All practice questions

Question 1981Question →

A manufacturing company determines that its weekly profit from producing $x$ batches of a product is constrained by resource availability. To meet these resource constraints, the number of batches $x$ must satisfy the inequality:

\frac{5(6 - x)}{3} - \frac{3(x + 2)}{4} > 2x - 11

What is the greatest number of whole batches the company can produce while satisfying this constraint?

Show answer & explanation

Answer: 4

Answer

The greatest number of whole batches the company can produce is 4.

Solving the inequality yields

x < \frac{234}{53} \approx 4.415

. The greatest integer value that satisfies this condition is 4.

Step-by-Step Solution

1

Multiply both sides of the inequality by the least common multiple of the denominators, which is 12.

20(6 - x) - 9(x + 2) > 24x - 132

Multiplying by 12 eliminates the fractions, making the inequality easier to solve.

2

Distribute the constants on the left side of the inequality.

120 - 20x - 9x - 18 > 24x - 132

Distributing 20 to

(6 - x)

yields

120 - 20x

, and distributing

-9

to

(x + 2)

yields

-9x - 18

.

3

Combine like terms on the left side of the inequality.

102 - 29x > 24x - 132

Combining

120 - 18

gives

102

, and combining

-20x - 9x

gives

-29x

.

4

Subtract

24x

from both sides to group the variable terms on the left side.

102 - 53x > -132

This groups all terms containing the variable

x

on one side of the inequality.

5

Subtract 102 from both sides to isolate the variable term.

-53x > -234

This isolates the term containing

x

on the left side of the inequality.

6

Divide both sides by

-53

and reverse the inequality sign.

x < \frac{234}{53}

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

7

Evaluate the fraction as a decimal and determine the greatest integer value of

x

that satisfies the inequality.

x < 4.415

, which means the greatest integer is 4.

Since the company must produce a whole number of batches, we find the largest integer less than 4.415.

Key Concept

Solving multi-step linear inequalities involving fractional coefficients, distributing negative numbers, and reversing the inequality sign when multiplying or dividing by a negative number.

Alternative Method

Instead of solving algebraically, you can test integer values for

x

directly in the inequality. Testing

x = 4

gives

\frac{10}{3} - \frac{18}{4} = 3.33 - 4.5 = -1.17

, which is greater than

2(4) - 11 = -3

(True). Testing

x = 5

gives

\frac{5}{3} - \frac{21}{4} = 1.67 - 5.25 = -3.58

, which is not greater than

2(5) - 11 = -1

(False). This confirms 4 is the largest integer satisfying the inequality.

Estimated Time:2m 30s

Question 1982Question →

Which of the following is the completely factored form of the expression $3x(x - 2)^2 - 12x$ ?

3x(x^2 - 8)

-9x(x - 2)^2

3x^2(x - 4)

3x(x - 4)

12x(x - 1)(x - 3)

Show answer & explanation

Answer: $3x^2(x - 4)$

Answer

The completely factored form of the expression is

3x^2(x - 4)

.

The correct answer is

3x^2(x-4)

. Factoring out the greatest common factor

3x

from the terms

3x(x-2)^2

and

-12x

gives

3x[(x-2)^2-4]

. The expression within the brackets is a difference of squares that can be factored as

[(x-2)-2][(x-2)+2]

, which simplifies to

x(x-4)

. Multiplying this by the GCF

3x

yields

3x^2(x-4)

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF),

3x

, from both terms of the expression

3x(x - 2)^2 - 12x

.

3x[(x - 2)^2 - 4]

Both terms

3x(x-2)^2

and

12x

share the common factors

3

and

x

.

2

Factor the difference of squares inside the bracket,

(x-2)^2 - 4

, using the formula

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

where

a = x-2

and

b = 2

.

[(x-2)-2][(x-2)+2] = (x-4)(x)

Since

4 = 2^2

, the terms inside the brackets form a difference of squares.

3

Combine the factored parts and simplify the expression by multiplying the variable terms.

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

Multiplying

3x

by

x

requires adding their exponents (

1 + 1 = 2

).

Key Concept

Factoring polynomials using the greatest common factor (GCF) and difference of squares.

Question 1983Question →

When the expression $4x(x - 2y) - (2x - 3y)^2 + 5y(2x - y)$ is simplified to the form $Ax^2 + Bxy + Cy^2$ , where $A$ , $B$ , and $C$ are constants, what is the value of $B$ ?

Show answer & explanation

Answer: 14

Answer

The value of the coefficient

B

is 14.

Expanding the entire expression yields

4x^2 - 8xy - 4x^2 + 12xy - 9y^2 + 10xy - 5y^2

. Grouping and combining the

xy

terms gives

(-8 + 12 + 10)xy = 14xy

. Therefore, the coefficient

B

is 14.

Step-by-Step Solution

1

Expand the first term

4x^2 - 8xy

Distribute

4x

to both terms inside the parentheses:

4x(x) - 4x(2y) = 4x^2 - 8xy

.

