All practice questions

Question 2001Question →

If

x

and

y

are positive real numbers such that

(x^3 y^{-2})^k \cdot (x^{-1} y^4)^3 = x^6 y^6

what is the value of the exponent

k

?

Show answer & explanation

Answer: 3

Answer

The value of the exponent

k

is 3.

Applying the exponent rules simplifies the left side of the equation to

x^{3k-3}y^{12-2k}

. Equating the exponent of

x

to the right side gives

3k - 3 = 6

, which yields

k = 3

. This value is confirmed by equating the exponent of

y

, since

12 - 2(3) = 6

.

Step-by-Step Solution

1

Apply the power of a power rule

(a^m)^n = a^{mn}

to expand the terms in the expression.

(x^3 y^{-2})^k = x^{3k} y^{-2k}

and

(x^{-1} y^4)^3 = x^{-3} y^{12}

To remove the outer parentheses by multiplying the internal exponents of each variable by the outer exponent.

2

Multiply the terms together by applying the product rule for exponents,

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

.

(x^{3k} y^{-2k})(x^{-3} y^{12}) = x^{3k-3} y^{12-2k}

To combine the like bases of

x

and

y

into a single simplified expression.

3

Set the exponents of like bases equal to the exponents on the right-hand side of the equation,

x^6 y^6

.

3k - 3 = 6

and

12 - 2k = 6

Since the bases are equal and non-zero, their respective exponents must also be equal.

4

Solve the linear equation

3k - 3 = 6

for

k

.

3k = 9 \implies k = 3

To determine the numerical value of the variable

k

.

5

Verify the solution by solving the second linear equation,

12 - 2k = 6

.

-2k = -6 \implies k = 3

To ensure consistency across both variable exponents in the expression.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 2002Question →

A manufacturing plant operates two machines, Machine X and Machine Y, to produce identical components. Machine Y produces 12 fewer components per hour than Machine X. On a certain day, Machine X operated for 5 hours and Machine Y operated for 7 hours. If the two machines produced a combined total of 636 components, how many components did Machine X produce on that day?

46

60

230

270

300

Show answer & explanation

Answer: 300

Answer

300

To find the number of components Machine X produced, let

x

represent Machine X's hourly production rate. This makes Machine Y's hourly rate

x - 12

. Since Machine X operated for 5 hours and Machine Y operated for 7 hours to produce a total of 636 components, we can write the equation:

5x + 7(x - 12) = 636

. Distributing the 7 yields

5x + 7x - 84 = 636

, which simplifies to

12x - 84 = 636

. Adding 84 to both sides gives

12x = 720

, and dividing by 12 yields

x = 60

components per hour for Machine X. To find the total produced by Machine X, we multiply its hourly rate by the 5 hours it operated:

5 \times 60 = 300

. This matches the correct option.

Step-by-Step Solution

1

Define variables for the hourly rates of both machines.

Let

x

be the number of components Machine X produces per hour. Since Machine Y produces 12 fewer components per hour, Machine Y's hourly rate is

x - 12

.

Establishing algebraic expressions for each machine's rate allows us to set up a linear equation.

2

Set up an equation representing the total combined production.

The total production is the sum of the components produced by Machine X in 5 hours and Machine Y in 7 hours:

5x + 7(x - 12) = 636

.

The sum of the products of each machine's rate and its operating time equals the total combined production of 636 components.

3

Solve the equation for the hourly rate of Machine X,

x

.

Distribute the 7:

5x + 7x - 84 = 636

. Combine like terms:

12x - 84 = 636

. Add 84 to both sides:

12x = 720

. Divide by 12:

x = 60

.

Solving for

x

gives the hourly rate of Machine X.

4

Calculate the total components produced by Machine X.

Multiply the hourly rate of Machine X by its hours of operation:

5 \text{ hours} \times 60 \text{ components/hour} = 300

.

The question asks for the total components produced by Machine X, not its hourly rate.

Key Concept

Setting up and solving single-variable linear equations from verbal descriptions

Alternative Method

Instead of defining the variable as Machine X's rate, we could define

y

as Machine Y's hourly rate. Then Machine X's rate is

y + 12

. The total equation becomes

5(y + 12) + 7y = 636 \implies 5y + 60 + 7y = 636 \implies 12y = 576 \implies y = 48

. Since Machine X's hourly rate is

y + 12 = 60

, it produced

5 \times 60 = 300

components.

Estimated Time:1m 30s

Question 2003Question →

When $5$ is subtracted from $2$ times a number $x$ , the result is at least $9$ . Which of the following inequalities represents all possible values of $x$ ?

x \leq -2

x \geq -2

x \geq 3

x \geq 7

x \leq 7

Show answer & explanation

Answer: $x \geq 7$

Answer

The inequality

x \geq 7

represents all possible values of

x

.

The verbal statement translates to the inequality

2x - 5 \geq 9

. Adding

5

to both sides gives

2x \geq 14

, and dividing both sides by the positive number

2

results in

x \geq 7

. Since we divide by a positive number, the direction of the inequality does not change.

