ -intercept of $(a, 0)$ , where $a \neq 0$ , and a $y$ -intercept of $(0, 2a)$ . If the line passes through the point $(4, -3)$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2.5

Answer

The value of

a

is

2.5

.

Representing the intercepts as

(a, 0)

and

(0, 2a)

lets us find the slope of the line, which is

m = \frac{2a - 0}{0 - a} = -2

. The line can then be written as

y = -2x + 2a

. Substituting the point

(4, -3)

into the equation yields

-3 = -2(4) + 2a

, which simplifies to

2a = 5

, and therefore

a = 2.5

.

Step-by-Step Solution

1

Find the slope of the line in terms of the variable

a

.

The slope is

m = -2

.

Applying the slope formula to the points

(a, 0)

and

(0, 2a)

yields

m = \frac{2a - 0}{0 - a} = -2

.

2

Write the general equation of the line.

The equation is

y = -2x + 2a

.

The slope is

-2

and the

y

-intercept is

2a

, so the equation in slope-intercept form is

y = mx + b

.

3

Substitute the coordinates of the point

(4, -3)

to find

a

.

a = 2.5

.

Substituting

x = 4

and

y = -3

into

y = -2x + 2a

gives

-3 = -8 + 2a

, which simplifies to

2a = 5

and

a = 2.5

.

Key Concept

Linear Equations and Graphing

Question 2119Question →

Two functions are defined as $f(x) = (x - 3)^2 - 16$ and $g(x) = |x - 1| - 8$ . If $g(f(x)) = 0$ , what is the sum of all positive values of $x$ ?

4

8

10

14

24

Show answer & explanation

Answer: $14$

Answer

The correct sum of all positive values of

x

is 14.

To find the sum of all positive values of

x

that satisfy

g(f(x)) = 0

, first substitute

f(x)

into

g(x)

to get the equation

|f(x) - 1| - 8 = 0

. This simplifies to

|f(x) - 1| = 8

. Solving this absolute value equation yields two cases:

f(x) - 1 = 8

or

f(x) - 1 = -8

.

For the first case,

f(x) = 9

. Substituting

f(x) = (x - 3)^2 - 16

gives

(x - 3)^2 - 16 = 9 \Rightarrow (x - 3)^2 = 25

. Taking the square root of both sides gives

x - 3 = 5

or

x - 3 = -5

, which results in

x = 8

or

x = -2

. The only positive solution from this case is

8

.

For the second case,

f(x) = -7

. Substituting

f(x) = (x - 3)^2 - 16

gives

(x - 3)^2 - 16 = -7 \Rightarrow (x - 3)^2 = 9

. Taking the square root of both sides gives

x - 3 = 3

or

x - 3 = -3

, which results in

x = 6

or

x = 0

. The only positive solution from this case is

6

(since

0

is not positive).

Adding the positive solutions gives

8 + 6 = 14

.

Step-by-Step Solution

1

Set up the composite equation

g(f(x)) = 0

.

|f(x) - 1| - 8 = 0 \Rightarrow |f(x) - 1| = 8

Substitute the expression for

f(x)

into

g(x)

to establish the relationship.

2

Solve the absolute value equation by separating it into two distinct cases.

f(x) - 1 = 8 \Rightarrow f(x) = 9

or

f(x) - 1 = -8 \Rightarrow f(x) = -7

An absolute value equation

|u| = c

splits into

u = c

and

u = -c

when

c \geq 0

.

3

Substitute the definition of

f(x) = (x - 3)^2 - 16

into the first case and solve for

x

.

(x - 3)^2 - 16 = 9 \Rightarrow (x - 3)^2 = 25 \Rightarrow x - 3 = \pm 5 \Rightarrow x = 8

or

x = -2

Isolate the squared binomial and take the square root of both sides to find all real solutions for this case.

4

Substitute the definition of

f(x) = (x - 3)^2 - 16

into the second case and solve for

x

.