2

Expand the squared binomial and apply the negative sign

-4x^2 + 12xy - 9y^2

Use the binomial expansion formula

(2x - 3y)^2 = 4x^2 - 12xy + 9y^2

, then multiply each term by

-1

.

3

Expand the third term

10xy - 5y^2

Distribute

5y

to both terms inside the parentheses:

5y(2x) - 5y(y) = 10xy - 5y^2

.

4

Combine the coefficients of the like terms

0x^2 + 14xy - 14y^2

Sum the coefficients for each corresponding variable group:

(4 - 4)x^2 + (-8 + 12 + 10)xy + (-9 - 5)y^2

.

Key Concept

Simplifying Algebraic Expressions and Combining Like Terms

Question 1984Question →

In the standard $(x, y)$ coordinate plane, the line defined by the equation $3x - 4y = k$ is tangent to the circle defined by the equation $x^2 + y^2 - 2x - 4y = 4$ . If $k > 0$ , what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

10

The correct answer is 10. Completing the square for the circle's equation gives

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

, showing the center is

(1, 2)

and the radius is

3

. The distance from

(1, 2)

to the line

3x - 4y - k = 0

is

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

. For tangency, this distance must equal the radius:

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

, which gives

|k + 5| = 15

. Solving this absolute value equation gives

k = 10

or

k = -20

. Since

k

must be positive, the value is 10.

Step-by-Step Solution

1

Complete the square for the circle's equation

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

.

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

, which represents a circle with center

(1, 2)

and radius

R = 3

.

To find the center and radius of the circle, which are needed to use the distance formula.

2

Express the distance

d

from the center

(1, 2)

to the line

3x - 4y - k = 0

using the formula

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

.

d = \frac{|3(1) - 4(2) - k|}{\sqrt{3^2 + (-4)^2}} = \frac{|-5 - k|}{5} = \frac{|k + 5|}{5}

.

A line is tangent to a circle if and only if the distance from the center of the circle to the line equals the radius.

3

Set the distance equal to the radius (

3

) and solve for

k

.

\frac{|k + 5|}{5} = 3 \Rightarrow |k + 5| = 15

, which yields

k + 5 = 15 \Rightarrow k = 10

, or

k + 5 = -15 \Rightarrow k = -20

.

To find the values of

k

that make the line tangent to the circle.

4

Select the positive value of

k

.

k = 10

.

The problem specifies that

k > 0

.

Key Concept

Systems of Linear and Non-Linear Equations

Estimated Time:2m 30s

Question 1985Question →

For a certain geometric sequence, the first term is $2^3$ and the common ratio is $2^2$ . Which of the following expressions represents the 3rd term of this sequence?

2^5

2^9

2^7

2^{10}

2^{12}

Show answer & explanation

Answer: $2^7$

Answer

The 3rd term of the sequence is

2^7

.

The correct answer is

2^7

. The

n

-th term of a geometric sequence is given by

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

. For a first term

a_1 = 2^3

and a common ratio

r = 2^2

, the 3rd term is

a_3 = 2^3 \cdot (2^2)^{3-1} = 2^3 \cdot 2^4

. Applying the exponent addition rule for multiplying bases of the same value yields

2^{3+4} = 2^7

.

Step-by-Step Solution

1

Identify the formula for the

n

-th term of a geometric sequence.

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

To find any specific term in a geometric sequence, the general term formula is used.

2

Substitute the given values (

a_1 = 2^3

,

r = 2^2

, and

n = 3

) into the formula.

a_3 = 2^3 \cdot (2^2)^{3-1} = 2^3 \cdot (2^2)^2

This sets up the calculation for the 3rd term of the sequence.

3

Simplify the expression using exponent rules: first compute

(2^2)^2

, then multiply by

2^3

.

a_3 = 2^3 \cdot 2^4 = 2^{3+4} = 2^7

Power of a power rule states

(x^a)^b = x^{ab}

, and product of powers rule states

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

.

Key Concept

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

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

and applying laws of exponents.

Estimated Time:45s

Question 1986Question →

The functions $f$ and $g$ are defined for all real numbers by $f(x) = 2x - 5$ and $g(x) = (x + 3)^2$ . What is the value of $f(g(-1))$ ?

15

16

3

27

-28

Show answer & explanation

Answer: 3

Answer

3

To evaluate the composite function

f(g(-1))

, first evaluate the inner function

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

. Then, substitute this result into the outer function to get

f(4) = 2(4) - 5 = 3

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = -1

.

g(-1) = 4

In the composition

f(g(-1))

, the inner function must be evaluated first. Substituting

-1

into

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

gives

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

.

2

Evaluate the outer function

f(x)

at the output of the inner function.

f(4) = 3

Substitute the output

g(-1) = 4

into

f(x) = 2x - 5

, which yields

f(4) = 2(4) - 5 = 8 - 5 = 3

.

Key Concept

Function Composition and Evaluation

Estimated Time:1m 0s

Question 1987Question →

A clothing boutique sells two types of shirts: t-shirts and polo shirts. The number of shirts sold at the boutique's two locations, Downtown and Uptown, on a weekend is represented by matrix

S

:

S = \begin{pmatrix} 40 & 30 \\ 50 & 20 \\end{pmatrix}

where the rows represent the locations (Downtown and Uptown, respectively) and the columns represent the shirt types (t-shirts and polo shirts, respectively).

The selling price and the production cost per shirt (in dollars) are represented by matrix

C

:

C = \begin{pmatrix} 15 & 6 \\ 25 & 10 \\end{pmatrix}

where the rows represent the shirt types (t-shirts and polo shirts, respectively) and the columns represent the selling price and production cost, respectively.

Which of the following matrices represents the total selling revenue and total production cost for the two locations?

\begin{pmatrix} 1350 & 540 \\ 1250 & 500 \\end{pmatrix}

\begin{pmatrix} 600 & 180 \\ 1250 & 200 \\end{pmatrix}

\begin{pmatrix} 900 & 570 \\ 1500 & 950 \\end{pmatrix}

\begin{pmatrix} 1850 & 740 \\ 950 & 380 \\end{pmatrix}

\begin{pmatrix} 780 & 1300 \\ 870 & 1450 \\end{pmatrix}

Show answer & explanation

Answer: \begin{pmatrix} 1350 & 540 \\ 1250 & 500 \\end{pmatrix}

Answer

The matrix representing the total selling revenue and total production cost for the two locations is the matrix with top row 1350 and 540, and bottom row 1250 and 500.

The correct matrix is obtained by multiplying the sales matrix

S

by the price/cost matrix

C

. Standard matrix multiplication pairs the quantity of each shirt type sold at each store with its respective selling price and production cost, yielding the correct total revenue and total production cost for both locations.

Step-by-Step Solution

1

Identify the dimensions and layout of the sales matrix

S

and the price/cost matrix

C

.

Matrix

S

is a

2 \times 2

matrix representing sales. Matrix

C

is a

2 \times 2

matrix representing price and cost. The product

SC

is defined because the number of columns in

S

(2) equals the number of rows in

C

(2).

Before multiplying, we must verify that the dimensions are compatible and that the product yields the desired real-world quantities.

2

Calculate the entries for the first row of the resulting matrix

SC

(Downtown location) by taking the dot product of the first row of

S

with the columns of

C

.

Revenue (Row 1, Column 1):

40 \times 15 + 30 \times 25 = 600 + 750 = 1350

.
Cost (Row 1, Column 2):

40 \times 6 + 30 \times 10 = 240 + 300 = 540

.

This yields the total revenue and production cost for the Downtown location by combining sales and prices/costs for both shirt types.

3

Calculate the entries for the second row of the resulting matrix

SC

(Uptown location) by taking the dot product of the second row of

S

with the columns of

C

.

Revenue (Row 2, Column 1):

50 \times 15 + 20 \times 25 = 750 + 500 = 1250

.
Cost (Row 2, Column 2):

50 \times 6 + 20 \times 10 = 300 + 200 = 500

.

This yields the total revenue and production cost for the Uptown location.

Key Concept

Matrix multiplication

Question 1988Question →

An infinite geometric series of positive terms has a sum of $9$ . The sum of the first two terms of the series is $8$ . What is the first term of this series?

2

3

4

6

12

Show answer & explanation

Answer: 6

Answer

The first term of the series is 6.

To find the first term of the geometric series, we set up a system of equations using the formulas for the sum of the first two terms,

S_2 = a(1+r) = 8

, and the sum of an infinite geometric series,

S_\infty = \frac{a}{1-r} = 9

. Expressing the first term as

a = 9(1-r)

and substituting this into the first equation yields

9(1-r)(1+r) = 8

, which simplifies to

9(1-r^2) = 8

. Solving for the ratio yields

r^2 = \frac{1}{9}

, which means

r = \frac{1}{3}

since all terms must be positive. Substituting

r = \frac{1}{3}

back into

a = 9(1-r)

gives the first term as

6

.

Step-by-Step Solution

1

Define variables and identify the given formulas.

Let the first term of the geometric series be

a

and the common ratio be

r

. Since the series has positive terms, we require

a > 0

and

0 < r < 1

.

This sets up the system of equations using standard geometric sequence notation.

2

Translate the given problem conditions into algebraic equations.

Equation 1 (sum of the first two terms):

S_2 = a + ar = a(1+r) = 8

.
Equation 2 (sum to infinity):

S_\infty = \frac{a}{1-r} = 9

.

We must express the two mathematical relationships given in the problem statement.

3

Express the first term

a

in terms of the common ratio

r

using Equation 2.

a = 9(1-r)

This allows for substitution into Equation 1 to solve for

r

.

4

Substitute the expression for

a

into Equation 1 and simplify.

9(1-r)(1+r) = 8 \implies 9(1-r^2) = 8 \implies 9 - 9r^2 = 8

.

Substituting simplifies the system from two variables to a single quadratic variable in terms of

r

.

5

Solve the quadratic equation for

r

.