Step-by-Step Solution

1

Translate the verbal phrase into an algebraic inequality.

2x - 5 \geq 9

'5 subtracted from 2 times a number

x

' translates to

2x - 5

, and 'at least 9' means greater than or equal to 9.

2

Add 5 to both sides of the inequality to isolate the variable term.

2x \geq 14

Adding 5 to both sides maintains the inequality and simplifies the left side.

3

Divide both sides by 2.

x \geq 7

Dividing by a positive number does not change the direction of the inequality sign.

Key Concept

Solving linear inequalities by translating verbal statements into algebraic forms and applying inverse operations.

Question 2004Question →

Let the function $h$ be defined by $h(x) = 3x - 5$ , and let the function $g$ be defined by $g(x) = x^2 + c$ for some constant $c$ . If $h(g(2)) = 10$ , what is the value of $c$ ?

Show answer & explanation

Answer: 1

Answer

The value of the constant

c

is

1

.

The correct answer is

1

. Evaluating the inner function

g(2)

gives

2^2 + c = 4 + c

. Applying the outer function

h(x)

to this expression gives

h(4 + c) = 3(4 + c) - 5

. Simplifying this expression results in

12 + 3c - 5 = 7 + 3c

. We are given that this value equals

10

, so

7 + 3c = 10

. Subtracting 7 from both sides gives

3c = 3

, which yields

c = 1

.

Step-by-Step Solution

1

Evaluate

g(2)

in terms of

c

g(2) = 4 + c

We substitute

x = 2

into the definition

g(x) = x^2 + c

to get

2^2 + c = 4 + c

.

2

Substitute

g(2)

into the definition of

h(x)

to express

h(g(2))

h(g(2)) = 3(4 + c) - 5

Since the composition is

h(g(2))

, we evaluate the function

h

at the input value

g(2) = 4 + c

.

3

Set

h(g(2)) = 10

and solve the linear equation for

c

c = 1

Expanding and simplifying the equation

3(4 + c) - 5 = 10

gives

12 + 3c - 5 = 10

, which reduces to

7 + 3c = 10

. Subtracting 7 gives

3c = 3

, so

c = 1

.

Key Concept

Function Composition and Parameter Evaluation

Estimated Time:1m 15s

Question 2005Question →

What is the smallest integer value of $x$ that satisfies the inequality $11 - 3x < 2$ ?

Show answer & explanation

Answer: 4

Answer

The smallest integer value of

x

that satisfies the inequality is 4.

Solving the inequality

11 - 3x < 2

leads to

-3x < -9

. Dividing by

-3

and reversing the inequality sign gives

x > 3

. The smallest integer that is strictly greater than 3 is 4.

Step-by-Step Solution

1

Isolate the variable term on one side of the inequality by subtracting 11 from both sides.

-3x < -9

Subtracting 11 from both sides of the inequality keeps the relationship balanced.

2

Divide both sides of the inequality by the coefficient of

x

, which is

-3

.

x > 3

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

3

Determine the smallest integer that is strictly greater than 3.

4

Since the inequality is strict (

x > 3

), 3 is not included in the solution set. The smallest integer greater than 3 is 4.

Key Concept

Solving linear inequalities and reversing the inequality sign when dividing by a negative number.

Question 2006Question →

A sporting goods company sells soccer balls and basketballs at two locations, North and South. The matrix $A$ represents the inventory of these balls at each location:

A = \begin{bmatrix} 50 & 30 \\ 40 & 60 \end{bmatrix}

where the rows represent the locations (Row 1: North, Row 2: South) and the columns represent the ball types (Column 1: soccer balls, Column 2: basketballs).

The matrix $B$ represents the price, in dollars, of each type of ball under two different pricing structures (Column 1: regular price, Column 2: sale price):

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

where Row 1 represents soccer balls and Row 2 represents basketballs.

Which of the following matrices represents the total revenue, in dollars, the company would collect from selling its entire inventory at each location under both pricing structures?

\begin{bmatrix} 1,000 & 450 \\ 1,000 & 1,080 \end{bmatrix}

\begin{bmatrix} 1,600 & 1,500 \\ 1,970 & 1,830 \end{bmatrix}

\begin{bmatrix} 250 & 210 \\ 700 & 480 \end{bmatrix}

\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

\begin{bmatrix} 1,750 & 1,380 \\ 2,300 & 1,720 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}$

Answer

\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

To find the total revenue under both pricing structures at each location, we perform the matrix multiplication

AB

. The entry in row

i

, column

j

of the resulting matrix corresponds to the revenue generated at location

i

under pricing structure

j

. Applying the row-by-column multiplication rule gives the correct matrix containing

1,750

,

1,290

,

2,300

, and

1,680

.

Step-by-Step Solution

1

Set up the matrix multiplication to compute the total revenue.

The total revenue matrix is represented by the product

AB

, where we multiply the

2 \times 2

inventory matrix

A

by the

2 \times 2

price matrix

B

.

Multiplying the quantities of items by their respective unit prices yields the total revenue.

