Algebra Problems of the Day (Integrated Algebra Regents, August 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


21. Given:

A = {1, 3, 5, 7, 9}
B = {2, 4, 6, 8, 10}
C = {2, 3, 5, 7}
D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Which statement is false?
1) A ∪ B ∪ C = D
2) A ∩ B ∩ C = { }
3) A ∪ C = {1, 2, 3, 5, 7}
4) A ∩ C = {3, 5, 7}

Answer: 3) A ∪ C = {1, 2, 3, 5, 7}

1) A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, which is set D.
(1) is True.

2) A ∩ B ∩ C = {1, 3, 5, 7, 9} ∩ {2, 4, 6, 8, 10} ∩ {2, 3, 5, 7}
= { } ∩ {2, 3, 5, 7} = { }.
(2) is True.

3) A ∪ C = {1, 3, 5, 7, 9} ∪ {2, 3, 5, 7}
= {1, 2, 3, 5, 7, 9} =/= {1, 2, 3, 5, 7, 9}.
(3) is False.

4) A ∩ C = {1, 3, 5, 7, 9} ∩ {2, 3, 5, 7} = {3, 5, 7}.
(4) is True.

22. Which expression is equivalent to ( 2x⁶ - 18x⁴ + 2x² ) / 2x²

1) x³ - 9x²
2) x⁴ - 9x²
3) x³ - 9x² + 1
4) x⁴ - 9x² + 1

Answer: 4) x⁴ - 9x² + 1

Divide the coefficient of each term in the numerator by 2, and reduce the exponent of the variable by 2. The rule for dividing variables with exponents is to keep the base and subtract the exponent. In this case, each term will have two fewer factors of x.

2x⁶ / 2x² = x⁴. Eliminate Choices (1) and (3).

18x⁴ / 2x² = 9x².

2x² / 2x² = 1. Eliminate Choice (2).

23. In a given linear equation, the value of the independent variable decreases at a constant rate while the value of the dependent variable increases at a constant rate. The slope of this line is

1) positve
2) negative
3) zero
4) undefined

Answer: 2) negative

If one variable is increasing while the other is decreasing, the slope is negative.

If the slope were positive, both variables would be increasing or both variables would be decreasing.

A zero slope would require that the dependent variable NOT change at all but instead remain constant. For example, y = 3.

An undefined slope would require that the independent variable NOT change at all but instead remain constant. For example, x = 2.

24. The volume of a cylindrical can is 32π cubic inches. If the height of the can is 2 inches, what is its radius, in inches?

1) 8
2) 2
3) 16
4) 4

Answer: 4) 4

The Volume of a cylinder uses the formula V = π r² h.
V = π r² h = 32π
π r² (2) = 32π
r² = 16
r = 4

25. The expression ( 14 + x ) / ( x² - 4 ) is undefined when x is
1) -14, only
2) 2, only
3) -2 or 2
4) -14, -2, or 2

Answer: 3) -2 or 2

A fraction is undefined if its denominator is equal to zero. You cannot divide by 0.

The numerator can be zero, and you can divide 0 as many ways as you want. The quotient will always be 0, unless you are divide by zero.

For what values of x is the denominator equal to 0?

x² - 4 = 0
(x - 2)(x + 2) = 0
x - 2 = 0 or x + 2 = 0
x = 2 or x = -2

More to come. Comments and questions welcome.

More Regents problems.