9r^2 = 1 \implies r^2 = \frac{1}{9} \implies r = \frac{1}{3}

(since

r > 0

).

Finding the value of the common ratio is the final step before calculating the first term.

6

Substitute the value of

r

back into the expression for

a

.

a = 9\left(1 - \frac{1}{3}\right) = 9\left(\frac{2}{3}\right) = 6

.

This yields the value of the first term

a

to complete the problem.

Key Concept

Solving systems of non-linear equations using geometric sequence term and infinite sum formulas.

Alternative Method

Instead of substituting

a = 9(1-r)

into

a(1+r) = 8

, we can divide the two equations:

\frac{a(1+r)}{a/(1-r)} = \frac{8}{9} \implies (1-r)(1+r) = \frac{8}{9} \implies 1 - r^2 = \frac{8}{9}

. This directly yields

r^2 = \frac{1}{9}

without needing to isolate

a

first.

Estimated Time:2m 0s

Question 1989Question →

For a real number $x$ , $\frac{1}{2}$ minus $\frac{2}{3}$ of $x$ is greater than $\frac{5}{6}$ . Which of the following is the complete set of solutions for $x$ ?

x > -\frac{1}{2}

x > 2

x < -\frac{1}{2}

x < -\frac{3}{2}

x < -5

Show answer & explanation

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

Answer

x < -\frac{1}{2}

Subtracting

\frac{1}{2}

from both sides of the inequality

\frac{1}{2} - \frac{2}{3}x > \frac{5}{6}

gives

-\frac{2}{3}x > \frac{2}{6}

, which simplifies to

-\frac{2}{3}x > \frac{1}{3}

. Multiplying both sides by the negative fraction

-\frac{3}{2}

isolates

x

on the left and reverses the inequality sign from greater than (

>

) to less than (

<

). Performing the multiplication on the right yields

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

. Thus, the solution set is

x < -\frac{1}{2}

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic inequality.

\frac{1}{2} - \frac{2}{3}x > \frac{5}{6}

The phrase '

\frac{1}{2}

minus

\frac{2}{3}

of

x

' represents the expression

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

, and 'is greater than' translates to the inequality symbol

>

.

2

Subtract

\frac{1}{2}

from both sides of the inequality.

-\frac{2}{3}x > \frac{1}{3}

To isolate the variable term on the left, we subtract

\frac{1}{2}

(which is equivalent to

\frac{3}{6}

) from

\frac{5}{6}

to get

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

.

3

Multiply both sides of the inequality by

-\frac{3}{2}

and reverse the inequality sign.

x < -\frac{1}{2}

Multiplying both sides of an inequality by a negative number requires reversing the direction of the inequality sign from

>

to

<

to keep the inequality true.

Key Concept

Solving multi-step linear inequalities involving multiplication or division by a negative number.

Question 1990Question →

Consider the function $f$ defined on the domain $[\frac{2}{3}, \infty)$ by the equation $f(x) = 2 + \sqrt{3x - 2}$ . If $f^{-1}$ represents the inverse of $f$ , what is the only real value of $x$ for which $f(x) = f^{-1}(x)$ ?

Show answer & explanation

Answer: 6

Answer

The only real value of

x

for which

f(x) = f^{-1}(x)

is 6.

For a strictly increasing function, the intersection of

f(x)

and

f^{-1}(x)

must occur on the line

y = x

. Equating

f(x) = x

gives

2 + \sqrt{3x - 2} = x

. Isolating the radical term yields

\sqrt{3x - 2} = x - 2

. Squaring both sides produces

3x - 2 = x^2 - 4x + 4

, which simplifies to the quadratic equation

x^2 - 7x + 6 = 0

. Factoring this equation gives

(x - 6)(x - 1) = 0

, yielding the solutions

x = 6

and

x = 1

. Substituting these back into the original equation shows that

x = 6

is a valid solution (

2 + \sqrt{16} = 6

), whereas

x = 1

is extraneous because it results in

2 + 1 = 3 \neq 1

. Furthermore, the domain of

f^{-1}

is the range of

f

, which is

[2, \infty)

, meaning

f^{-1}(1)

is undefined. Thus, the only real solution is 6.

Step-by-Step Solution

1

Equate the function to

x

using properties of increasing functions and their inverses.

Since

f(x)

is strictly increasing on its domain

[\frac{2}{3}, \infty)

, the graph of

f(x)

and the graph of its inverse

f^{-1}(x)

can only intersect on the line of symmetry

y = x

. Therefore, the equation

f(x) = f^{-1}(x)

is equivalent to

f(x) = x

.

This simplifies the relation by eliminating the need to solve a high-degree polynomial equation derived from direct composition or substitution.

2

Set up the equation

f(x) = x

and isolate the radical term.

2 + \sqrt{3x - 2} = x \implies \sqrt{3x - 2} = x - 2

Isolating the square root term is a necessary prerequisite step before squaring both sides.

3

Square both sides and rewrite the equation as a standard quadratic equation.