2

Calculate the entries for the first row of the product matrix, representing the North location.

Row 1, Column 1 (North, Regular):

50(20) + 30(25) = 1,000 + 750 = 1,750

.
Row 1, Column 2 (North, Sale):

50(15) + 30(18) = 750 + 540 = 1,290

.

The first row of matrix

A

is multiplied by the columns of matrix

B

to find revenues under both pricing structures at the North location.

3

Calculate the entries for the second row of the product matrix, representing the South location.

Row 2, Column 1 (South, Regular):

40(20) + 60(25) = 800 + 1,500 = 2,300

.
Row 2, Column 2 (South, Sale):

40(15) + 60(18) = 600 + 1,080 = 1,680

.

The second row of matrix

A

is multiplied by the columns of matrix

B

to find revenues under both pricing structures at the South location.

4

Construct the final product matrix from the calculated values.

AB = \begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

Placing the computed row values in their respective positions yields the completed revenue matrix.

Key Concept

Matrix multiplication is used to combine quantity data and unit cost data systematically across multiple categories and scenarios.

Question 2007Question →

A sporting goods chain has two locations. Location 1 sells 10 soccer balls and 15 basketballs. Location 2 sells 8 soccer balls and 20 basketballs. This is represented by the sales matrix $A = \begin{bmatrix} 10 & 15 \\ 8 & 20 \\end{bmatrix}$ . Soccer balls cost $15 each and basketballs cost$ 25 each, represented by the price matrix $B = \begin{bmatrix} 15 \\ 25 \\end{bmatrix}$ . What is the product $AB$ , which represents the total revenue generated by each location?

\begin{bmatrix} 150 \\ 500 \\end{bmatrix}

\begin{bmatrix} 350 \\ 725 \\end{bmatrix}

\begin{bmatrix} 525 \\ 520 \\end{bmatrix}

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

\begin{bmatrix} 525 & 620 \\end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 525 \\ 620 \\end{bmatrix}$

Answer

The column matrix with elements 525 in the first row and 620 in the second row

The correct answer is the column matrix containing 525 and 620. This is found by multiplying each row of the sales matrix (representing each location) by the column of the price matrix (representing prices of the items):

10(15) + 15(25) = 525

for the first row, and

8(15) + 20(25) = 620

for the second row.

Step-by-Step Solution

1

Set up the matrix multiplication of the

2 \times 2

sales matrix

A

and the

2 \times 1

price matrix

B

.

AB = \begin{bmatrix} 10 & 15 \\ 8 & 20 \\end{bmatrix} \begin{bmatrix} 15 \\ 25 \\end{bmatrix}

To find the total revenue for each location, we multiply the sales of each item by its corresponding price.

2

Multiply the first row of

A

by the column of

B

to find the element in the first row of the product matrix (revenue for Location 1).

10(15) + 15(25) = 150 + 375 = 525

This calculates the total revenue for Location 1 by summing the revenue from soccer balls (

10 \times 15

) and basketballs (

15 \times 25

).

3

Multiply the second row of

A

by the column of

B

to find the element in the second row of the product matrix (revenue for Location 2).

8(15) + 20(25) = 120 + 500 = 620

This calculates the total revenue for Location 2 by summing the revenue from soccer balls (

8 \times 15

) and basketballs (

20 \times 25

).

4

Combine the results into a

2 \times 1

resulting matrix.

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

Since matrix

A

has dimensions

2 \times 2

and matrix

B

has dimensions

2 \times 1

, their product

AB

must have dimensions

2 \times 1

.

Key Concept

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix, matching corresponding elements and adding the products.

Alternative Method

Instead of matrix multiplication, the revenue can be computed using standard arithmetic: Location 1 revenue is

10 \times \$15 + 15 \times \$25 = \$525

and Location 2 revenue is

8 \times \$15 + 20 \times \$25 = \$620

. The resulting column matrix is

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

.

Estimated Time:1m 0s

Question 2008Question →

The first term of a geometric sequence is $3$ , and the second term is $6$ . What is the $5$ th term of this sequence?

15

24

48

96

1296

Show answer & explanation

Answer: $48$

Answer

The fifth term of the sequence is

48

.

The first term of the geometric sequence is

a_1 = 3

, and the second term is

a_2 = 6

. The common ratio

r

is

6 \div 3 = 2

. The

n

th term of a geometric sequence is given by

a_n = a_1 \cdot r^{n-1}

. For the fifth term (

n=5

), the calculation is

3 \cdot 2^{5-1} = 3 \cdot 2^4 = 3 \cdot 16 = 48

. This matches the correct value.

Step-by-Step Solution

1

Find the common ratio (

r

) of the geometric sequence.

r = \frac{6}{3} = 2

The common ratio of a geometric sequence is found by dividing any term by the preceding term.

2

Apply the formula for the

n

th term of a geometric sequence,

a_n = a_1 \cdot r^{n-1}

, to find the fifth term.

a_5 = 3 \cdot 2^{5-1} = 3 \cdot 2^4

We substitute

a_1 = 3

,

r = 2

, and

n = 5

into the standard formula.

