Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.
More Regents problems.
August 2012
Part I: Each correct answer will receive 2 credits.
26. What is the solution of 2 / (x + 1) = (x + 1) / 2
1) -1 and -3
2) -1 and 3
3) 1 and - 3
4) 1 and 3
Answer: 3) 1 and - 3
Cross-multiply and solve the resulting quadratic equation.
(x + 1)(x + 1) = 4
x2 + 2x + 1 = 4
x2 + 2x - 3 = 0
(x - 1)(x + 3) = 0
x - 1 = 0 or x + 3 = 0
x = 1 or x = -3
27. The total score in a football game was 72 points. The winning team scored 12 points more than the losing team. How many points did the winning team score?
1) 30
2) 42
3) 54
4) 60
Answer: 2) 42
As much as I'd like to say that 42 is the Ultimate Answer to the Question of Life, the Universe and Everything, we still have to solve this problem to see if it's the solution to this problem.
If x is the number of points the losing team scored, then x + 12 is the number of points the winning team scored because they scored 12 more points. The sum of those two expressions is x + x + 12.
2x + 12 = 72
2x = 60
x = 30
The losing team scored 30. The winning team scored 30 + 12 = 42.
28. What is the perimeter of the figure shown below, which consists of
an isosceles trapezoid and a semicircle?
1) 20 + 3π
2) 20 + 6π
3) 26 + 3π
4) 26 + 6π
Answer: 1) 20 + 3π
Add the three sides of the trapezoid that are shown. Do not add the dotted line because it is not part of the perimeter. It is inside. Then calculate the circumference of the semicircle.
The trapezoid sides are 4 + 6 + 10 = 20. Eliminate Choices (3) and (4).
The semicircle has a diameter of 6 because the trapezoid is isosceles. The Circumference of a circle is πd or 2πr, so the circumference of a semicircle (1/2 of a circle) is half as much: 1/2 πd or πr.
We know the diameter is 6, so the circumference is 3π. Choice (1).
29. The probability that it will rain tomorrow is 1/2. The probability that our team will win tomorrow’s basketball game is 3/5. Which expression represents the probability that it will rain and that our team will not win the game?
1) 1/2 + 3/5
2) 1/2 + 2/5
3) 1/2 * 3/5
4) 1/2 * 2/5
Answer: 4) 1/2 * 2/5
The most important word here is not. You were given the probability that the team will win. We need the probability that they will NOT win. Since there are only two possiblities, winning and not winning, and because those two probabilities must add up to 1, the probability of not winning is 1 - 3/5, which is 2/5.
The probability of two independent events occurring is the product of the two individual events happening separately.
30. The formula for the volume of a pyramid is V = 1/3 Bh. What is h expressed in terms of B and V?
1) h = 1/3 VB
2) h = V / (3B)
3) h = (3V)/B
4) h = 3VB
Answer: 3) -2 or 2
Use inverse operations to isolate the h.
3V = Bh
3V/B = h
End of Part I.
More to come. Comments and questions welcome.
More Regents problems.
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