(x - 3)^2 - 16 = -7 \Rightarrow (x - 3)^2 = 9 \Rightarrow x - 3 = \pm 3 \Rightarrow x = 6

or

x = 0

Isolate the squared binomial and take the square root of both sides to find all real solutions for this case.

5

Identify the positive solutions and calculate their sum.

The positive solutions are

8

and

6

. The sum is

8 + 6 = 14

.

Exclude non-positive values (

-2

is negative, and

0

is neither positive nor negative) and add the remaining values.

Key Concept

Function composition involves substituting one function into another, and evaluating the resulting composite equation requires solving multi-step equations including absolute value and quadratic relations.

Question 2120Question →

An arithmetic sequence $a_1, a_2, a_3, \dots$ has a first term $a_1$ and a common difference $d$ , where both $a_1$ and $d$ are non-zero. Let $S_n$ represent the sum of the first $n$ terms of this sequence. If the ratio $\frac{S_{3n}}{S_n}$ is equal to a constant value $C$ for all positive integers $n$ , what is the ratio of the tenth term, $a_{10}$ , to the first term, $a_1$ ?

10

21

19

9

46

Show answer & explanation

Answer: 19

Answer

The ratio of the tenth term to the first term is 19.

The sum of the first

n

terms of an arithmetic sequence is

S_n = \frac{n}{2}[2a_1 + (n-1)d]

. For

n=1

, the ratio is

\frac{S_3}{S_1} = 3 + \frac{3d}{a_1}

. For

n=2

, the ratio is

\frac{S_6}{S_2} = \frac{6a_1 + 15d}{2a_1 + d}

. Setting these equal because the ratio is constant for all positive integers

n

gives

3 + \frac{3d}{a_1} = \frac{6a_1 + 15d}{2a_1 + d}

, which simplifies to

\frac{3a_1+3d}{a_1} = \frac{6a_1+15d}{2a_1+d}

. Cross-multiplying and simplifying yields

3d^2 = 6a_1 d

. Since

d \neq 0

, we have

d = 2a_1

. The tenth term is

a_{10} = a_1 + 9d = a_1 + 9(2a_1) = 19a_1

. Thus, the ratio of the tenth term to the first term is

19

.

Step-by-Step Solution

1

Write the formula for the sum of the first

n

terms of an arithmetic sequence,

S_n = \frac{n}{2}[2a_1 + (n-1)d]

, and find the expressions for

S_1

and

S_3

.

S_1 = a_1

and

S_3 = 3a_1 + 3d

.

To evaluate the ratio

\frac{S_{3n}}{S_n}

for the case where

n = 1

.

2

Write the expressions for

S_2

and

S_6

using the arithmetic sum formula.

S_2 = 2a_1 + d

and

S_6 = 6a_1 + 15d

.

To evaluate the ratio

\frac{S_{3n}}{S_n}

for the case where

n = 2

.

3

Equate the ratio for

n = 1

to the ratio for

n = 2

since the ratio

\frac{S_{3n}}{S_n}

must be constant for all

n

.

\frac{3a_1 + 3d}{a_1} = \frac{6a_1 + 15d}{2a_1 + d}

.

To set up an algebraic equation to find the relationship between the first term

a_1

and the common difference

d

.

4

Solve the equation for

d

in terms of

a_1

by cross-multiplying and simplifying.

d = 2a_1

.

Cross-multiplying gives

(3a_1 + 3d)(2a_1 + d) = a_1(6a_1 + 15d)

, which expands to

6a_1^2 + 9a_1 d + 3d^2 = 6a_1^2 + 15a_1 d

. Subtracting

6a_1^2

and

9a_1 d

from both sides yields

3d^2 = 6a_1 d

. Since

d \neq 0

, dividing by

3d

gives

d = 2a_1

.

5

Substitute

d = 2a_1

into the formula for the tenth term,

a_{10} = a_1 + 9d

, and calculate the ratio

\frac{a_{10}}{a_1}

.

a_{10} = 19a_1

, so the ratio is

19

.

To determine the final value of the requested ratio.

Key Concept

Relating arithmetic sequence term and sum formulas through systems of algebraic equations.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept