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.
21. A spinner is divided into eight equal sections. Five sections are red
and three are green. If the spinner is spun three times, what is the probability that it lands on red exactly twice?
1) 25/64
2) 45/512
3) 75/512
4) 225/512
Answer: 4) 225/512
Exactly two reds mean we want exactly one green as well. You could draw a sample space or tree diagram, but it would unfortunately be huge with 8 * 8 * 8 = 512 branches. Note: Eliminate Choice (1).
P(R,R,G) = 5/8 * 5/8 * 3/8 = 75/512.
However, the order doesn't matter and P(R,G,R) and P(G,R,R) are always 75/512 each.
The total is (75+75+75)/512 = 225/512.
22. What is the range of f(x) = | x - 3| + 2?
1) {x | x > 3}
2) {y | y > 2}
3) {x | x ∈ real numbers}
4) {y | y ∈ real numbers}
Answer: 2) {y | y > 2}
The range is concerned with the y-values on the graph, the output of the function. Eliminate Choices (1) and (3) which concern themselves with the domain, not the range.
The minimum point of this Absolute Value graph is (3, 2). All the output values must be greater than or equal to 2. Values less than 2 CANNOT appear in the range.
23. Which calculator output shows the strongest linear relationship
between x and y?
1) Lin Reg
y = a + bx
a = 59.026
b = 6.767
r = .8643
2) Lin Reg
y = a + bx
a = .7
b = 24.2
r = .8361
3) Lin Reg
y = a + bx
a = 2.45
b = .95
r = .6022
4) Lin Reg
y = a + bx
a = -2.9
b = 24.1
r = -.8924
Answer: 4) Lin Reg y = a + bx a = -2.9 b = 24.1 r = -.8924
My apologies for the formatting. I've included an image.
The strongest linear relationship will be the one with the coefficient of correlation crossest to 1 or -1.
-.8924 is closer to -1 than .8643 is to 1. So the answer is Choice (4).
You would get Choice (3) 22.7 if you forgot about the 1/2.
24. If log x2 - log 2a = log 3a, then log x expressed in terms of log a is
equivalent to
1) 1/2 log 5a
2) 1/2 log 6 + log a
3) log 6 + log a
4) log 6 + 2 log a
Answer: 2) 1/2 log 6 + log a
Cards on the table: my first attempt didn't get me to any of the four choices and I wasn't sure of the mistake I'd made. So I first solved for x, and then found log x. I may have taken the long way around but I got there.
log x2 - log 2a = log 3a
log (x2/(2a)) = log 3a
x2/(2a) = 3a
x2 = 6a2
x = a*(6)^(1/2)
log x = log (a*(6)^(1/2))
log x = log a + log (6)^(1/2)
log x = log a + 1/2 log 6
More to come. Comments and questions welcome.
More Regents problems.
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