Question 1

Solve the problem.
-A computer company uses the linear model y $y = - 33 x + 35,836$ to predict the decline in sales of a computer after it has been on the market more than one year. If x is the number of months after the first year and y is the
Number of computers sold during that month, how many computers will be sold 9 months after the first year?

Accepted Answer

The answer of Solve the problem.
-A computer company uses the...

Question 2

Indicate whether the statement is true always, sometimes, or never.
-When i is raised to an odd power, the result $$\text { is } - 1$$

Accepted Answer

The powers of i cycle through four values: i, -1, -i, and 1. When i is raised to an odd power, the result is either i or -i, but never -1.

Question 3

Indicate whether the statement is true always, sometimes, or never.
-The product of a pair of complex conjugates is a real number.

Accepted Answer

The product of a pair of complex conjugates is always a real number because the imaginary parts cancel out, leaving a real number.

Question 4

Indicate whether the statement is true always, sometimes, or never.
-The difference between two real numbers is a real number.

Accepted Answer

The difference between any two real numbers is always a real number due to the closure property of real numbers under subtraction.

Question 5

Indicate whether the statement is true always, sometimes, or never.
-The sum of two imaginary numbers is an imaginary number.

Accepted Answer

The sum of two imaginary numbers is sometimes an imaginary number. If the imaginary parts add up to a non-zero number, the result is imaginary. However, if they add up to zero, the result is a real number, not an imaginary one.

Question 6

Indicate whether the statement is true always, sometimes, or never.
-When i is raised to an even power, the result is -1.

Accepted Answer

When $i$ is raised to an even power, the result alternates between $1$ and $-1$, depending on the specific even power. For example, $i^2 = -1$, but $i^4 = 1$. Therefore, the statement is true only sometimes.

Question 7

Identify the number as real, complex, pure imaginary, or nonreal complex. More than one of these descriptions may
apply.
-$- 10 + i$

Accepted Answer

The answer of Identify the number as real, complex, pure...

Question 8

Solve the problem.
-A toy company uses the linear model y $y = - 2 x + 568$ to predict the decline in sales of a toy after it has been on the market more than one year. If x is the number of months after the first year and y is the number of toys sold in
Hundreds during that month, how many toys will be sold 11 months after the first year?

Accepted Answer

To find the number of toys sold 11 months after the first year, substitute $x = 11$ into the equation $y = -2x + 568$. This gives $y = -2(11) + 568 = -22 + 568 = 546$. Since $y$ represents the number of toys sold in hundreds, $546$ means $54,600$ toys.

Question 9

Identify the number as real, complex, pure imaginary, or nonreal complex. More than one of these descriptions may
apply.
-$5 - 3 i$

Accepted Answer

The answer of Identify the number as real, complex, pure...

Question 10

Solve the problem.
-A bank loaned out $54,000, part of it at the rate of 2.75% per year and the rest at a rate of 3.25% per year. If the interest received was $1595.00, how much was loaned at 2.75%?

Accepted Answer

Let x be the amount loaned at 2.75% and y be the amount loaned at 3.25%. We have two equations: x + y = 54000 (total amount loaned) and 0.0275x + 0.0325y = 1595 (total interest). Solving these equations simultaneously, we find x = $32,000.

Question 11

Indicate whether the statement is true always, sometimes, or never.
-A complex number is an imaginary number.

Accepted Answer

A complex number is sometimes an imaginary number. Complex numbers include all imaginary numbers (of the form bi where b is not zero and i is the square root of -1) and all real numbers (when b=0), making them a broader category. An imaginary number is a complex number, but a complex number is not always purely imaginary; it can also have a real component.

Question 12

Identify the number as real, complex, pure imaginary, or nonreal complex. More than one of these descriptions may
apply.
-$- 2$

Accepted Answer

$-2$ is a real number because it can be found on the real number line. All real numbers are also considered complex numbers, where the imaginary part is 0. Thus, $-2$ is both real and complex.

Question 13

Solve the problem.
-Kevin invested part of his $5000 bonus in a certificate of deposit that paid 2% annual simple interest, and the remainder in a mutual fund that paid 4% annual simple interest. If his total interest for that year was $120, how
Much did Kevin invest in the mutual fund?

Accepted Answer

Let the amount invested in the certificate of deposit be $x, and the amount invested in the mutual fund be $5000 - x. The interest from the certificate of deposit is 0.02x, and the interest from the mutual fund is 0.04(5000 - x). The total interest is $120, so the equation is 0.02x + 0.04(5000 - x) = 120. Solving this equation for x gives the amount invested in the certificate of deposit, and subtracting this from $5000 gives the amount invested in the mutual fund, which is $1000.

Question 14

Solve the problem.
-Your home state uses a linear model y $y = 45 ( x - 70 ) + 1610$ to predict the number of vacationers (y) as compared to the average temperature for that week (x). Find the number of vacationers predicted for a week with an
Average temperature of 60 degrees.

Accepted Answer

Substituting $x = 60$ into the equation $y = 45(x - 70) + 1610$, we get $y = 45(60 - 70) + 1610 = 45(-10) + 1610 = -450 + 1610 = 1160$ vacationers.