3x - 2 = (x - 2)^2 \implies 3x - 2 = x^2 - 4x + 4 \implies x^2 - 7x + 6 = 0

Squaring eliminates the radical to yield a standard quadratic equation that can be solved analytically.

4

Solve the quadratic equation by factoring.

(x - 6)(x - 1) = 0 \implies x = 6 \text{ or } x = 1

Factoring the quadratic trinomial yields the two candidate values for

x

.

5

Check the candidate solutions in the original equation to eliminate extraneous roots.

For

x = 1

:

2 + \sqrt{3(1)-2} = 3 \neq 1

(extraneous). For

x = 6

:

2 + \sqrt{3(6)-2} = 2 + 4 = 6

(valid). Also, the domain of

f^{-1}

is the range of

f

, which is

[2, \infty)

, thus excluding

x = 1

.

Squaring both sides can introduce extraneous roots. We must verify that the solutions satisfy the original radical equation and fall within the domains of both

f

and

f^{-1}

.

Key Concept

Applying the symmetry of inverse functions about the line

y=x

to solve composition-based equations, while rigorously accounting for domain restrictions and extraneous roots.

Alternative Method

Find the algebraic formula for

f^{-1}(x)

by setting

y = 2 + \sqrt{3x-2}

. Subtracting 2 and squaring both sides gives

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

for

y \geq 2

. Solving for

x

yields

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

for

x \geq 2

. Equating

f(x) = f^{-1}(x)

results in

2 + \sqrt{3x - 2} = \frac{(x - 2)^2 + 2}{3}

. Multiplying by 3 and isolating the radical term gives

3\sqrt{3x - 2} = x^2 - 4x

. Squaring both sides results in a fourth-degree polynomial equation:

9(3x - 2) = (x^2 - 4x)^2 \implies x^4 - 8x^3 + 16x^2 - 27x + 18 = 0

. This factors into

(x - 6)(x - 1)(x^2 - x + 3) = 0

. Since the domain of

f^{-1}(x)

is restricted to

x \geq 2

, the root

x = 1

is rejected, and the quadratic factor

x^2 - x + 3 = 0

has no real roots, leaving

x = 6

as the unique real solution.

Estimated Time:3m 0s

Question 1991Question →

A right triangle has an area of $6$ square units. The lengths of its legs, $a$ and $b$ , satisfy the quadratic equation $a^2 - 2.75ab + 1.5b^2 = 0$ . If the length of the hypotenuse is an integer, what is the perimeter of the triangle?

9

10

12

14

15

Show answer & explanation

Answer: 12

Answer

The perimeter of the triangle is 12.

The correct answer is 12. By solving the quadratic relationship between the legs, we find two possible ratios: one where one leg is twice the other, and one where one leg is 0.75 times the other. Using the area of 6, the first case yields non-integer side lengths, while the second case yields legs of length 3 and 4. This results in an integer hypotenuse of 5, giving a perimeter of 3 + 4 + 5 = 12.

Step-by-Step Solution

1

Express the relation

a^2 - 2.75ab + 1.5b^2 = 0

as a quadratic in terms of the ratio

r = \frac{a}{b}

.

The equation becomes

r^2 - 2.75r + 1.5 = 0

.

To find the relationship between the two legs of the triangle by solving for their ratio.

2

Apply the quadratic formula to solve for

r

.

r = 2

or

r = 0.75

, meaning either

a = 2b

or

a = 0.75b

.

To determine the two possible linear relationships between the legs of the right triangle.

3

Substitute each ratio case into the area formula

\text{Area} = \frac{1}{2}ab = 6

, which simplifies to

ab = 12

.

For

a = 2b

, we get

b = \sqrt{6}

and

a = 2\sqrt{6}

. For

a = 0.75b

, we get

b = 4

and

a = 3

.

To calculate the actual leg lengths for both geometric cases.

4

Calculate the hypotenuse

c = \sqrt{a^2 + b^2}

for both cases to check which yields an integer value.

The first case yields

c = \sqrt{30}

, which is not an integer. The second case yields

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

, which is an integer.

To satisfy the constraint that the hypotenuse must be an integer, identifying the correct leg lengths as

3

and

4

.

5

Calculate the perimeter of the triangle for the valid case.

The perimeter is

3 + 4 + 5 = 12

.

To find the final requested value.

Key Concept

Solving quadratic relationships and applying the quadratic formula in geometric constraints.

Estimated Time:3m 0s

Question 1992Question →

If $\sqrt{3x + 19} - x = 3$ , what is the value of $x$ ?

-10

-5

-2

2

5

Show answer & explanation

Answer: 2

Answer

The value of

x

is 2.

To solve the equation, isolate the radical to obtain

\sqrt{3x + 19} = x + 3

. Squaring both sides yields

3x + 19 = x^2 + 6x + 9

. Rearranging terms results in the quadratic equation

x^2 + 3x - 10 = 0

, which factors into

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

. The potential solutions are

x = 2

and

x = -5

. Testing these in the original equation shows that

2

is a valid solution because

\sqrt{3(2) + 19} - 2 = 3

. Testing

-5

results in

\sqrt{3(-5) + 19} - (-5) = 7

, which does not equal

3

. Thus, the only real solution is

2

.