3

Evaluate the expression.

a_5 = 3 \cdot 16 = 48

Calculate

2^4 = 16

first, then multiply by

3

according to the order of operations.

Key Concept

Finding a specific term in a geometric sequence using the general formula

Question 2009Question →

The matrices $M$ and $N$ are defined as follows:

M = \begin{bmatrix} 4 & -2 \\ 1 & 3 \end{bmatrix}, \quad N = \begin{bmatrix} 2 & 5 \\ 0 & -1 \end{bmatrix}

If the product matrix $P$ is defined by the equation $P = MN$ , what is the value of the element in the first row and second column of $P$ ?

Show answer & explanation

Answer: 22

Answer

The value of the element in the first row and second column of

P

is 22.

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

P = MN

, we calculate the dot product of the first row of matrix

M

(

[4, -2]

) and the second column of matrix

N

(

\begin{bmatrix} 5 \\ -1 \end{bmatrix}

):

4(5) + (-2)(-1) = 20 + 2 = 22

.

Step-by-Step Solution

1

Determine the row of the left matrix and the column of the right matrix needed for the target element.

To find the element in row 1, column 2 of

P = MN

, use the first row of

M

, which is

[4, -2]

, and the second column of

N

, which is

\begin{bmatrix} 5 \\ -1 \end{bmatrix}

.

The entry in row

i

and column

j

of a product matrix is the dot product of row

i

of the first matrix and column

j

of the second matrix.

2

Compute the dot product of the selected row and column.

(4 \times 5) + (-2 \times -1) = 20 + 2

Multiply corresponding elements and sum the products.

3

Simplify to get the final numeric answer.

22

Adding 20 and 2 yields 22.

Key Concept

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix.

Question 2010Question →

One of the solutions to the quadratic equation $0.5x^2 - x + c = 0$ is $1 + i$ , where $i = \sqrt{-1}$ and $c$ is a constant. What is the value of $c$ ?

0

1 - 2i

1

-1

1 + i

Show answer & explanation

Answer: 1

Answer

The constant

c

is equal to

1

.

Substituting the given solution

x = 1 + i

into the equation

0.5x^2 - x + c = 0

gives

0.5(1+i)^2 - (1+i) + c = 0

. Since

(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i

, the equation simplifies to

0.5(2i) - 1 - i + c = 0

, which is

i - 1 - i + c = 0

. Simplifying this yields

-1 + c = 0

, so the value of

c

is

1

.

Step-by-Step Solution

1

Substitute the solution

x = 1 + i

into the quadratic equation

0.5x^2 - x + c = 0

.

0.5(1+i)^2 - (1+i) + c = 0

Since

1 + i

is a solution to the equation, substituting it for

x

must satisfy the equation.

2

Expand the squared term

(1+i)^2

using the binomial square formula and the identity

i^2 = -1

.

(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i

Expanding

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

with

a=1

and

b=i

, and replacing

i^2

with

-1

simplifies the term.

3

Substitute

2i

back into the equation and simplify.

0.5(2i) - (1+i) + c = 0 \Rightarrow i - 1 - i + c = 0 \Rightarrow -1 + c = 0

Multiplying

0.5

by

2i

yields

i

, distributing the negative sign across

(1+i)

yields

-1 - i

, and combining like terms leaves

-1 + c = 0

.

4

Solve the linear equation for

c

.

c = 1

Adding

1

to both sides isolates the variable

c

.

Key Concept

Solving quadratic equations with complex roots by substitution and algebraic manipulation.

Question 2011Question →

A storage tank contains $18$ gallons of water and is being filled at a constant rate of $\frac{2}{3}$ gallons per minute. A second storage tank contains $33$ gallons of water and is being drained at a constant rate of $\frac{5}{6}$ gallons per minute. If both tanks begin their processes at the same time, they will contain the same amount of water after $m$ minutes. What is the value of $2m + 3$ ?

23

26

33

71

183

Show answer & explanation

Answer: 23

Answer

23

The correct answer is 23. By representing the volume of the first tank as

18 + \frac{2}{3}m

and the second tank as

33 - \frac{5}{6}m

, setting them equal gives

18 + \frac{2}{3}m = 33 - \frac{5}{6}m

. Solving this equation by finding a common denominator for the fraction coefficients yields

\frac{9}{6}m = 15

, which simplifies to

\frac{3}{2}m = 15

, and thus

m = 10

. Substituting

m = 10

into the expression

2m + 3

results in

2(10) + 3 = 23

.

Step-by-Step Solution

1

Set up the equation representing the water volume in both tanks over time.

18 + \frac{2}{3}m = 33 - \frac{5}{6}m

The first tank starts with

18

gallons and increases by

\frac{2}{3}

gallons per minute, while the second starts with

33

gallons and decreases by

\frac{5}{6}

gallons per minute. We set their volumes equal to find when they contain the same amount.

2

Isolate the variable terms on one side and the constant terms on the other side.