Question 15

Indicate whether the statement is true always, sometimes, or never.
-The product of two imaginary numbers is a real number.

Accepted Answer

The product of two imaginary numbers is sometimes a real number. For example, the product of the imaginary numbers $i$ and $i$ is $-1$, which is real. However, not all products of imaginary numbers result in real numbers, hence "sometimes" is correct.

Question 16

Solve the problem.
-Don James wants to invest $63,000 to earn $1547 per year. He can invest in B-rated bonds paying 3.9% per year or in a Certificate of Deposit (CD) paying 1.3% per year. How much money should be invested in each to realize
Exactly $1547 in interest per year?

Accepted Answer

The answer of Solve the problem.
-Don James wants to invest...

Question 17

Indicate whether the statement is true always, sometimes, or never.
-The product of a pair of complex conjugates (with b  0) is the difference of the squares of the real and imaginary parts.

Accepted Answer

The product of a pair of complex conjugates (a + bi) and (a - bi) is always a^2 + b^2, which is the sum of the squares of the real and imaginary parts, not the difference.

Question 18

Solve the problem.
-A person's emotional quotient (EQ) is found by multiplying emotional age by 100 and dividing by chronological age. Using this information, if a 5 yr old person has an EQ of 102 what is that person's emotional
Age?

Accepted Answer

The EQ is calculated as (Emotional Age / Chronological Age) * 100 = EQ. Given an EQ of 102 for a 5-year-old, we solve for Emotional Age: (Emotional Age / 5) * 100 = 102. Simplifying, Emotional Age = (102/100) * 5 = 5.1 years.

Question 19

Indicate whether the statement is true always, sometimes, or never.
-$$\text { To find the quotient } \frac { 5 + 8 \mathrm { i } } { 5 - 8 \mathrm { i } } \text {, multiply numerator and denominator by } 5 + 8 \mathrm { i } \text {. }$$

Accepted Answer

To find the quotient of complex numbers in the form $\frac{a+bi}{c+di}$, one common method is to multiply both the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. In this case, the conjugate of $5 - 8i$ is $5 + 8i$, so multiplying both the numerator and denominator by $5 + 8i$ is the correct method to simplify the expression. This approach is always used to simplify complex number divisions.

Question 20

Indicate whether the statement is true always, sometimes, or never.
-A real number is a complex number.

Accepted Answer

Every real number can be considered a complex number with an imaginary part of 0. Therefore, a real number is always a complex number.

Quiz 2: Equations and Inequalities

Indicate whether the statement is true always, sometimes, or never. -When i is raised to an odd power, the result $\text { is } - 1$

Indicate whether the statement is true always, sometimes, or never. -The product of a pair of complex conjugates is a real number.

Indicate whether the statement is true always, sometimes, or never. -The difference between two real numbers is a real number.

Indicate whether the statement is true always, sometimes, or never. -The sum of two imaginary numbers is an imaginary number.

Indicate whether the statement is true always, sometimes, or never. -When i is raised to an even power, the result is -1.

Identify the number as real, complex, pure imaginary, or nonreal complex. More than one of these descriptions may apply. - $- 10 + i$

Identify the number as real, complex, pure imaginary, or nonreal complex. More than one of these descriptions may apply. - $5 - 3 i$

Solve the problem. -A bank loaned out $54,000, part of it at the rate of 2.75% per year and the rest at a rate of 3.25% per year. If the interest received was $1595.00, how much was loaned at 2.75%?

Indicate whether the statement is true always, sometimes, or never. -A complex number is an imaginary number.

Identify the number as real, complex, pure imaginary, or nonreal complex. More than one of these descriptions may apply. - $- 2$

Solve the problem. -Kevin invested part of his $5000 bonus in a certificate of deposit that paid 2% annual simple interest, and the remainder in a mutual fund that paid 4% annual simple interest. If his total interest for that year was $120, how Much did Kevin invest in the mutual fund?

Solve the problem. -Your home state uses a linear model y $y = 45 ( x - 70 ) + 1610$ to predict the number of vacationers (y) as compared to the average temperature for that week (x) . Find the number of vacationers predicted for a week with an Average temperature of 60 degrees.

Indicate whether the statement is true always, sometimes, or never. -The product of two imaginary numbers is a real number.

Solve the problem. -Don James wants to invest $63,000 to earn $1547 per year. He can invest in B-rated bonds paying 3.9% per year or in a Certificate of Deposit (CD) paying 1.3% per year. How much money should be invested in each to realize Exactly $1547 in interest per year?

Indicate whether the statement is true always, sometimes, or never. -The product of a pair of complex conjugates (with b  0) is the difference of the squares of the real and imaginary parts.

Solve the problem. -A person's emotional quotient (EQ) is found by multiplying emotional age by 100 and dividing by chronological age. Using this information, if a 5 yr old person has an EQ of 102 what is that person's emotional Age?

Indicate whether the statement is true always, sometimes, or never. - $\text { To find the quotient } \frac { 5 + 8 \mathrm { i } } { 5 - 8 \mathrm { i } } \text {, multiply numerator and denominator by } 5 + 8 \mathrm { i } \text {. }$

Indicate whether the statement is true always, sometimes, or never. -A real number is a complex number.