Step-by-Step Solution

1

Isolate the radical expression on one side of the equation.

\sqrt{3x + 19} = x + 3

Isolating the radical allows us to eliminate it by squaring both sides.

2

Square both sides of the equation.

3x + 19 = x^2 + 6x + 9

Squaring a square root removes the radical. Remember to expand

(x+3)^2

correctly as

x^2 + 6x + 9

.

3

Rearrange the terms into standard quadratic form,

ax^2 + bx + c = 0

.

x^2 + 3x - 10 = 0

Moving all terms to one side sets up the quadratic equation for factoring.

4

Factor the quadratic expression.

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

Factoring allows us to find the potential values of

x

by setting each factor to zero.

5

Solve for the potential values of

x

.

x = -5

or

x = 2

Setting

x + 5 = 0

gives

x = -5

, and setting

x - 2 = 0

gives

x = 2

.

6

Check both potential solutions in the original equation to identify any extraneous solutions.

For

x = 2

:

\sqrt{3(2) + 19} - 2 = \sqrt{25} - 2 = 5 - 2 = 3

(valid). For

x = -5

:

\sqrt{3(-5) + 19} - (-5) = \sqrt{4} + 5 = 2 + 5 = 7 \neq 3

(extraneous).

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original equation.

Key Concept

Solving radical equations and checking for extraneous solutions.

Question 1993Question →

Let matrix $X = \begin{bmatrix} 5 & -3 \\ 2 & 1 \end{bmatrix}$ and matrix $Y = \begin{bmatrix} 1 & 4 \\ -2 & 3 \end{bmatrix}$ . If $Z = X - Y$ , what is the value of the element in the first row and second column of $Z$ ?

Show answer & explanation

Answer: -7

Answer

The correct answer is

-7

.

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

Z = X - Y

, we subtract the element in the first row and second column of matrix

Y

from the corresponding element in matrix

X

. The element in the first row and second column of

X

is

-3

, and the element in the first row and second column of

Y

is

4

. Subtracting these values gives

-3 - 4 = -7

.

Step-by-Step Solution

1

Identify the elements in the first row and second column for both matrices.

x_{1,2} = -3

and

y_{1,2} = 4

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

Z

, we must use the corresponding elements from matrices

X

and

Y

.

2

Subtract the element of

Y

from the element of

X

.

-3 - 4 = -7

Since

Z = X - Y

, each element

z_{i,j}

of the resulting matrix is calculated as

x_{i,j} - y_{i,j}

.

Key Concept

Matrix Subtraction

Question 1994Question →

What is the sum of the two solutions to the quadratic equation $2x^2 - 7x - 4 = 0$ ?

Show answer & explanation

Answer: 3.5

Answer

The sum of the solutions is

3.5

.

The sum of the solutions of the quadratic equation

2x^2 - 7x - 4 = 0

is

3.5

. According to Vieta's formulas, the sum of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is given by

-\frac{b}{a}

. Here,

a = 2

and

b = -7

, so the sum is

-\frac{-7}{2} = 3.5

. Alternatively, solving the equation using the quadratic formula yields the roots

4

and

-0.5

, which sum to

3.5

.

Step-by-Step Solution

1

Identify the coefficients from the quadratic equation

2x^2 - 7x - 4 = 0

.

a = 2

,

b = -7

, and

c = -4

.

A quadratic equation in standard form is written as

ax^2 + bx + c = 0

.

2

Apply the sum of roots formula

-\frac{b}{a}

.

Sum

= -\frac{-7}{2} = 3.5

.

By Vieta's formulas, the sum of the roots of

ax^2 + bx + c = 0

is

-\frac{b}{a}

.

Key Concept

Sum of roots of a quadratic equation using Vieta's formulas

Alternative Method

Solve the quadratic equation by factoring or using the quadratic formula. Factoring

2x^2 - 7x - 4 = 0

yields

(2x + 1)(x - 4) = 0

, which gives solutions

x = -0.5

and

x = 4

. Adding these solutions together gives

-0.5 + 4 = 3.5

.

Estimated Time:1m 0s

Question 1995Question →

The polynomial $P(x)$ is defined by $P(x) = (2x^2 - 3x + 5)^2 - 4x(x^3 - 2x^2 - 7x + 1)$ . When $P(x)$ is written in standard form, what is the coefficient of the $x^2$ term?

Show answer & explanation

Answer: 57

Answer

The coefficient of the

x^2

term is 57.

Expanding

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

yields

(2x^2 - 3x + 5)(2x^2 - 3x + 5) = 4x^4 - 12x^3 + 29x^2 - 30x + 25

. Distributing the

-4x

term yields

-4x(x^3 - 2x^2 - 7x + 1) = -4x^4 + 8x^3 + 28x^2 - 4x

. Combining the

x^2

terms from both expressions gives

29x^2 + 28x^2 = 57x^2

. Thus, the coefficient of the

x^2

term is 57.