\frac{2}{3}m + \frac{5}{6}m = 33 - 18

Adding

\frac{5}{6}m

to both sides and subtracting

18

from both sides groups like terms together.

3

Find a common denominator to add the fraction coefficients.

\frac{4}{6}m + \frac{5}{6}m = 15 \Rightarrow \frac{9}{6}m = 15 \Rightarrow \frac{3}{2}m = 15

A common denominator of

6

is used to add the fractions

\frac{2}{3}

and

\frac{5}{6}

.

4

Solve for

m

by multiplying both sides by the reciprocal of the coefficient.

m = 15 \times \frac{2}{3} = 10

Multiplying by

\frac{2}{3}

isolates

m

on the left side.

5

Evaluate the expression

2m + 3

using the value of

m

.

2(10) + 3 = 20 + 3 = 23

Substitute

m = 10

into the expression and follow the correct order of operations (multiply first, then add).

Key Concept

Solving linear equations with fractional coefficients and translating word problems into algebraic equations.

Question 2012Question →

A triangle has a base of $2x + 4$ inches and a height of $x - 3$ inches. What is the coefficient of $x$ when the expression representing the area of the triangle, in square inches, is written in standard form?

Show answer & explanation

Answer: -1

Answer

The coefficient of

x

is

-1

.

The area of a triangle is calculated using

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

. Substituting the given expressions, we get

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

. First, we can multiply the

\frac{1}{2}

by

(2x + 4)

, which simplifies to

x + 2

. Next, we expand

(x + 2)(x - 3)

using FOIL to get

x^2 - 3x + 2x - 6

. Combining like terms yields

x^2 - x - 6

. The coefficient of the

x

term in this simplified expression is

-1

.

Step-by-Step Solution

1

State the formula for the area of a triangle.

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

To establish the mathematical relationship.

2

Substitute the given values into the formula.

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

To express the area in terms of the variable

x

.

3

Distribute the fraction to the first binomial.

x + 2

Distributing

\frac{1}{2}

to

(2x + 4)

simplifies the expression before multiplying.

4

Expand the product of the binomials.

x^2 - 3x + 2x - 6

Using the FOIL method to multiply

(x + 2)

and

(x - 3)

.

5

Combine the linear terms.

x^2 - x - 6

To simplify the polynomial and write it in standard form.

6

Identify the coefficient of

x

.

-1

The coefficient of the

x

term in

x^2 - x - 6

is

-1

.

Key Concept

Multiplying binomials and applying formulas in geometric contexts

Estimated Time:1m 0s

Question 2013Question →

If the expression $\frac{(x^2 y^3)^4}{(x^{-1} y^2)^3}$ is written in the equivalent form $x^a y^b$ , what is the value of $a - b$ ?

Show answer & explanation

Answer: 5

Answer

5

Simplifying the numerator yields

x^8 y^{12}

and simplifying the denominator yields

x^{-3} y^6

. Dividing these expressions by subtracting the exponents of like bases results in

x^{8 - (-3)} y^{12 - 6} = x^{11} y^6

. Comparing this to the expression

x^a y^b

shows that

a = 11

and

b = 6

. The value of

a - b

is

11 - 6 = 5

.

Step-by-Step Solution

1

Simplify the numerator of the expression.

x^8 y^{12}

Apply the power of a power and power of a product properties of exponents:

(x^2 y^3)^4 = x^{2 \cdot 4} y^{3 \cdot 4}

.

2

Simplify the denominator of the expression.

x^{-3} y^6

Apply the power of a power and power of a product properties of exponents:

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

.

3

Simplify the quotient by dividing the simplified numerator by the simplified denominator.

x^{11} y^6

Use the quotient property of exponents,

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

, to subtract the exponents:

8 - (-3) = 11

and

12 - 6 = 6

.

4

Identify the values of

a

and

b

and compute the difference

a - b

.

5

By comparing

x^{11} y^6

to the target form

x^a y^b

, we find

a = 11

and

b = 6

. Subtracting

b

from

a

yields

11 - 6 = 5

.

Key Concept

Properties of exponents including power of a power, power of a product, and quotient rules.

Question 2014Question →

For real numbers $x$ where the functions are defined, let $f(x) = \frac{x}{x^2 - 4}$ and $g(x) = \frac{x+3}{x-1}$ . What is the set of all real numbers $x$ for which the composite function $h(x) = f(g(x))$ is undefined?

\left\{1\right\}

\left\{-2, 1, 2\right\}

\left\{1, 5\right\}

\left\{-\frac{1}{3}, 1, 5\right\}

\left\{-\frac{1}{3}, -2, 1, 2, 5\right\}

Show answer & explanation

Answer: $\left\{-\frac{1}{3}, 1, 5\right\}$

Answer

The set containing negative one-third, one, and five

The correct answer is the set containing negative one-third, one, and five. The composite function

f(g(x))

is undefined when the inner function

g(x)

is undefined (which occurs at

x = 1

) or when the output of

g(x)

equals a value that makes the outer function

f(x)

undefined. The outer function

f(x)

is undefined when

x^2 - 4 = 0

, which gives

x = \pm 2

. Setting

g(x) = 2

leads to

x = 5

, and setting

g(x) = -2

leads to

x = -\frac{1}{3}

. Collecting all these values gives the complete set of restrictions.

