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) an = 4 + 2.5n
2) an = 4 + 2.5(n - 1)
3) an = 4(2.5)n
4) an = 4(2.5)n-1
Answer: 4) an = 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 =ex 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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