Step-by-Step Solution

1

Expand the squared trinomial

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

4x^4 - 12x^3 + 29x^2 - 30x + 25

Expanding the first part of the expression by multiplying the trinomial by itself.

2

Distribute the term

-4x

to the trinomial

(x^3 - 2x^2 - 7x + 1)

-4x^4 + 8x^3 + 28x^2 - 4x

Expanding the second part of the polynomial expression while distributing the negative sign to all terms.

3

Combine the expanded expressions and isolate the

x^2

terms

29x^2 + 28x^2 = 57x^2

Adding the coefficients of the terms of degree 2 to find the combined coefficient.

Key Concept

Operations on Polynomials

Question 1996Question →

For all non-zero real numbers

y

, the expression

\frac{(y^3)^2 \cdot (y^{-4})^a}{y^5}

is equivalent to

y^{-7}

. What is the value of

a

?

2

-3

-4

\frac{1}{12}

\frac{3}{2}

Show answer & explanation

Answer: $2$

Answer

The correct value of

a

is 2.

Applying the exponent rules systematically allows us to simplify the expression. First,

(y^3)^2

becomes

y^6

and

(y^{-4})^a

becomes

y^{-4a}

using the power of a power rule. Second, we combine the terms in the numerator using the product rule to get

y^{6-4a}

. Third, we divide by

y^5

using the quotient rule to obtain

y^{6-4a-5} = y^{1-4a}

. Setting this equal to the target expression

y^{-7}

gives the equation

1-4a = -7

. Solving for

a

gives

-4a = -8

, which simplifies to

2

.

Step-by-Step Solution

1

Apply the power of a power rule,

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

, to the exponential terms in the numerator.

(y^3)^2 = y^6

and

(y^{-4})^a = y^{-4a}

This simplifies nested exponent terms into single base terms.

2

Apply the product rule of exponents,

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

, to combine the numerator terms.

y^6 \cdot y^{-4a} = y^{6-4a}

This simplifies the numerator to a single power of

y

.

3

Apply the quotient rule of exponents,

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

, to divide by the denominator.

\frac{y^{6-4a}}{y^5} = y^{(6-4a) - 5} = y^{1-4a}

This simplifies the entire rational expression into a single exponential expression.

4

Equate the simplified exponent to the exponent of the equivalent expression and solve the linear equation for

a

.

1-4a = -7 \implies -4a = -8 \implies a = 2

Since the bases are equal and non-zero, their exponents must be equal for the expressions to be equivalent.

Key Concept

Properties of Exponents in Algebraic Expressions

Estimated Time:1m 30s

Question 1997Question →

If the quadratic expression $6x^2 - 7x - 5$ is factored completely into the product of two linear binomials of the form $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a > c > 0$ , what is the value of the expression $ad - bc$ ?

Show answer & explanation

Answer: 13

Answer

The value of the expression

ad - bc

is 13.

Step-by-Step Solution

1

Factor the quadratic expression

6x^2 - 7x - 5

.

(3x - 5)(2x + 1)

Find two numbers that multiply to

6 \times (-5) = -30

and add to

-7

, which are

-10

and

3

. Rewrite the middle term and factor by grouping:

6x^2 - 10x + 3x - 5 = 2x(3x - 5) + 1(3x - 5) = (3x - 5)(2x + 1)

.

2

Determine the values of the coefficients

a

,

b

,

c

, and

d

.

a = 3

,

b = -5

,

c = 2

, and

d = 1

The expression is factored into the form

(ax + b)(cx + d)

where

a > c > 0

. Comparing the factors

(3x - 5)

and

(2x + 1)

, we see the coefficients of

x

are

3

and

2

. Since

3 > 2 > 0

, we have

a = 3

and

c = 2

. This leaves

b = -5

and

d = 1

.

3

Calculate the value of

ad - bc

.

13

Substitute

a = 3

,

b = -5

,

c = 2

, and

d = 1

into the expression:

ad - bc = (3)(1) - (-5)(2) = 3 + 10 = 13

.

Key Concept

Factoring quadratic polynomials of the form

ax^2 + bx + c

with

a > 1

.

Question 1998Question →

A system of equations is given below:

\begin{aligned} y &= (x - 1)^2 - 6 \\ y &= x - 1 \end{aligned}

If $(x, y)$ is a solution to this system in the first quadrant, what is the value of $x + y$ ?

5

-3

7

4

9

Show answer & explanation

Answer: 7

Answer

The sum of the coordinates of the first quadrant solution is 7.

To solve the system, substitute the expression for

y

from the linear equation into the quadratic equation to get

x - 1 = (x - 1)^2 - 6

. Substituting

u = x - 1

yields

u^2 - u - 6 = 0

. Factoring the quadratic expression gives

(u - 3)(u + 2) = 0

, so

u = 3

or

u = -2

. Since

u = x - 1

, this means

x = 4

or

x = -1

. Evaluating both cases gives the intersection points

(4, 3)

and

(-1, -2)

. Only

(4, 3)

is in the first quadrant. Adding these coordinates together yields

4 + 3 = 7

.