Step-by-Step Solution

1

Determine the domain restriction of the inner function

x \neq 1

For the composite function

h(x) = f(g(x))

to be defined, the input

x

must first be in the domain of the inner function

g(x) = \frac{x+3}{x-1}

. Thus, the denominator cannot be zero, which gives

x - 1 \neq 0 \implies x \neq 1

.

2

Determine the restricted inputs for the outer function

u \neq 2

and

u \neq -2

The outer function

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

is undefined when its denominator is zero. Solving

u^2 - 4 = 0

yields

u^2 = 4 \implies u = 2

or

u = -2

. Therefore, the output of the inner function,

g(x)

, cannot equal

2

or

-2

.

3

Solve the equation for the first restricted value of the outer function

x = 5

Setting the inner function equal to

2

gives

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

. Multiplying both sides by

x-1

results in

x+3 = 2(x-1) \implies x+3 = 2x-2 \implies x = 5

.

4

Solve the equation for the second restricted value of the outer function

x = -\frac{1}{3}

Setting the inner function equal to

-2

gives

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

. Multiplying both sides by

x-1

results in

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

.

5

Combine all restricted values of

x

S = \left\{-\frac{1}{3}, 1, 5\right\}

The values of

x

for which the composite function is undefined are

x = 1

(from the inner function's domain restriction) along with

x = 5

and

x = -\frac{1}{3}

(which make the inner function's output equal to the outer function's restricted inputs).

Key Concept

Domain of Composite Functions

Estimated Time:2m 30s

Question 2015Question →

For all real values of $x$ , which of the following inequalities represents the complete solution set to the inequality $\frac{5 - 2x}{3} - \frac{3x - 1}{4} \leq \frac{x + 2}{2}$ ?

x \leq \frac{11}{23}

x \geq \frac{5}{23}

x \geq -\frac{2}{17}

x \geq \frac{11}{23}

x \leq \frac{5}{23}

Show answer & explanation

Answer: $x \geq \frac{11}{23}$

Answer

The complete solution set is the set of all real numbers greater than or equal to 11/23.

The correct answer is found by clearing the denominators with the least common multiple of 12, carefully expanding the terms to get

20 - 8x - 9x + 3 \leq 6x + 12

, simplifying to

23 - 17x \leq 6x + 12

, grouping terms to get

-23x \leq -11

, and dividing by

-23

which flips the sign to yield all real values greater than or equal to 11/23.

Step-by-Step Solution

1

Multiply all terms of the inequality by the least common multiple of the denominators (3, 4, and 2), which is 12, to clear the fractions.

4(5 - 2x) - 3(3x - 1) \leq 6(x + 2)

Multiplying by a positive number allows us to eliminate denominators without changing the direction of the inequality.

2

Distribute the coefficients on both sides of the inequality, paying close attention to the distribution of the negative sign over the second term.

20 - 8x - 9x + 3 \leq 6x + 12

Distributing

-3

to both

3x

and

-1

yields

-9x

and

+3

respectively.

3

Combine the constant terms and the variable terms on the left side of the inequality.

23 - 17x \leq 6x + 12

Simplifying the expressions on each side makes the inequality easier to isolate.

4

Isolate the variable terms on the left and the constant terms on the right by subtracting

6x

and 23 from both sides.

-23x \leq -11

Grouping like terms together is necessary to solve for the variable.

5

Divide both sides of the inequality by

-23

and reverse the direction of the inequality sign.

x \geq \frac{11}{23}

Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign from

\leq

to

\geq

.

Key Concept

Solving linear inequalities involving fractions and distributing negative coefficients, specifically applying the rule that multiplying or dividing by a negative number reverses the inequality direction.

Alternative Method

Instead of clearing the fractions first, write each fraction as separate terms:

\frac{5}{3} - \frac{2}{3}x - \frac{3}{4}x + \frac{1}{4} \leq \frac{1}{2}x + 1

. Then, collect the constant terms on one side and the variable terms on the other side using decimal or fractional conversions, and isolate the variable.

Estimated Time:2m 0s

Question 2016Question →

A catering service is organizing two events: a wedding and a corporate gala. The matrix $P$ represents the number of Deluxe packages ( $d$ ) and Premium packages ( $p$ ) ordered for each event:

P = \begin{bmatrix} 50 & 40 \\ 80 & x \end{bmatrix}

where the rows represent the Wedding and the Gala respectively, and the columns represent Deluxe and Premium packages respectively.

The matrix $R$ represents the requirements (in units) of protein and vegetables for each package type:

R = \begin{bmatrix} 3 & 2 \\ 4 & y \end{bmatrix}

where the rows represent Deluxe and Premium packages respectively, and the columns represent protein and vegetables respectively.

