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, June 2013
Part I: Each correct answer will receive 2 credits.
11.In the right triangle shown below, what is the measure of angle S, to
the nearest minute?
1) 28°1'
2) 28°4'
3) 61°56'
4) 61°93'
Answer: 2) 28°4'
A minute is 1/60 of a degree, the same way that a minute is 1/60 of an hour. It is a "minute" (pronounce this last one as "my NEWT") portion.
Since there are only 60 minutes in a degrees, you can eliminate choice (4).
You are given the opposite side from S and the hypotenuse, so use Sine.
Sin x = 8/17
x = sin-1 (8/17) = 28.072..., but that's a decimal you have to convert the decimal to Sexigesmal:
0.072 * 60 = 4.32
So angle S has a measure of 28°4'.
Note that if you'd done cosine by mistake, you would've gotten choice (3).
12. Which ordered pair is in the solution set of the system of equations
shown below?
y2 - x2 + 32 = 0
3y - x = 0
1) (2,6)
2) (3,1)
3) (-1,-3)
4) (-6,-2)
Answer: 4) (-6,-2)
Eliminate Choice (1) and (3) immediately. The x coordinate should be three times the y coordinate, not the other way around.
From there, you can check choices (2) and (4), but it should be obvious that 3 and 1 are too small to add 32 and make zero.
(-2)2 - (-6)2 + 32 = 4 - 36 + 32 = 0
13. Susie invests $500 in an account that is compounded continuously computations.
at an annual interest rate of 5%, according to the formula A = Pert
where A is the amount accrued, P is the principal, r is the rate of
interest, and t is the time, in years. Approximately how many years
will it take for Susie’s money to double?
1) 1.4
2) 6.0
3) 13.9
4) 14.7
Answer: 3) 13.9
Substitute what you are given into the equation A = Pert, which gives you 1000 = (500)e(.05)t.
You can take try the choices or you can solve it algebraically.
(500)e(.05)(6.0) = 674.93
(500)e(.05)(13.9) = 1001.85
(500)e(.05)(14.7) = 1042.74
You want all the y values that are less than or equal to 0. There are no y values greater than 0.
Algebraically, solve for t:
1000 = (500)e(.05)t
2 = e.05t
ln 2 = ln e.05t
ln 2 = .05t
t = (ln 2) / .05 = 13.86...
14. If n is a negative integer, then which statement is always true?
1) 6n-2 < 4n-1
2) n/4 > -6n-1
3) 6n-1 < 4n-1
4) 4n-1 > (6n)-1
Answer: 3) 6n-1 < 4n-1
Let's look at the cases one at a time. Notice that the negative exponents only go with the variable, except in the last instance where it is next to parentheses.
6n-2 < 4n-1 means 6/n2 < 4/n.
Since n is a negative integer, the square of n will be be positive. Therefore 6/n2 will be positive, but 4/n will be negative. So the statement can never be true.
n/4 > -6n-1 means n/4 > -6/n.
n/4 will always be negative but -6/n will always be positive. So this statement can never be true.
6n-1 < 4n-1 means that 6/n < 4/n.
Since n is negative, this is always true.
4n-1 > (6n)-1 means 4/n > 1/6n, or 4/n > 4/24n.
Since n is negative, 4/24n would be much closer to 0 than 4/n. So this statement can never be true.
15. The expression 4 + (k=2)∑(5), 3(k-x) is equal to
1) 58 - 4x
2) 46 - 4x
3) 58 - 12x
4) 46 - 12x
Answer: 4) 46 - 12x
Write out the summation:
4 + 3(2 - x) + 3(3 - x) + 3(4 - x) + 3(5 - x)
This first thing you should see is that there will be 12x, not 4x, so eliminate Choices (1) and (2).
4 + 6 - 3x + 9 - 3x + 12 - 3x + 15 - 3x = 4 + 6 + 9 + 12 + 15 - 12x = 46 - 12x
More to come. Comments and questions welcome.
More Regents problems.
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