Step-by-Step Solution

1

Substitute

y = x - 1

into the quadratic equation to set up an equation in terms of

x

.

x - 1 = (x - 1)^2 - 6

To find the

x

-coordinates of the intersection points.

2

Solve the equation for

x

by substituting

u = x - 1

.

u = u^2 - 6 \Rightarrow u^2 - u - 6 = 0 \Rightarrow (u - 3)(u + 2) = 0

. This gives

u = 3

or

u = -2

.

To find the values of the substituted variable

u

.

3

Find the corresponding

x

and

y

values for both cases.

Case 1:

x - 1 = 3 \Rightarrow x = 4

, which gives

y = 3

. Point is

(4, 3)

. Case 2:

x - 1 = -2 \Rightarrow x = -1

, which gives

y = -2

. Point is

(-1, -2)

.

To determine the full coordinates of all intersection points.

4

Identify the first quadrant solution and calculate the sum of its coordinates.

The first quadrant solution is

(4, 3)

. The sum of the coordinates is

4 + 3 = 7

.

To answer the question's requirement for the sum of coordinates in the first quadrant.

Key Concept

Solving systems of linear and quadratic equations by substitution and identifying quadrant-specific solutions.

Question 1999Question →

The equation of a circle is $x^2 + y^2 = 25$ , and the equation of a line is $2x - y = 5$ . If the line intersects the circle at the points $P$ and $Q$ , what is the sum of the $y$ -coordinates of $P$ and $Q$ ?

-18

-6

-2

2

4

Show answer & explanation

Answer: -2

Answer

The sum of the

y

-coordinates is

-2

.

Substituting the linear equation into the circle equation yields a quadratic equation with roots

x = 0

and

x = 4

. Evaluating the linear equation at these values gives the

y

-coordinates

-5

and

3

. Adding these

y

-coordinates results in

-2

.

Step-by-Step Solution

1

Rearrange the linear equation to express

y

in terms of

x

.

y = 2x - 5

Expressing one variable in terms of the other allows for substitution into the quadratic circle equation.

2

Substitute

y = 2x - 5

into the circle equation

x^2 + y^2 = 25

and simplify.

x^2 + (2x - 5)^2 = 25 \implies x^2 + 4x^2 - 20x + 25 = 25 \implies 5x^2 - 20x = 0

This substitution reduces the system of equations to a single quadratic equation in terms of

x

.

3

Solve the quadratic equation

5x^2 - 20x = 0

for

x

.

5x(x - 4) = 0 \implies x_1 = 0

and

x_2 = 4

Finding the roots of this quadratic equation gives the

x

-coordinates of the intersection points.

4

Substitute the

x

-values back into the linear equation

y = 2x - 5

to find the corresponding

y

-coordinates.

For

x_1 = 0

,

y_1 = 2(0) - 5 = -5

. For

x_2 = 4

,

y_2 = 2(4) - 5 = 3

. The intersection points are

P(0, -5)

and

Q(4, 3)

.

This step determines the coordinates of the two points of intersection.

5

Calculate the sum of the

y

-coordinates of the points

P

and

Q

.

y_1 + y_2 = -5 + 3 = -2

This yields the final value requested by the question.

Key Concept

Solving a system of linear and circular equations by substitution

Estimated Time:2m 0s

Question 2000Question →

What is the value of the larger real solution to the equation $\frac{10}{x} - \frac{3}{x-1} = 2$ ?

Show answer & explanation

Answer: 2.5

Answer

The larger real solution is

2.5

.

Multiplying the equation by the least common denominator

x(x-1)

yields

10(x-1) - 3x = 2x(x-1)

. Simplifying this leads to

2x^2 - 9x + 10 = 0

. Factoring the quadratic yields

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

, which gives the solutions

x = 2.5

and

x = 2

. The larger of these two solutions is

2.5

.

Step-by-Step Solution

1

Multiply the entire equation by the least common denominator, which is

x(x-1)

, for

x \neq 0

and

x \neq 1

.

10(x-1) - 3x = 2x(x-1)

This clears the fractions from the rational equation.

2

Expand both sides of the equation.

10x - 10 - 3x = 2x^2 - 2x

Expanding the terms allows us to combine like terms.

3

Combine like terms on the left side and move all terms to one side to set the quadratic equation equal to zero.

2x^2 - 9x + 10 = 0

Setting the quadratic equation to zero is required to solve it by factoring.

4

Factor the quadratic equation by grouping.

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

Factoring allows us to find the roots of the quadratic equation.

5

Solve for

x

by setting each factor equal to zero.

x = 2.5

or

x = 2

By the zero product property, at least one of the factors must be zero.

6

Check for extraneous solutions and select the larger real value.

Both

2

and

2.5

are valid because they do not make the original denominators zero. The larger value is

2.5

.

The question specifically asks for the larger of the two real solutions.

Key Concept

Solving rational equations by clearing denominators

Estimated Time:1m 30s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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