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 2011
Part I: Each correct answer will receive 2 credits.
16. If x^{2} + 2 = 6x is solved by completing the square, an intermediate
step would be
1) (x + 3)^{2} = 7
2) (x  3)^{2} = 7
3) (x  3)^{2} = 11
4) (x  6)^{2} = 34
Answer: 2) (x  3)^{2} = 7
First, 6x needs to be subtracted from both sides. To complete the square, you will end up with half of 6, which is 3. Eliminate Choice (1) and (4).
x^{2} + 2 = 6x
x^{2}  6x + 2 = 0
x^{2}  6x + 9 + 2 = 0 + 9
x^{2}  6x + 9 = 9  2
(x  3^{2} = 7
This is Choice (2).
17. Three marbles are to be drawn at random, without replacement, from
a bag containing 15 red marbles, 10 blue marbles, and 5 white marbles.
Which expression can be used to calculate the probability of drawing
2 red marbles and 1 white marble from the bag?
1) _{15}C_{2} * _{5}C_{1} / _{30}C_{3}
2) _{15}P_{2} * _{5}P_{1} / _{30}C_{3}
3) _{15}C_{2} * _{5}C_{1} / _{30}P_{3}
4) _{15}P_{2} * _{5}P_{1} / _{30}P_{3}
Answer: 1) _{15}C_{2} * _{5}C_{1} / _{30}C_{3}
This problem is a lot easier than it looks as soon as you see the choices. This is purely a Combination problem, not a permutation problem.
Therefore, the answer is Choice (1).
18. The expression x^{(2/5)} is equivalent to
Answer: 4) [See image]
The negative exponent means that the x will move to the denominator. Eliminate Choices (1) and (2).
The denominator of 5 in the exponent means that the expression is the fifth root, which is Choice (4).
19. On January 1, a share of a certain stock cost $180. Each month
thereafter, the cost of a share of this stock decreased by onethird.
If x represents the time, in months, and y represents the cost of the
stock, in dollars, which graph best represents the cost of a share over
the following 5 months?
Answer: 3) [See image]
If something drops in 1/3 of its value every month, that is exponential decay. Eliminate Choices (1) and (2) which show linear graphs.
Choice (3) shows the stock losing 1/3 of its value. This is the correct response.
Choice (4) shows the stock reduced to 1/3 of its value. It has lost 2/3.
20. In the diagram below of right triangle JTM, JT = 12, JM = 6, and
m∠JMT = 90
What is the value of cot J ?
1) √(3)/3
2) 2
3) √(3)
4) 2 √(3) / 3
Answer: 1) √(3)/3
You want cotangent, which means we need the adjacent over the opposite (the inverse of tangent). We know the adjacent is 6. We don't know the opposite, but we could find it.
If you realized that this is half of an equilateral traingle, and therefore a 306090 right triangle, they you know that MT is JM time %radic;(3). If you didn't realize that, you can perform Pythagorean Theorem to get the same result.
So cot J = 6 / 6 %radic;(3) = 1 / %radic;(3) = 1 / %radic;(3) * %radic;(3) / %radic;(3) = %radic;(3) / 3, which is Choice (1).
More to come. Comments and questions welcome.
More Regents problems.
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