The total units of ingredients required for each event are represented by the product matrix $T = P \times R$ .

If the total vegetables required for the wedding is 220 units, and the total protein required for the corporate gala is 560 units, what is the value of the sum $x + y$ ?

11

54.5

82.8

83

145.5

Show answer & explanation

Answer: 83

Answer

83

The correct option is the value 83. Setting up the product matrix

T = P \times R

, the entry in row 1, column 2 (representing wedding vegetables) is

50(2) + 40y = 220

, which simplifies to

y = 3

. The entry in row 2, column 1 (representing gala protein) is

80(3) + 4x = 560

, which simplifies to

x = 80

. Adding these values together gives

x + y = 83

.

Step-by-Step Solution

1

Set up the matrix product expression

T = P \times R = \begin{bmatrix} 50 & 40 \\ 80 & x \end{bmatrix} \begin{bmatrix} 3 & 2 \\ 4 & y \end{bmatrix} = \begin{bmatrix} 50(3) + 40(4) & 50(2) + 40(y) \\ 80(3) + 4x & 80(2) + xy \end{bmatrix}

We multiply row elements of the first matrix by column elements of the second matrix to obtain the entries of the product matrix.

2

Isolate the equation for vegetables required for the wedding

50(2) + 40y = 220 \implies 100 + 40y = 220 \implies 40y = 120 \implies y = 3

The wedding corresponds to the first row of matrix

P

, and vegetables correspond to the second column of matrix

R

. Their dot product equals the total vegetables for the wedding.

3

Isolate the equation for protein required for the corporate gala

80(3) + 4x = 560 \implies 240 + 4x = 560 \implies 4x = 320 \implies x = 80

The corporate gala corresponds to the second row of matrix

P

, and protein corresponds to the first column of matrix

R

. Their dot product equals the total protein for the gala.

4

Find the sum of the variables

x + y = 80 + 3 = 83

We add the solved values of

x

and

y

to obtain the requested sum.

Key Concept

Matrix Multiplication and Word Problem Translation

Estimated Time:3m 0s

Question 2017Question →

If $3 - 2i$ is a root of the quadratic equation $x^2 + bx + c = 0$ , where $b$ and $c$ are real numbers and $i = \sqrt{-1}$ , what is the value of $b + c$ ?

-11

-5

-1

7

19

Show answer & explanation

Answer: 7

Answer

7

The correct answer is

7

. Since the quadratic equation has real coefficients, the roots must be complex conjugates. The conjugate of the root

3 - 2i

is

3 + 2i

. Vieta's formulas show that the sum of the roots is

-b

, meaning

(3 - 2i) + (3 + 2i) = 6 = -b

, which yields

b = -6

. The product of the roots is

c

, meaning

(3 - 2i)(3 + 2i) = 9 - 4i^2 = 9 - 4(-1) = 13

. Adding these coefficients together gives

b + c = -6 + 13 = 7

.

Step-by-Step Solution

1

Identify the second root of the quadratic equation

The second root is

3 + 2i

.

Since the quadratic equation has real coefficients, the Complex Conjugate Theorem dictates that if a complex number is a root, its complex conjugate must also be a root.

2

Determine the coefficient

b

using the sum of the roots

b = -6

According to Vieta's formulas, the sum of the roots of the equation

x^2 + bx + c = 0

is equal to

-b

. Thus,

(3 - 2i) + (3 + 2i) = 6 = -b

, which simplifies to

b = -6

.

3

Determine the constant term

c

using the product of the roots

c = 13

According to Vieta's formulas, the product of the roots of the equation

x^2 + bx + c = 0

is equal to

c

. Thus,

(3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - 4i^2

. Substituting

i^2 = -1

yields

9 - 4(-1) = 9 + 4 = 13

.

4

Calculate the value of

b + c

b + c = 7

Substitute the calculated values of

b

and

c

into the expression:

-6 + 13 = 7

.

Key Concept

The Complex Conjugate Theorem states that complex roots of polynomials with real coefficients occur in conjugate pairs. Vieta's formulas state that for a quadratic equation

x^2 + bx + c = 0

, the sum of the roots is

-b

and the product of the roots is

c

.

Question 2018Question →

A square playground has a side length of $2x^3$ meters. A square sandbox with a side length of $x^3 - 2x$ meters is built inside the playground. Which of the following expressions represents the area, in square meters, of the playground that is NOT covered by the sandbox?

3x^6 - 4x^4 + 4x^2

3x^6 - 4x^2

3x^6 + 4x^4 + 4x^2

3x^6 + 4x^4 - 4x^2

3x^6 + 4x^3 - 4x^2

Show answer & explanation

Answer: $3x^6 + 4x^4 - 4x^2$

Answer

The correct expression is

3x^6 + 4x^4 - 4x^2

.

The expression

3x^6 + 4x^4 - 4x^2

is correct because the area of the playground is

(2x^3)^2 = 4x^6

and the area of the sandbox is

(x^3 - 2x)^2 = x^6 - 4x^4 + 4x^2

. Subtracting the sandbox area from the playground area requires distributing the negative sign, resulting in

4x^6 - x^6 - (-4x^4) - 4x^2 = 3x^6 + 4x^4 - 4x^2

.

