Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2012)

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

More Regents problems.

Algebra 2/Trigonometry Regents, January 2012

Part I: Each correct answer will receive 2 credits.


16. What is the solution set of the equation 3x⁵ - 48x = 0?

1) {0, ±2}
2) {0, ±2, 3}
3) {0, ±2, ±2i}
4) {±2, ±2i}

Answer: 3) {0, ±2, ±2i}

If you factor 3x from each term, you are left with the Difference of Squares.

3x⁵ - 48x = 0
3x(x⁴ - 16) = 0
3x(x² - 4)(x² + 4) = 0
3x(x - 2)(x + 2)(x² + 4) = 0

The sum of two squares has complex conjugates:

3x(x - 2)(x + 2)(x² + 4) = 0
3x(x - 2)(x + 2)(x - 2i)(x + 2i) = 0
x = 0 or x = 2 or x = -2 or x = 2i or x = -2i

17. A sequence has the following terms: a₁ = 4, a₂ = 10, a₃ = 25, a₄ = 62.5. Which formula represents the nth term in this sequence?


1) a_n = 4 + 2.5n
2) a_n = 4 + 2.5(n - 1)
3) a_n = 4(2.5)ⁿ
4) a_n = 4(2.5)^n-1

Answer: 4) a_n = 4(2.5)^n-1

The series goes from 4 to 10 to 25, so it is NOT arithmetic because it doesn't have a common difference. Eliminate Choices (1) and (2).

The Common Ratio is 2.5, which you can get from the two remaining choices, or from dividing 10/4 = 2.5 and 25/10 = 2.5.

If the first term is 4, then the exponent has to be zero because (2.5)^{0 = 1 and 4(1) = 4. If the exponent has to be zero for the first term, then you need to subtract 1 from n, which is n - 1. So Choice (4) is the answer.}

18. In parallelogram BFLO, OL = 3.8, LF = 7.4, and m∠O = 126. If diagonal BL is drawn, what is the area of triangle BLF?

1) 11.4
2) 14.1
3) 22.7
4) 28.1

Answer: 1) 11.4

The area of a triangle can be found using the formula A = 1/2 ab sin C, where a and b are two side and C is the angle between them.

In this case, the sides are LO and OB (which is congurnet to LF) and the angle is O.

A = 1/2 ab sin C = 1/2(3.8)(7.4)(sin 126) = 11.37... = 11.4, which is Choice (1).

You would get Choice (3) 22.7 if you forgot about the 1/2.

19. Which statement about the graph of the equation y =e^x not true?

1) It is asymptotic to the x-axis.
2) The domain is the set of all real numbers.
3) It lies in Quadrants I and II.
4) It passes through the point (e,1).

Answer: 4) It passes through the point (e,1).

The ordered pair is Choice (4) is reversed. It goes through the point (1, e). When x = 1, y = e¹.

Choice (1) is true. As the exponent goes toward negative infinity, the graph gets closer to the x-axis without ever touching it.

Choice (2) is true. The domain is all real numbers. There is no real value of x that is excluded from the domain. The range, on the other hand, is limited to y > 0.

Choice (3) is true. The entire graph lies in the first and second quadrant because all values of x are possible but only positive values of y are possible.

20. What is the number of degrees in an angle whose measure is 2 radians?

1) 360/π
2) π/360
3) 360
4) 90

Answer: 1) 360/π

180 degrees is π radians which is 3.141592... radians. So 90 degrees is half of that, and 360 is double that. Neither one is a whole number, let alone exactly 2. Eliminate Choices (3) and (4).

Use the above conversion:

180 / π = x / 2
π x = 360
x = 360 / π

More to come. Comments and questions welcome.

More Regents problems.

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