Step-by-Step Solution

1

Calculate the area of the square playground.

\text{Area}_{\text{playground}} = (2x^3)^2 = 4x^6

The area of a square is equal to the square of its side length, and applying the exponent rules yields

(2x^3)^2 = 2^2 \cdot (x^3)^2 = 4x^6

.

2

Calculate the area of the square sandbox.

\text{Area}_{\text{sandbox}} = (x^3 - 2x)^2 = x^6 - 4x^4 + 4x^2

Using the binomial squaring formula

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

, we square each term and compute the middle product, adding exponents for

x^3 \cdot x^1 = x^4

.

3

Subtract the sandbox's area from the playground's area.

4x^6 - (x^6 - 4x^4 + 4x^2) = 4x^6 - x^6 + 4x^4 - 4x^2 = 3x^6 + 4x^4 - 4x^2

Distribute the negative sign to all three terms inside the parentheses and combine the like terms of

x^6

.

Key Concept

Polynomial subtraction and squaring binomials with variables containing exponents.

Alternative Method

Evaluate the expression for a small integer value of

x

. If

x = 2

, the playground side length is

2(2)^3 = 16

, giving an area of

16^2 = 256

. The sandbox side length is

2^3 - 2(2) = 4

, giving an area of

4^2 = 16

. The remaining area is

256 - 16 = 240

. Substituting

x = 2

into the correct expression

3x^6 + 4x^4 - 4x^2

yields

3(64) + 4(16) - 4(4) = 192 + 64 - 16 = 240

, confirming its correctness.

Estimated Time:2m 0s

Question 2019Question →

A parabola is defined by the equation $y = 2x^2 - 5x + 1$ and a line is defined by the equation $y = x - 3$ . If the parabola and the line intersect at the points $(x_1, y_1)$ and $(x_2, y_2)$ , what is the value of $y_1 + y_2$ ?

Show answer & explanation

Answer: -3

Answer

The sum of the y-coordinates of the intersection points is -3.

By setting the two equations equal to each other, we obtain a quadratic equation in terms of

x

. Solving this equation gives the x-coordinates of the intersection points. Substituting these x-values back into the linear equation yields the corresponding y-coordinates. Summing these y-coordinates gives the final value of

-3

.

Step-by-Step Solution

1

Equate the equations for the line and the parabola to find the x-coordinates of their intersection points.

2x^2 - 5x + 1 = x - 3

Since both equations are equal to

y

, their right-hand sides must be equal at the points of intersection.

2

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

2x^2 - 6x + 4 = 0

Grouping all terms on one side of the equation allows us to solve for $x.

3

Divide the entire equation by 2 to simplify.

x^2 - 3x + 2 = 0

Simplifying the quadratic equation makes factoring easier.

4

Factor the quadratic equation to solve for

x

.

(x - 1)(x - 2) = 0

, yielding

x = 1

and

x = 2

.

Finding the roots of the quadratic equation gives the x-coordinates of the intersection points.

5

Substitute each x-coordinate back into the linear equation

y = x - 3

to find the corresponding y-coordinates.

For

x_1 = 1

:

y_1 = 1 - 3 = -2

. For

x_2 = 2

:

y_2 = 2 - 3 = -1

.

The intersection points must satisfy both equations in the system.

6

Calculate the sum of the two y-coordinates,

y_1 + y_2

.

y_1 + y_2 = -2 + (-1) = -3

The question asks for the value of the sum of the y-coordinates.

Key Concept

Solving systems of linear and quadratic equations by substitution

Question 2020Question →

A line in the standard $(x,y)$ coordinate plane passes through the point $(0, 4)$ and has a slope of $3$ . Which of the following equations represents this line?

y = 3x + 4

y = 4x + 3

y = 3x - 4

y = -3x + 4

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

Show answer & explanation

Answer: $y = 3x + 4$

Answer

y = 3x + 4

The slope-intercept form of a linear equation is

y = mx + b

. Here, the slope

m

is given as

3

, and the y-intercept

b

is given as

4

since the line crosses the y-axis at the point

(0,4)

. Substituting these values into the form gives the equation

y = 3x + 4

.

Step-by-Step Solution

1

Identify the slope-intercept form of a linear equation.

The slope-intercept form is

y = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

To write the equation of a line, we can substitute the known slope and y-intercept into this general form.

2

Determine the values of

m

and

b

from the given information.

The slope

m = 3

. The point

(0, 4)

lies on the y-axis, meaning the y-intercept

b = 4

.

The y-intercept is the y-coordinate of the point where the line crosses the y-axis, which occurs when

x = 0

.

3

Substitute the values of

m

and

b

into the slope-intercept equation.

y = 3x + 4

Replacing

m

with

3

and

b

with

4

yields the equation representing the line.

Key Concept

Writing linear equations in slope-intercept form using the slope and a point.

Answer

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